define these number

• Mathematicians
• Math Lessons
• Square Roots
• Math Calculators

Types of Numbers – Difference and Classification

JUMP TO TOPIC

Natural Numbers

Whole numbers, rational numbers, irrational numbers, real numbers, imaginary numbers, complex numbers, prime numbers and composite numbers, transcendental numbers, discrete and continuous numbers, sets of numbers, practice questions.

Numbers are strings of digits used to represent a quantity. The magnitude of a number indicates the size of the quantity. It can be either large or small. They exist in different forms, such as 3, 999, 0.351, 2/5, etc.

Types of Numbers in Math

Natural numbers or counting numbers are the most basic types of numbers you learned for the first time as toddlers. They start from 1 and go to infinity, i.e., 1, 2, 3, 4, 5, 6, and so on. They are also called positive integers. In the set form, they can be written as:

{1, 2, 3, 4, 5, …}

Natural numbers are represented by the symbol N .

Whole numbers are the set of natural numbers, including zero. This means they start from 0 and go up to 1, 2, 3, and so on, i.e.

{0, 1, 2, 3, 4, 5, …}

Whole numbers are represented by the symbol W .

Integers are the set of all whole numbers and the negatives of natural numbers. They contain all the numbers which lie between negative infinity and positive infinity. They can be positive, zero, or negative but cannot be written in decimal or fraction. Integers can be written in set form as

{…, -3, -2, -1, 0, 1, 2, 3, …}

We can say that all whole numbers and natural numbers are integers, but not all integers are natural numbers or whole numbers.

The symbol Z represents integers.

A fraction represents parts of a whole piece. It can be written in the form a/b , where both a and b are whole numbers, and b can never be equal to 0. All fractions are rational numbers, but not all rational numbers are fractions.

Fractions are further reduced to proper and improper fractions. Improper fractions are ones in which the numerator is greater than the denominator while the opposite is true in proper functions, i.e., the denominator is greater than the numerator. Examples of proper fractions are 3/7 and 99/101, while 7/3 and 101/99 are improper fractions. This means the improper fractions are always greater than 1.

All terminating decimals and repeating decimals can be written as fractions. You can write the terminating decimal 1.25 as 125/100 = 5/4. A repeating decimal 0.3333 can be written as 1/3.

You can write rational numbers in fraction form. The word “rational” is derived from the word “ratio, ” as rational numbers are the two integers’ ratios. For example, 0.7 is a rational number because it can be written as 7/10. Other examples of rational numbers are -1/3, 2/5, 99/100, 1.57, etc.

Consider a rational number p/q , where p and q are two integers. Here, the numerator p can be any integer (positive or negative), but the denominator q can never be 0, as the fraction is undefined. Also, if q = 1, then the fraction is an integer.

The symbol Q represents rational numbers.

Irrational numbers cannot be written in fraction form, i.e., they cannot be written as the ratio of the two integers. A few examples of irrational numbers are √2, √5, 0.353535…, π, and so on. You can see that the digits in irrational numbers continue for infinity with no repeating pattern.

The symbol Q represents irrational numbers.

Real numbers are the set of all rational and irrational numbers. This includes all the numbers which can be written in decimal form. All integers are real numbers, but not all real numbers are integers. Real numbers include all the integers, whole numbers, fractions, repeating decimals, terminating decimals, and so on.

The symbol R represents real numbers.

Numbers other than real numbers are imaginary or complex numbers. When we square an imaginary number, it gives a negative result, which means it is a square root of a negative number, for example, √-2 and √-5. When we square these numbers, the results are -2 and -5. The square root of negative one is represented by the letter i , i.e.

What is the square root of -16? Write your answer in terms of the imaginary number i .

• Step 1: Write the square root form.
• Step 2: Separate -1.
• Step 3: Separate square roots.

√(16) × √(-1)

• Step 4: Solve the square root.
• Step 5: Write in the form of i.

Sometimes you get an imaginary solution to the equations.

Solve the equation,

x 2 + 2 = 0

• Step 1: Take the constant term on other side of the equation.
• Step 2: Take the square root on both sides.

√ x 2 = +√-2 or -√-2

• Step 3: Solve.

x = √(2) × √(-1)

x = +√2 i or -√2 i

• Step 4: Verify the answers by plugging values in the original equation and see if we get 0.

(+√2 i ) 2 + 2 = -2 + 2 = 0                     (as i = √-1 and square of i is -1)

(-√2 i ) 2 + 2 = -2 + 2 = 0                      (as i = √-1 and square of i is -1)

Just because their name is “imaginary” does not mean they are useless. They have many applications. One of the greatest applications of imaginary numbers is their use in electric circuits. The calculations of current and voltage are done in terms of imaginary numbers. These numbers are also used in complex calculus computations. In some places, the imaginary number is also represented by the letter j .

An imaginary number is combined with a real number to obtain a complex number. It is represented as a + bi , where the real part and b are the complex part of the complex number. Real numbers lie on a number line, while complex numbers lie on a two-dimensional flat plane.

Prime and composite numbers are opposite of each other. Prime numbers are the type of integers with no factors other than themselves and 1, for example, 2, 3, 5, 7, and so on. The number 4 is not a prime number because it is divisible by 2. Similarly, 12 is also not a prime number because it is divisible by 2, 3, and 4. Therefore, 4 and 12 are the examples of composite numbers.

The numbers which can never be the zero (or root) of a polynomial equation with rational coefficients are called transcendental numbers. Not all irrational numbers are transcendental numbers, but all transcendental numbers are irrational numbers.

Classification of Numbers

The family of numbers we saw above can be classified in different categories as well. It is like a family has 20 members, but they live in two joint family houses of 10 members each, which means 10 members live in the same house. We can say two or more types of numbers can fall under one category.

The types of countable numbers are referred to as discrete numbers, and the types of numbers that cannot be counted are called continuous numbers. All natural numbers, whole numbers, integers, and rational numbers are discrete. This is because each of their sets is countable. The set of real numbers is too big and cannot be counted, so it is classified as continuous numbers. If we randomly take the two closest real numbers, there still exist infinitely more real numbers between them; hence they cannot be counted.

Numbers can also be classified in the form of sets. Every type of number is a subset of another type of number. For example, natural numbers are the subset of whole numbers. Similarly, whole numbers are the subset of integers. The set of rational numbers contains all integers and fractions. The sets of rational numbers and irrational numbers form the real numbers. The real numbers fall under complex numbers with the imaginary part as 0. We can classify these numbers in a hierarchical chart as below:

Natural numbers can be further reduced to even, odd, prime, co-prime, composite, and perfect square numbers.

Previous Lesson  |  Main Page | Next Lesson

• Daily Crossword
• Word Puzzle
• Word Finder
• Word of the Day
• Synonym of the Day
• Word of the Year
• Language stories
• All featured
• Gender and sexuality
• All pop culture
• Writing hub
• Grammar essentials
• Commonly confused
• All writing tips
• Pop culture
• Writing tips

Advertisement

[ nuhm -ber ]

Synonyms: figure , digit

The number of children experiencing homelessness in the city has risen alarmingly.

• a word or symbol, or a combination of words or symbols, used in counting or in noting a total.

house number;

license number;

jersey number.

Number 5's order is ready to be delivered.

Did you call the right number?

The governor put forth a plan to increase the number of eligible voters.

Their number was more than 20,000.

Numbers flocked to the city to see the parade.

• metrical feet; verse .
• musical periods, measures, or groups of notes.
• numbers pool ( def 1 ) .

We won't make a decision until we see the numbers.

• Obsolete. arithmetic.

The garrison is not up to its full number.

The comic routine followed the dance number.

For her third number she played a nocturne.

• any of a collection of poems or songs.
• a tune or arrangement for singing or dancing.
• a distinct part of an extended musical work or one in a sequence of compositions.
• conformity in music or verse to regular beat or measure; rhythm.
• a single part of a book published in a series of parts.

The bookcase contained several numbers of a popular magazine.

Synonyms: edition

• Grammar. a category of noun, verb, or adjective inflection found in many languages, such as English, Latin, and Arabic, used to indicate whether a word has one or more than one referent. There may be a two-way distinction in number, as between singular and plural, three-way, as between singular, dual, and plural, or a more complex system.

Why don't you go talk to the attractive number standing at the bar?

Put those leather numbers in the display window.

Number is the basis of science.

verb (used with object)

Number each of the definitions.

• to ascertain the number of; count .

The manuscript already numbers 425 pages.

I number myself among his friends.

to number one's blessings.

They numbered the highlights of their trip at length.

The sick old man's days are numbered.

• to live or have lived (a number of years).

The players were numbered off into two teams.

verb (used without object)

Casualties numbered in the thousands.

Several eminent scientists number among his friends.

• a concept of quantity that is or can be derived from a single unit, the sum of a collection of units, or zero. Every number occupies a unique position in a sequence, enabling it to be used in counting. It can be assigned to one or more sets that can be arranged in a hierarchical classification: every number is a complex number ; a complex number is either an imaginary number or a real number , and the latter can be a rational number or an irrational number ; a rational number is either an integer or a fraction , while an irrational number can be a transcendental number or an algebraic number See complex number imaginary number real number rational number irrational number integer fraction transcendental number algebraic number See also cardinal number ordinal number
• the symbol used to represent a number; numeral

a telephone number

she was number seven in the race

a large number of people

• one of a series, as of a magazine or periodical; issue
• a self-contained piece of pop or jazz music
• a self-contained part of an opera or other musical score, esp one for the stage

he was not one of our number

who's that nice little number?

that little number is by Dior

roll another number

• a grammatical category for the variation in form of nouns, pronouns, and any words agreeing with them, depending on how many persons or things are referred to, esp as singular or plural in number and in some languages dual or trial
• any number of several or many
• by numbers military (of a drill procedure, etc) performed step by step, each move being made on the call of a number
• do a number on someone slang. to manipulate or trick someone
• get someone's number or have someone's number informal. to discover someone's true character or intentions
• in numbers in large numbers; numerously
• one's number is up informal. one is finished; one is ruined or about to die
• without number or beyond number of too great a quantity to be counted; innumerable
• to assign a number to
• to add up to; total
• also intr to list (items) one by one; enumerate

they were numbered among the worst hit

his days were numbered

/ nŭm ′ bər /

• A member of the set of positive integers. Each number is one of a series of unique symbols, each of which has exactly one predecessor except the first symbol in the series (1), and none of which are the predecessor of more than one number.
• A member of any of the further sets of mathematical objects defined in terms of such numbers, such as negative integers, real numbers, and complex numbers.
• The grammatical category that classifies a noun , pronoun , or verb as singular or plural . Woman, it , and is are singular; women, they , and are are plural.

Discover More

Grammar note, confusables note, other words from.

• num·ber·a·ble adjective
• num·ber·er noun
• de·num·ber verb (used with object)
• mis·num·ber verb
• pre·num·ber verb (used with object) noun
• re·num·ber verb (used with object)
• sub·num·ber noun

Word History and Origins

Origin of number 1

Idioms and Phrases

In a number of states, like Ohio, Iowa, and North Carolina, early voting has already begun.

The island is home to any number of artistic residents from around the world.

We're going to run things here by the numbers.

The class involves calisthenics by the numbers.

The committee really did a number on the mayor's proposal.

She could do a number on anything from dentistry to the Bomb.

It's time for you to get on stage and do your number.

Whenever I call, he does his number about being too busy to talk.

He was only interested in her fortune, but she got his number fast.

That bullet had his number on it.

• one is (was, will be) in serious trouble.

Convinced that her number was up anyway, she refused to see doctors.

The night sky was filled with stars without number.

More idioms and phrases containing number

Synonym study, example sentences.

Administration officials note that a number of former employees also have praised the president extensively and that the president has overwhelming support in his own party.

By Sunday, that number will rise to nearly 20 states, including Wisconsin, Georgia, Indiana, Virginia and Rhode Island.

Increasing numbers of rank-and-file Democrats are beginning to question that approach.

Baron Cohen spoke to Kardashian West and helped attract a number of other celebrities, Steyer said.

The exact number of people posting the messages was not clear.

“Our members continue to face a number of challenges,” she said.

The number of dissenters though is unprecedented in the modern era.

Starting under Theodore Roosevelt and Howard Taft, embassies headed by career diplomats increased in number.

The number of diplomats was pitiful (45 appointees in 1860), as was the amount of money allocated to them.

Jett sees this number as a marker of how much the president allows professionals to do the job.

The country is well inhabited, for it contains fifty-one cities, near a hundred walled towns, and a great number of villages.

We had six field-pieces, but we only took four, harnessed wit twice the usual number of horses.

There are a number of bacilli, called acid-fast bacilli, which stain in the same way as the tubercle bacillus.

Five of the number had studied with Liszt before, and the young men are artists already before the public.

I do not think the average number of passengers on a corresponding route in our country could be so few as twenty.

Related Words

Definitions and idiom definitions from Dictionary.com Unabridged, based on the Random House Unabridged Dictionary, © Random House, Inc. 2023

Idioms from The American Heritage® Idioms Dictionary copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.

• More from M-W
• To save this word, you'll need to log in. Log In

Definition of number

 (Entry 1 of 2)

called also numbers game

Number is regularly used with count nouns

while amount is mainly used with mass nouns.

The use of amount with count nouns has been frequently criticized; it usually occurs when the number of things is thought of as a mass or collection

or when money is involved.

Definition of number  (Entry 2 of 2)

transitive verb

intransitive verb

• whole number

Examples of number in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'number.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Middle English nombre, numbre, noumer, borrowed from Anglo-French nombre, numbre (also continental Old French nombre ), going back to Latin numerus "numerical sum or symbol, quantity, aggregate" — more at nimble

Middle English nombren, noumbren, nowmeryn, in part derivative of nombre number entry 1 , in part borrowed from Anglo-French nombrer, numbrer, going back to Latin numerāre "to add up, count, reckon, compute," derivative of numerus "numerical sum or symbol, quantity, aggregate" — more at number entry 1

14th century, in the meaning defined at sense 1a(1)

14th century, in the meaning defined at transitive sense 1

Phrases Containing number

• a good number of
• agree in number
• algebraic number
• a number of
• any number of
• atomic number
• Avogadro's number
• beyond number
• Brinell hardness number
• call number
• cardinal number
• cetane number
• chromosome number
• complex number
• compound number
• conjugate complex number
• coordination number
• counting number
• denominate number
• do a number on
• even number
• Fibonacci number
• go number one
• have someone's number
• imaginary number
• index number
• irrational number
• license number
• look out for number one
• Mach number
• magnetic quantum number
• mass number
• mixed number
• natural number
• number cruncher
• number line
• number one pick
• number plate
• number sign
• number theory
• octane number
• one of our / your / their number
• opposite number
• ordinal number
• perfect number
• pre - number
• prime number
• public enemy number one
• quantum number
• rational number
• real number
• registration number
• Reynolds number
• round number
• serial number
• Social Security number
• someone's number is up
• telephone number
• triangular number
• wave number
• went number two
• without number
• wrong number

Articles Related to number

A Countdown of Words with Numbers 10-1

From 'hang ten' to 'one-horse town'

Dictionary Entries Near number

number agreement

Cite this Entry

“Number.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/number. Accessed 5 Jun. 2024.

Kids Definition

Kids definition of number.

Kids Definition of number  (Entry 2 of 2)

More from Merriam-Webster on number

Nglish: Translation of number for Spanish Speakers

Britannica English: Translation of number for Arabic Speakers

Britannica.com: Encyclopedia article about number

Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!

Can you solve 4 words at once?

Word of the day.

See Definitions and Examples »

Get Word of the Day daily email!

Popular in Grammar & Usage

What's the difference between 'fascism' and 'socialism', more commonly misspelled words, commonly misspelled words, how to use em dashes (—), en dashes (–) , and hyphens (-), absent letters that are heard anyway, popular in wordplay, 8 words for lesser-known musical instruments, the words of the week - may 31, 9 superb owl words, 10 words for lesser-known games and sports, etymologies for every day of the week, games & quizzes.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.1: Our Number System

• Last updated
• Save as PDF
• Page ID 130901

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

When we are children, we learn to count objects using ordinary numbers like \(1,2,3,4,\) so we can describe quantities in the world around us. As we get older, we learn larger and larger numbers, so we can make sense of statements like "the population of the world is about 8 billion people ."

We also learn about other types of numbers, such as \(0\), negative numbers, and non-whole numbers, which help us describe even more precise quantities, such as "I walked \(2.3\) miles today." In this section, we'll briefly review the meaning of these numbers and the multiple ways to express them.

In this section, you will learn to:

• Accurately describe the meaning of the base \(10\) number system, including both whole numbers and non-whole numbers
• Correctly perform base \(10\) addition with any numbers and explain why it works
• Round numbers correctly to any place

The Base 10 Number System

You're already an expert at addition — you've been adding numbers for at least a decade, if not much longer. But when's the last time you really thought about how addition works?

Many of us routinely rely on technology such as smartphones or calculators to perform basic addition. It's faster and less error-prone than using mental math or pencil and paper, particularly when dealing with large numbers. The purpose of this section is not to discourage use of technology, but rather to peek behind the curtain of how humans, the creators of technology, designed the number system we all use every day.

Below, you'll encounter the first example in this book. Take a few minutes to think about the answer to the question in the example before viewing the solution. This example, as well as the many others throughout this book, will be most impactful to your learning if you attempt to solve them before looking ahead at the solution. In fact, neuroscience has shown that the more you struggle initially, the more likely you'll be to remember the reasoning behind the solution, which in turn leads to durable and applicable mathematical skill.

Example \(\PageIndex{1}\)

Your seven-year-old cousin is learning to add numbers in school, and she asks you for help. She gives you the following problem:

\[\begin{array}{cccc}& 3& 7 & 4 \\ + & 2 & 5& 8\\ \hline &&&\\ \end{array} \]

How would you explain the solution to her? 

The most common method used to solve this problem is explained below. This might not be how you learned how to do this, but for illustration, here it is:

• Add the two numbers in the rightmost column;
• If they add up to more than \(10\), put the ones digit of the sum in that column, and "carry" the \(1\) to the column to the left;
• Repeat this procedure in the next column to the left, remembering to add in any numbers that had been carried;
• Keep going from right to left until you finish.

In terms of what you'd tell your cousin, you might say: "First, add together the \(4\) and the \(8\) in the rightmost column. That answer is \(12\), so you write down the \(2\) underneath, and then carry the \(1\)." You could write it like this:

\[\begin{array}{cccc} & 3& 7 & \overset{1}{4} \\ + & 2 & 5& 8\\ \hline &&&2\\ \end{array}\]

From there, you could explain: "Next, add together all of the numbers in the next column. That is \(1 + 7 + 5\), which is \(13\). Just like before, write the \(3\) down below, and then carry the \(1\)." You could write it like this:

\[\begin{array}{cccc}& 3& \overset{1}{7} & 4 \\+ & 2 & 5& 8\\ \hline&&3&2\\ \end{array}\]

And then you could say: "Since you're out of columns, just add up the last numbers in that column, and put the answer underneath. That would give you \(1 + 3 + 2 = 6\). So the answer is \(632\)." Writing that would look like this:

\[\begin{array}{cccc}& \overset{1}{3}& 7 & 4 \\+ &2 & 5& 8\\ \hline&6&3&2\\ \end{array} \]

If you said that, your cousin would know exactly how to solve this sort of problem, and with practice she could likely solve many similar problems. It's a great explanation of how the process works.

But your cousin wants to know why the process works. She asks, "why do you carry the \(1\)?"

Definition: Base 10 System

We use the base \(10\) or decimal number system. In this system, each digit is a single whole number between \(0\) and \(9\) that represents a particular power of \(10\), with the power increasing from right to left. We typically refer to these powers of ten as place values.

What does this mean when referring to a particular number? For example, consider the number \(4027\), which would be read as "four thousand and twenty-seven." If we break this down into its place values, we have that \(4\) is in the thousands place, \(0\) is in the hundreds place, \(2\) is in the tens place, and \(7\) is in the ones place. Therefore, we could choose to rewrite this number as follows:

\[4027 = (4 \times 1000) + (0 \times 100) + (2 \times 10) + (7 \times 1)\]

Writing numbers in this way is more time-consuming than writing them the normal way. However, it allows us to see a pattern that emerges. In order to see that pattern more fully, recall the following definition:

Definition: Exponent

An exponent represents repeated multiplication a certain number of times. Exponents are written as superscripts. For example,

\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\]

Notice that above, there are \(5\) copies of the number \(2\) being multiplied together. In general,

\[a^b = a \times a \times \cdots \times a \quad (b \; times)\]

A nonzero number to the \(0\) power is defined to equal \(1\); for example \(2^0 = 1\).

We introduce this definition because it allows us to describe the pattern that occurs in base \(10\) numbers. Observe — or better yet, check with your phone or a calculator — that applying larger and larger exponents to \(10\) simply increases the number of \(0\)'s following the \(1\). For example, computing \(10^3\) gives us

\[10^3 = 10 \times 10 \times 10 = 1000\]

which has \(3\) zeros following the \(1\). This patterns occurs in general. The follow table shows the first several occurrences of this pattern:

In the United States, commas are used to separate groups of three digits in large numbers, starting from the ones place. That is, the number \(10000\) could be written \(10,000\), and both would be said as "ten thousand." In this text, we will typically omit commas unless it is helpful for comprehension.

Let's take this pattern and use it to further inspect the number \(4027\). We have

\[ \begin{array}{lll} 4027 &=& (4 \times 1000) + (0 \times 100) + (2 \times 10) + (7 \times 1) \\ 4027 &=& (4 \times 10^3) + (0 \times 10^2) + (2 \times 10^1) + (7 \times 10^0) \\ \end{array}\]

This is why our number system is called base 10. Each successive digit, going from right to left, corresponds to a power of \(10\). When we write a number as we did above on the right side of the equations, we call it the base 10 expansion of that number. The numbers \(4, 0, 2\) and \(7\) are called the digits of the number.

Example \(\PageIndex{2}\)

Why do we use \(10\) as the base for our number system?

The answer lies at the ends of your arms! Most humans have \(10\) fingers, so it's very convenient to count to \(10\) and then start over again. If we had \(8\) fingers, our number system would likely be very different, and the numeral \(9\) might not even exist! This is also why our fingers are sometimes called digits themselves.

Other base systems are used in particular contexts. For example, binary numbers are written in base 2, in which each successive place corresponds to another power of \(2\). This is the number system used in computer and electronics programming, though it is hidden from most device users. We also see base \(12\) and base \(60\) showing up in our computations of time: when counting seconds and minutes, we count to \(59\) and then start over again at \(0\).

Nevertheless, base \(10\) is the most common way to express numbers for the vast majority of people in our world today.

Now that we better understand the base \(10\) system, we can return to your cousin's question: why do we carry the 1 when we add? The answer is quite simple: because of our number system, we can only write a single digit in a given place. Remember your cousin's addition problem:

\[ \begin{array}{cccc} & 3& 7 & 4 \\+ & 2 & 5& 8\\ \hline &&&\\ \end{array}\]

In adding the \(4+8\), we get \(12\), but we can't put a \(12\) in the ones place, because \(12\) consists of two digits. Therefore, we have to remember what the number \(12\) actually means. We can rewrite \(12\) using the base \(10\) expansion to uncover that meaning:

\[ 12 = (1 \times 10^1) + (2 \times 10^0)\]

We can write the \(2\) in the ones place, since it's only one digit. However, the additional \(1\) in the tens place has to be added to the other values in the tens place, which in this case are \(7\) and \(5\). Since the sum of the digits in a single column exceed \(9\), the excess has to be "carried over" to the next column, which represents the next larger power of \(10\).

While these observations may seem very basic, these insights on how we represent numbers can have a big impact on how comfortable we feel working with more complicated types of numbers and making judgments about numerical questions.

Decimal numbers are written using the same logic and patterns that whole numbers do. While technically anything written in base 10 is a "decimal number," people often use the word "decimal" to describe numbers that are not whole numbers, such as \(98.7\) or \(510.106\). Non-whole numbers exist between whole numbers, and the section of the number after the decimal point describes precisely where the number lies on a number line.

Let's investigate a number between \(0\) and \(1\) first: the number \(0.439\). The \(0\) to the left of the decimal point tells us that this number is between \(0\) and \(1\), and the part after the decimal point describes its value. The place values of this number work just like they do for whole numbers, except that instead of counting in groups of \(10\), \(100\), \(1000\), and so on, they stand for the reciprocals of those values. We use negative exponents to denote these reciprocals since it is shorter to write.

Definition: Negative Exponents

A negative exponent on a number indicates that you should find the reciprocal of that number. That is,

\[2^{-5} = \frac{1}{2^5} = \frac{1}{32}\]

In general,

\[a^{-b} = \frac{1}{a \times a \times \cdots \times a} \quad (b \; times)\]

Let's use this definition in context to help us understand how decimals can be expressed using base 10 notation.

Example \(\PageIndex{3}\)

Rewrite \(10^{-1}\), \(10^{-2}\), and \(10^{-3}\) without any exponents, and convert them to decimal form.

Using the definition of negative exponents along with the original definition of exponents, we see that

\[10^{-1} = \frac{1}{10^1} = \frac{1}{10}\]

\[ 10^{-2} = \frac{1}{10^2} = \frac{1}{100}\]

\[ 10^{-3} = \frac{1}{10^3} = \frac{1}{1000}\]

These numbers are valid fractions, but it's often easier to work with decimals in calculations. Each of these fractions corresponds to a decimal value in the following way. The pronunciation is included as well.

\[\frac{1}{10} = 0.1 = \text{one tenth} \]

\[\frac{1}{100} = 0.01 = \text{one hundredth} \]

\[\frac{1}{1000} = 0.001 = \text{one thousandth} \]

Note that the reciprocal of a number is named similarly to the number itself, with a "th" afterwards.

Of course, we could keep going with this pattern to ten thousandths, hundred thousandths, and so on.

These negative powers of \(10\) form the basis for how we write decimals, just as the positive powers of \(10\) give us a way to write any whole number. A final example is given next. Remember, you'll learn better if you try it first before looking at the answer!

Example \(\PageIndex{4}\)

Write \(0.439\) in expanded form, using negative powers of \(10\).

Each successive place in this number corresponds to a negative power of \(10\) as we saw in the previous example. The \(4\) following the decimal point is in the \(0.1\) place, also known as the \(\frac{1}{10}\) place or the \(10^{-1}\) place. Next, the \(3\) is in the \(0.01\) place, also known as the \(\frac{1}{100}\) place or the \(10^{-2}\) place. And last, the \(9\) is in the \(0.001\) place, also known as the \(\frac{1}{1000}\) place or the \(10^{-3}\) place.

We can therefore expand this number in the following way:

\[0.439 = 4 \times 0.1 + 3 \times 0.01 + 9 \times 0.001 = 4 \times 10^{-1} + 3 \times 10^{-2} + 9 \times 10^{-3}\]

Note the similarities and differences with a base \(10\) expansion of a whole number.

Now that we've got some tools to describe and compare whole numbers and decimals, we'll introduce one last important concept for the chapter.

Rounding is a skill you've likely seen before, but may need to practice. Let's get started with two examples.

Example \(\PageIndex{5}\)

The population of Bluffington is \(57,489\) people. Express this fact as a sentence in which the population is rounded to the nearest thousand.

To round the number \(57,489\) to the nearest thousand, the first step is to locate the thousands place. After rounding, every number after the thousands place will be a \(0\).

In this case, the \(7\) is in the thousands place, because it stands for \(7 \times 1000\) in the base 10 expansion of \(57,489\). Since the number in the thousands place is a \(7\), it tells us that the number \(57,489\) is between \(57,000\) and \(58,000\), and will be rounded to one of those two values.

The question is: which of \(57,000\) and \(58,000\) is \(57,489\) closer to? The answer is \(57,000\), because the number \(57,489\) is less than \(57,500\), which is the halfway point between \(57,000\) and \(58,000\).

So, as a sentence, we could write, "The population of Bluffington is approximately \(57,000\)  people."

Try the next exercise on your own before looking at the answer.

Example \(\PageIndex{6}\)

In performing a calculation, you need to enter \(45 \div 40\) into your calculator. This gives you the answer \(1.125\). However, question you are answering with this calculation asks for an amount of money, in dollars and cents. Round the number \(1.125\) to the correct decimal place and value so it expressed as a number of dollars and cents.

To answer this question, you'll need to know that money amounts in the United States are typically expressed as a number of dollars followed by a number of cents, where a dollar is composed of one hundred cents. That means that each cent is one hundredth  of a dollar, so we will need to round to the hundredths place.

The hundredths place refers to the second number after a decimal point in a decimal number. In the number \(1.125\), the current value of the hundredths place is a \(2\). This tells us the number \(1.125\) is between \(1.12\) and \(1.13\).

However, we still need to ask what \(1.125\) is closest to: either \(1.12\) or \(1.13\). In reality, it's exactly halfway between these two values, but we adopt (as do most textbooks) the habit of "rounding up" from the halfway point. Therefore, the number \(1.125\) rounded to the nearest hundredth is \(1.13\). As a dollar amount, you would write \(\$1.13\).

These two examples illustrate the general principles of rounding. The process in general is summed up in the Procedure box below.

Rounding Numbers

To round a number to a given place value, follow these steps:

• Identify the the original digit in the place value you're rounding to.
• If the digit to the right is \(4\) or less, keep the original digit in the place to which you are rounding.
• If the digit to the right is \(5\) or more, increase the original digit by \(1\).
• All digits to the right of the original digit either become \(0\) (if you are rounding to a whole number) or go away (if you are rounding to a decimal value).

One more example will help to clarify. Consider the number \(34.5718\). We label the place values of this number in the table below:

Let's practice rounding this number in various ways:

• Nearest ten or "tens place": The digit in the tens place currently is a \(3\). The digit to the right is a \(4\). Since this \(4\) is smaller than \(5\), keep the \(3\) as it is, and replace the \(4\) with a \(0\). We also get rid of everything after the decimal point. Therefore, the answer in this case is \(30\).
• Nearest whole or "ones place": This means "round to the ones place." The ones digit is currently a \(4\). The digit to the right of the ones place is currently a \(5\). This means that we need to "round up," so the \(4\) will increase by \(1\) to become a \(5\). The rest of the digits after the decimal point go away. Therefore, the answer in this case is \(35\).
• Tenths place or "one decimal place": The digit currently in the tenths place is a \(5\). The digit to the right is a \(7\), which is greater than \(5\). Therefore, we increase the value in the tenths place from \(5\) to \(6\). We then delete everything following the digit in the tenths place. Therefore, the answer in this case is \(34.6\).
• Hundredths place or "two decimal places": The digit currently in the hundredths place is a \(7\). The digit to the right is a \(1\). Since \(1\) is less than \(5\), we keep the \(7\) as is, and delete everything after. Therefore, the answer is \(34.57\).
• Thousandths place or "three decimal places": The answer is \(34.572\). Do you see why?

Throughout this book, there will be specific rounding directions on many of the exercises. Two rules also apply throughout, and will be reinforced through examples:

• When computing a number of living things, round to the nearest whole.
• When computing an amount of money, always round to the hundredths place (two digits after the decimal point)

Other rounding directions will be provided as needed.

• Write the number \(35,023\) using its base \(10\) expansion, explicitly writing the power of ten correspond to each place. 

\[ \begin{array}{cccc} & 3& 7 & 4 \\ - & 2 & 5& 8\\ \hline &&&\\ \end{array} \]

Pretend you're explaining to your cousin how to do this by hand, showing her work. What would you say to her? Can you explain what happens when you "borrow?" You should write at least 5-6 sentences and include each step in the process.

• Write down the number that the US population counter says when you look at the website for the first time. It may be slowly increasing, write down the closest number that you can.
• Round that number to the millions place.
• Write a sentence that describes the number you found in the previous part. Your sentence should answer the question "about how many people live in the United States right now?

Number Systems

Number systems are systems in mathematics that are used to express numbers in various forms and are understood by computers. A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. Numbers have various categories like natural numbers, whole numbers, rational and irrational numbers, and so on. Similarly, there are various types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system.

In this article, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will learn the conversions between these number systems and solve examples for a better understanding of the concept.

What are Number Systems?

A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

Number Systems Definition

A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

Types of Number Systems

There are different types of number systems in which the four main types are as follows.

• Binary number system (Base - 2)
• Octal number system (Base - 8)
• Decimal number system (Base - 10)
• Hexadecimal number system (Base - 16)

We will study each of these systems one by one in detail after going through the following number system chart.

Number System Chart

Given below is a chart of the main four types of number system that we use to represent numbers.

Binary Number System

The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 10001 2 , 111101 2 , 1010101 2 are some examples of numbers in the binary number system.

Octal Number System

The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just like the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example, 35 8 , 23 8 , and 141 8 are some examples of numbers in the octal number system.

Decimal Number System

The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example, 723 10 , 32 10 , and 4257 10 are some examples of numbers in the decimal number system.

Hexadecimal Number System

The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B3 16 , 6F 16 , and 4B2A 16 are some examples of numbers in the hexadecimal number system.

Conversion of Number Systems

A number can be converted from one number system to another number system using number system formulas. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa, and so on. Let us see the steps required in converting number systems.

Steps for Conversion of Binary to Decimal Number System

To convert a number from the binary to the decimal system, we use the following steps.

• Step 1: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• Step 2: The exponents should start with 0 and increase by 1 every time we move from right to left.
• Step 3: Simplify each of the above products and add them.

Let us understand the steps with the help of the following example in which we need to convert a number from binary to decimal number system.

Example: Convert 100111 2 into the decimal system.

Step 1: Identify the base of the given number. Here, the base of 100111 2 is 2.

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 2 0 , 2 1 , 2 2 , and so on from right to left.

Step 3: We just simplify each of the above products and add them.

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.

100111 = (1 × 2 5 ) + (0 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 )

= (1 × 32) + (0 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)

= 32 + 0 + 0 + 4 + 2 + 1

Thus, 100111 2 = 39 10 .

Conversion of Decimal Number System to Binary / Octal / Hexadecimal Number System

To convert a number from the decimal number system to a binary/octal/hexadecimal number system, we use the following steps. The steps are shown on how to convert a number from the decimal system to the octal system.

Example: Convert 4320 10 into the octal system.

Step 1: Identify the base of the required number. Since we have to convert the given number into the octal system, the base of the required number is 8.

Step 2: Divide the given number by the base of the required number and note down the quotient and the remainder in the quotient-remainder form. Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.

Step 3: The given number in the octal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

Therefore, 4320 10 = 10340 8

Conversion from One Number System to Another Number System

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems, we first convert it into the decimal system, and then we convert it to the required systems by using the above-mentioned processes.

Example: Convert 1010111100 2 to the hexadecimal system.

Step 1: Convert this number to the decimal number system as explained in the above process.

Thus, 1010111100 2 = 700 10 → (1)

Step 2: Convert the above number (which is in the decimal system), into the required number system (hexadecimal).

Here, we have to convert 700 10 into the hexadecimal system using the above-mentioned process. It should be noted that in the hexadecimal system, the numbers 11 and 12 are written as B and C respectively.

Thus, 700 10 = 2BC 16 → (2)

From the equations (1) and (2), 1010111100 2 = 2BC 16

• Indian Numeral System
• International Number System
• Binary Calculator
• Binary to Octal Conversion
• Octal to Binary
• Decimal to Binary
• Binary to Decimal
• Decimal to Hexadecimal
• Hexadecimal to Decimal

Cuemath is one of the world's leading math learning platforms that offers LIVE 1-to-1 online math classes for grades K-12 . Our mission is to transform the way children learn math, to help them excel in school and competitive exams. Our expert tutors conduct 2 or more live classes per week, at a pace that matches the child's learning needs.

Number Systems Examples

Example 1: Convert 300 10 into the binary number system with base 2.

Solution: 300 10 is in the decimal system. We divide 300 by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2.

The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.

Thus, 300 10 = 100101100 2

Example 2: Convert 5BC 16 into the decimal system.

Solution: 5BC 16 is in the hexadecimal system. We know that B = 11 and C = 12 in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:

Thus, 5BC 16 = 1468 10

Example 3: Convert 144 8 into the hexadecimal system.

Solution: The base of 144 8 is 8. First, we will convert this number into the decimal system as follows:

Thus, 144 8 = 100 10 → (1). Now we will convert this into the hexadecimal system as follows:

Thus, 100 10 = 64 16 → (2)

From the equations (1) and (2), we can conclude that: 144 8 = 64 16

go to slide go to slide go to slide

Book a Free Trial Class

Practice Questions on Number Systems

go to slide go to slide

FAQs on Number Systems

What are number systems with examples.

A number system is a system of writing or expressing numbers. In mathematics, numbers are represented in a given set by using digits or symbols in a certain manner. Every number has a unique representation of its own and numbers can be represented in the arithmetic and algebraic structure as well. There are different types of number systems that have different properties, like the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. Some examples of numbers in different number systems are 10010 2 , 234 8 , 428 10 , and 4BA 16 .

What are the Different Types of Number Systems?

There are four main types of number systems:

What are the Conversion Rules of Number Systems?

To convert a number from binary/octal/hexadecimal system to a decimal number system, we use the following steps:

• Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base.
• The exponents should start with 0 and increase by 1 every time we move from right to left.
• Simplify each of the above products and add them.

To convert a number from decimal system to binary/octal/hexadecimal system, we use the following steps:

• Divide the given number by the base of the required number and note down the quotient and the remainder in the “quotient-remainder” form.
• Repeat this process (dividing the quotient again by the base) until we get the quotient less than the base.
• The given number in the decimal number system is obtained just by reading all the remainders and the last quotient from bottom to top.

To convert a number from one of the binary/octal/hexadecimal systems to one of the other systems:

• We first convert it into the decimal system.
• Then we convert it to the required system.

What are the Uses of Each Number System?

There are different purposes of each number system, such as:

• The binary number system is used to store the data in computers.
• The advantage of the octal number system is that it has fewer digits when compared to several other systems, hence, there would be fewer computational errors.
• The decimal number system is the system that we use in daily life.
• The hexadecimal number system is used in computers to reduce the large-sized strings of the binary system.

What is the Importance of Number Systems?

Number systems help in representing the numbers in a small symbol set. Binary numbers are mostly used in computers that use digits like 0 and 1 for calculating simple problems. The number systems also help in converting one number system to another.

How are Number Systems Classified?

The number systems can be classified mainly into two categories: Positional and Non-positional number systems. For positional number systems, each digit is associated with a weight and its examples are binary, octal, decimal, etc. In non-positional number systems, the digit values are independent of their positions and its examples are gray code, cyclic code, aroma code, etc.

Why are Different Number Systems Used in Computers?

Computers cannot understand human languages, so to understand the commands and instructions given to the computers by programmers, different number systems are used such as the binary system, the octal system, the decimal system, and so on.

Digit in Math – Definition with Examples

Digit definition in math, place value, solved examples, practice problems, frequently asked questions.

Digits are the single numbers used to represent values in math. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used in different combinations and repetitions to represent all the values in math. 

Any of the ten numbers from 0 to 9 can be represented by a symbol known as a digit.

An example of a two-digit (2-D) number is 65. It is made up of the 6 and 5.  

In ancient times, people did not have a number system or numerals to measure or count things. As the world expanded its roots into subjects like science and the trade between nations grew, an urgent requirement for a uniform numeric system came. Thus, digits were formed and combined for use in different situations.

We use international numerals like “123” and “65”, but the Romans used Roman numerals, and there have been many other numerals used throughout history.

Related Worksheets

In math, every digit in a number has a place value. Place value can be defined as the value represented by a digit in a number on the basis of its position in the number.

For example, the place value of 7 in 3743 is 7 hundreds or 700. However, the place value of 7 in 7432 is 7 thousands or 7000. Here, we can see that even though the 7 is the same in both numbers, its place value changes with the change in its position. 

Place value and face value are not the same. The face value of a digit  is the value of the digit whereas the place value of a digit is its place in the number. In simple words, the face value tells the actual value whereas the place value tells the value based on its position.  Hence, the face value of the digit never changes irrespective of its position in the number, whereas its place value changes with the change in its position. 

For example, the face value of 2 in both the numbers 283 and 823 is 2. Whereas, the place value of 2 is 200 in 283 and 20 in 823. 

Example of digit numbers:

• Two-digit Numbers

Two-digit numbers begin from 10 and end at 99. The tens place should be between 1 and 9.

Consider two numbers, 15 and 37. When these two numbers are added, they give a new number, 52.

15 + 37 = 52

• Four-digit Numbers

When four digits are written together, a four-digit number is formed. The range of these numbers is from 1000 to 9999.

Consider the numbers 1001, 2001, 5000, and 1040. When these numbers are added, they give a new four-digit number, 9042.

$1001 + 2001 + 5000 + 1040 = 9042$

Similar to the examples mentioned above, as the digits increase in a number, its value increases. For example, 10,000 is a 5-D number whose value is more than all 4 – D natural numbers.

Example 1: How many digits are there in 1458?

Solution : The number 1458 has four digits that are 1, 4, 5, and 8. 

Example 2: Using the digits 6, 6, 8, get the greatest $3-$ digit number.

Solution : The greatest $3-$digit number that can be formed with these is 866. 

Example 3: What is the place value of digit 4 in the number 84,527?Solution : The place value of 4 in 84,527 is 4000 (four thousand).

Digit in Math - Definition with Examples

Attend this quiz & Test your knowledge.

Which of these make the greatest three-digit number?

Which of these is at the ten thousands place in the number 783,425, which is a decimal number in 36.2.

What is the difference between decimals and digits?

The term digits refer to a collection of real numbers, including zero and all positive counting numbers. Fractions, negative integers, and decimals are not treated as digits. 

The decimal is a number that is placed on the right-hand side after the point (.) in a number.

For example, in the number 23.8, 8 is a decimal number.

Which digits are used to make numerals?

0 to 9 are used in different combinations to make numerals.

Are fractions a part of digits?

Fractions represent specific parts of a whole significant digit. They lie in between two significant numbers. They are a part of the number system but not necessarily a part of significant digits.

Who introduced the concept of a digit?

There is no specific name. It is believed that zero was first used in 4 CE by the Mayans.

RELATED POSTS

• Area in Math – Definition, Composite Figures, FAQs, Examples
• Flat Surface – Definition with Examples
• Lateral Face – Definition With Examples
• Repeated Addition – Definition, Examples, Practice Problems, FAQs
• Base Ten Number System – Definition With Examples

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Make study-time fun with 14,000+ games & activities, 450+ lesson plans, and more—free forever.

Parents, Try for Free Teachers, Use for Free

Common Number Sets

There are sets of numbers that are used so often they have special names and symbols:

Number Sets In Use

Here are some algebraic equations, and the number set needed to solve them:

We can take an existing set symbol and place in the top right corner:

• a little + to mean positive, or
• a little * to mean non zero, like this:

And we can always use set-builder notation .

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Praxis Core Math

Course: praxis core math   >   unit 1.

• Rational number operations | Lesson
• Rational number operations | Worked example
• Ratios and proportions | Lesson
• Ratios and proportions | Worked example
• Percentages | Lesson
• Percentages | Worked example
• Rates | Lesson
• Rates | Worked example
• Naming and ordering numbers | Lesson
• Naming and ordering numbers | Worked example

Number concepts | Lesson

• Number concepts | Worked example
• Counterexamples | Lesson
• Counterexamples | Worked example
• Pre-algebra word problems | Lesson
• Pre-algebra word problems | Worked example
• Unit reasoning | Lesson
• Unit reasoning | Worked example

What are number concepts?

What skills are tested.

• Using the properties of odd and even numbers
• Finding factors and greatest common factors
• Finding multiples and least common multiples
• Using number properties to solve problems

What's always true when adding or multiplying odd and even numbers?

• The sum of two odd numbers is even.
• The sum of two even numbers is even.
• The sum of an odd and an even number is odd.
• The product of two odd numbers is odd.
• The product of two even numbers is even.
• The product of an odd and an even number is even.

What are factors and how do we find them?

• 1 ‍   and 12 ‍  
• 2 ‍   and 6 ‍  
• 3 ‍   and 4 ‍  

What is prime factorization?

What is the greatest common factor.

• List the factors of each number and select the largest common factor.
• Write the prime factorization of each number and then multiply all common factors.
• 24 ‍   : 1 ‍   , 2 ‍   , 3 ‍   , 4 ‍   , 6 ‍   , 8 ‍   , 12 ‍   , 24 ‍  
• 36 ‍   : 1 ‍   , 2 ‍   , 3 ‍   , 4 ‍   , 6 ‍   , 9 ‍   , 12 ‍   , 18 ‍   , 36 ‍  
• 60 ‍   : 1 ‍   , 2 ‍   , 3 ‍   , 4 ‍   , 5 ‍   , 6 ‍   , 10 ‍   , 12 ‍   , 15 ‍   , 20 ‍   , 30 ‍   , 60 ‍  
• 24 = 2 × 2 × 2 × 3 ‍  
• 36 = 2 × 2 × 3 × 3 ‍  
• 60 = 2 × 2 × 3 × 5 ‍  

What are multiples?

• 3 × 1 = 3 ‍  
• 3 × 2 = 6 ‍  
• 3 × 3 = 9 ‍  
• 3 × 4 = 12 ‍  

What is the least common multiple?

• List the multiples of each number until we find a common multiple.
• Multiply together all the unique factors in the prime factorization of both numbers.
• 8 ‍   : 8 ‍   , 16 ‍   , 24 ‍   , 32 ‍   , 40 ‍   , 48 ‍   , 56 ‍  
• 10 ‍   : 10 ‍   , 20 ‍   , 30 ‍   , 40 ‍   , 50 ‍   , 60 ‍  
• 8 = 2 × 2 × 2 ‍  
• 10 = 2 × 5 ‍  
• (Choice A)   k ‍   is even. A k ‍   is even.
• (Choice B)   r ‍   is even. B r ‍   is even.
• (Choice C)   r ‍   can be either even or odd. C r ‍   can be either even or odd.
• (Choice D)   k ‍   can be either even or odd. D k ‍   can be either even or odd.
• (Choice E)   Both k ‍   and r ‍   are odd. E Both k ‍   and r ‍   are odd.
• (Choice A)   3 ‍   A 3 ‍  
• (Choice B)   5 ‍   B 5 ‍  
• (Choice C)   9 ‍   C 9 ‍  
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Things to remember

Want to join the conversation.

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

• Dictionaries home
• American English
• Collocations
• German-English
• Grammar home
• Practical English Usage
• Learn & Practise Grammar (Beta)
• Word Lists home
• My Word Lists
• Recent additions
• Resources home
• Text Checker

NEW words and meanings added: March 2024

Game on or game over ?

With the Summer Olympics coming up, the main focus for our latest release is on the world of sport.

Worrying about your team’s back four in the relegation six-pointer ? Did one of your team’s blueliners just score an empty-netter ? Or maybe you prefer T20™ , with the drama of one-dayers and super overs , or just like to pay your green fee and get started on the front nine ?

We’ve added over 170 new words and meanings from sport and other topics.

Our word lists are designed to help learners at any level focus on the most important words to learn.

Explore our general English and academic English lists.

Spread the Word

A recent addition to our online dictionary is the term culture war , which is used to describe the conflict between groups of people with different ideals and beliefs.

Topic Dictionaries

Our Topic Dictionaries are lists of topic-related words, like Animals and Health , that can help you expand your vocabulary. Each topic is divided into smaller subtopics and every word has a CEFR level.

Learn & Practise Grammar

Our grammar pages combine clear explanations with interactive exercises to test your understanding.

Learn more with these dictionary and grammar resources

We offer a number of premium products on this website to help you improve your english..

JOIN our community of language learners!

Connect with us TODAY to start receiving the language learning and assessment resources you need directly to your newsfeed and inbox.

Prime numbers are natural numbers that are divisible by only 1 and the number itself. In other words, prime numbers are positive integers greater than 1 with exactly two factors, 1 and the number itself.   Some of the prime numbers include 2, 3, 5, 7, 11, 13, etc. Always remember that 1 is neither prime nor composite. Also, we can say that except for 1, the remaining numbers are classified as prime and composite numbers . All prime numbers are odd numbers except 2, 2 is the smallest prime number and is the only even prime number.

Prime numbers are the natural numbers greater than 1 with exactly two factors, i.e. 1 and the number itself.

In this article, you will learn the meaning and definition of prime numbers, their history, properties,  list of prime numbers from 1 to 1000, chart, differences between prime numbers and composite numbers, how to find the prime numbers using formulas, along with video lesson and examples.

Learn: Mathematics

What are Prime Numbers?

A prime number is a positive integer having exactly two factors, i.e. 1 and the number itself. If p is a prime, then its only factors are necessarily 1 and p itself. Any number that does not follow this is termed a composite number, which can be factored into other positive integers. Another way of defining it is a positive number or integer, which is not a product of any other two positive integers other than 1 and the number itself. 

First Ten Prime Numbers

The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 .

Note: It should be noted that 1 is a non-prime number. It is a unique number.

Download PDF – Prime Numbers

History of prime numbers.

The prime number was discovered by Eratosthenes (275-194 B.C., Greece). He took the example of a sieve to filter out the prime numbers from a list of natural numbers and drain out the composite numbers.

Students can practise this method by writing the positive integers from 1 to 100, circling the prime numbers, and putting a cross mark on composites. This kind of activity refers to the Sieve of Eratosthenes .

Properties of Prime Numbers

Some of the properties of prime numbers are listed below:

• Every number greater than 1 can be divided by at least one prime number.
• Every even positive integer greater than 2 can be expressed as the sum of two primes.
• Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number.
• Two prime numbers are always coprime to each other.
• Each composite number can be factored into prime factors and individually all of these are unique in nature.

Prime Numbers Chart

Before calculators and computers, numerical tables were used for recording all of the primes or prime factorizations up to a specified limit and are usually printed. The most beloved method for producing a list of prime numbers is called the sieve of Eratosthenes. This method results in a chart called Eratosthenes chart, as given below. The chart below shows the prime numbers up to 100 , represented in coloured boxes.

Video Lesson on Prime Numbers

A prime number is a whole number greater than 1 whose only factors are 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. It should be noted that 1 is a non-prime number. Conferring to the definition of prime number, which states that a number should have exactly two factors, but number 1 has one and only one factor. Thus 1 is not considered a Prime number. To learn more about prime numbers watch the video given below.

List of Prime Numbers 1 to 100

There are several primes in the number system. As we know, the prime numbers are the numbers that have only two factors which are 1 and the number itself. 

The list of prime numbers from 1 to 100 are given below:

Thus, there are 25 prime numbers between 1 and 100, i.e. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 . All these numbers are divisible by only 1 and the number itself. Hence, these numbers are called prime numbers. Also, these are the first 25 prime numbers.

Prime Numbers 1 to 200

Here is the list of prime numbers from 1 to 200, which we can learn and crosscheck if there are any other factors for them.

Prime Numbers 1 to 1000

There are a total of 168 prime numbers between 1 to 1000. They are:

Also, get the list of prime numbers from 1 to 1000 along with detailed factors here.

Facts About Prime Numbers

The table below shows the important points about prime numbers. These will help you to solve many problems in mathematics.

Click here to learn more about twin prime numbers .

How to Find Prime Numbers?

The following two methods will help you to find whether the given number is a prime or not. Method 1: We know that 2 is the only even prime number. And only two consecutive natural numbers which are prime are 2 and 3. Apart from those, every prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a natural number. For example: 6(1) – 1 = 5 6(1) + 1 = 7 6(2) – 1 = 11 6(2) + 1 = 13 6(3) – 1 = 17 6(3) + 1 = 19 6(4) – 1 = 23 6(4) + 1 = 25 (multiple of 5) … Method 2: To know the prime numbers greater than 40, the below formula can be used. n 2 + n + 41, where n = 0, 1, 2, ….., 39 For example: (0) 2 + 0 + 0 = 41 (1) 2 + 1 + 41 = 43 (2) 2 + 2 + 41 = 47 …..

Is 1 a Prime Number?

Conferring to the definition of the prime number, which states that a number should have exactly two factors for it to be considered a prime number. But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered a Prime number.

Examples: 2, 3, 5, 7, 11, etc.

In all the positive integers given above, all are either divisible by 1 or itself, i.e. precisely two positive integers.

Prime Numbers vs Composite Numbers

A few differences between prime numbers and composite numbers are tabulated below:

Prime Numbers Related Articles

Solved examples on prime numbers.

Is 10 a Prime Number?

No, because it can be divided evenly by 2 or 5, 2×5=10, as well as by 1 and 10.

Alternatively,

Using method 1, let us write in the form of 6n ± 1. 

10 = 6(1) + 4 = 6(2) – 2

This is not of the form 6n + 1 or 6n – 1.

Hence, 10 is not a prime number.

Is 19 a Prime Number?

Let us write the given number in the form of 6n ± 1. 6(3) + 1 = 18 + 1 = 19 Therefore, 19 is a prime number.

Find if 53 is a prime number or not.

The only factors of 53 are 1 and 53.

Let us write the given number in the form of 6n ± 1.

6(9) – 1 = 54 – 1 = 53

So, 53 is a prime number.

Check if 64 is a prime number or not.

The factors of 64 are 1, 2, 4, 8, 16, 32, 64.

It has factors other than 1 and 64.

Hence, it is a composite number and not a prime number.

Example 5:  

Which is the greatest prime number between 1 to 10?

As we know, the first 5 prime numbers are 2, 3, 5, 7, 11.

There are 4 prime numbers between 1 and 10 and the greatest prime number between 1 and 10 is 7.

Practice Problems

• Identify the prime numbers from the following numbers: 34, 27, 29, 41, 67, 83
• Which of the following is not a prime number? 2, 19, 91, 57
• Write the prime numbers less than 50.

Frequently Asked Questions on Prime Numbers

What are prime numbers in maths, how to find prime numbers.

To find whether a number is prime, try dividing it with the prime numbers 2, 3, 5, 7 and 11. If the number is exactly divisible by any of these numbers, it is not a prime number, otherwise, it is a prime. Alternatively, we can find the prime numbers by writing their factors since a prime number has exactly two factors, 1 and the number itself.

What are the examples of prime numbers?

As we know, prime numbers are whole numbers greater than 1 with exactly two factors, i.e. 1 and the number itself. Some of the examples of prime numbers are 11, 23, 31, 53, 89, 179, 227, etc.

What is the smallest prime number?

2 is the smallest prime number. Also, it is the only even prime number in maths.

What is the largest prime number so far?

Which is the largest 4 digit prime number.

The largest 4 digits prime number is 9973, which has only two factors namely 1 and the number itself.

What are prime numbers between 1 and 50?

The list of prime numbers between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Why 1 is not a prime number?

As per the definition of prime numbers, 1 is not considered as the prime number since a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

Visit BYJU’S for all Maths related queries and study materials

Your result is as below

Request OTP on Voice Call

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Post My Comment

By just helped me understand prime numbers in a better way. I have learnt many concepts in mathematics and science in a very easy and understanding way

I understand I lot by this website about prime numbers

when are classes mam or sir. thank you. from: lakshita singh. have a good day.

This is a very nice app .,i understand many more things on this app .thankyou so much teachers 🙏🙏🙏

Thanks for video I learn a lot by watching this website

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

• Patient Care & Health Information
• Diseases & Conditions
• Sleep apnea

Sleep apnea is a potentially serious sleep disorder in which breathing repeatedly stops and starts. If you snore loudly and feel tired even after a full night's sleep, you might have sleep apnea.

The main types of sleep apnea are:

• Obstructive sleep apnea (OSA), which is the more common form that occurs when throat muscles relax and block the flow of air into the lungs
• Central sleep apnea (CSA) , which occurs when the brain doesn't send proper signals to the muscles that control breathing
• Treatment-emergent central sleep apnea , also known as complex sleep apnea, which happens when someone has OSA — diagnosed with a sleep study — that converts to CSA when receiving therapy for OSA

If you think you might have sleep apnea, see your health care provider. Treatment can ease your symptoms and might help prevent heart problems and other complications.

Products & Services

• A Book: Future Care
• A Book: Mayo Clinic Family Health Book, 5th Edition
• Mayo Clinic Store — Solutions for a Healthier Living
• Newsletter: Mayo Clinic Health Letter — Digital Edition
• Sleep Apnea Products from Mayo Clinic Store

The symptoms of obstructive and central sleep apneas overlap, sometimes making it difficult to determine which type you have. The most common symptoms of obstructive and central sleep apneas include:

• Loud snoring.
• Episodes in which you stop breathing during sleep — which would be reported by another person.
• Gasping for air during sleep.
• Awakening with a dry mouth.
• Morning headache.
• Difficulty staying asleep, known as insomnia.
• Excessive daytime sleepiness, known as hypersomnia.
• Difficulty paying attention while awake.
• Irritability.

When to see a doctor

Loud snoring can indicate a potentially serious problem, but not everyone who has sleep apnea snores. Talk to your health care provider if you have symptoms of sleep apnea. Ask your provider about any sleep problem that leaves you fatigued, sleepy and irritable.

There is a problem with information submitted for this request. Review/update the information highlighted below and resubmit the form.

From Mayo Clinic to your inbox

Sign up for free and stay up to date on research advancements, health tips, current health topics, and expertise on managing health. Click here for an email preview.

Error Email field is required

Error Include a valid email address

To provide you with the most relevant and helpful information, and understand which information is beneficial, we may combine your email and website usage information with other information we have about you. If you are a Mayo Clinic patient, this could include protected health information. If we combine this information with your protected health information, we will treat all of that information as protected health information and will only use or disclose that information as set forth in our notice of privacy practices. You may opt-out of email communications at any time by clicking on the unsubscribe link in the e-mail.

Thank you for subscribing!

You'll soon start receiving the latest Mayo Clinic health information you requested in your inbox.

Sorry something went wrong with your subscription

Please, try again in a couple of minutes

Obstructive sleep apnea

Obstructive sleep apnea occurs when the muscles that support the soft tissues in your throat, such as your tongue and soft palate, temporarily relax. When these muscles relax, your airway is narrowed or closed, and breathing is momentarily cut off.

This type of sleep apnea happens when the muscles in the back of the throat relax. These muscles support the soft palate, the triangular piece of tissue hanging from the soft palate called the uvula, the tonsils, the side walls of the throat and the tongue.

When the muscles relax, your airway narrows or closes as you breathe in. You can't get enough air, which can lower the oxygen level in your blood. Your brain senses that you can't breathe, and briefly wakes you so that you can reopen your airway. This awakening is usually so brief that you don't remember it.

You might snort, choke or gasp. This pattern can repeat itself 5 to 30 times or more each hour, all night. This makes it hard to reach the deep, restful phases of sleep.

Central sleep apnea

This less common form of sleep apnea occurs when your brain fails to send signals to your breathing muscles. This means that you make no effort to breathe for a short period. You might awaken with shortness of breath or have a difficult time getting to sleep or staying asleep.

Risk factors

Sleep apnea can affect anyone, even children. But certain factors increase your risk.

Factors that increase the risk of this form of sleep apnea include:

• Excess weight. Obesity greatly increases the risk of OSA . Fat deposits around your upper airway can obstruct your breathing.
• Neck circumference. People with thicker necks might have narrower airways.
• A narrowed airway. You might have inherited a narrow throat. Tonsils or adenoids also can enlarge and block the airway, particularly in children.
• Being male. Men are 2 to 3 times more likely to have sleep apnea than are women. However, women increase their risk if they're overweight or if they've gone through menopause.
• Being older. Sleep apnea occurs significantly more often in older adults.
• Family history. Having family members with sleep apnea might increase your risk.
• Use of alcohol, sedatives or tranquilizers. These substances relax the muscles in your throat, which can worsen obstructive sleep apnea.
• Smoking. Smokers are three times more likely to have obstructive sleep apnea than are people who've never smoked. Smoking can increase the amount of inflammation and fluid retention in the upper airway.
• Nasal congestion. If you have trouble breathing through your nose — whether from an anatomical problem or allergies — you're more likely to develop obstructive sleep apnea.
• Medical conditions. Congestive heart failure, high blood pressure and type 2 diabetes are some of the conditions that may increase the risk of obstructive sleep apnea. Polycystic ovary syndrome, hormonal disorders, prior stroke and chronic lung diseases such as asthma also can increase risk.

Risk factors for this form of sleep apnea include:

• Being older. Middle-aged and older people have a higher risk of central sleep apnea.
• Being male. Central sleep apnea is more common in men than it is in women.
• Heart disorders. Having congestive heart failure increases the risk.
• Using narcotic pain medicines. Opioid medicines, especially long-acting ones such as methadone, increase the risk of central sleep apnea.
• Stroke. Having had a stroke increases the risk of central sleep apnea.

Complications

Sleep apnea is a serious medical condition. Complications of OSA can include:

Daytime fatigue. The repeated awakenings associated with sleep apnea make typical, restorative sleep impossible, in turn making severe daytime drowsiness, fatigue and irritability likely.

You might have trouble concentrating and find yourself falling asleep at work, while watching TV or even when driving. People with sleep apnea have an increased risk of motor vehicle and workplace accidents.

You might also feel quick-tempered, moody or depressed. Children and adolescents with sleep apnea might perform poorly in school or have behavior problems.

High blood pressure or heart problems. Sudden drops in blood oxygen levels that occur during OSA increase blood pressure and strain the cardiovascular system. Having OSA increases your risk of high blood pressure, also known as hypertension.

OSA might also increase your risk of recurrent heart attack, stroke and irregular heartbeats, such as atrial fibrillation. If you have heart disease, multiple episodes of low blood oxygen (hypoxia or hypoxemia) can lead to sudden death from an irregular heartbeat.

• Type 2 diabetes. Having sleep apnea increases your risk of developing insulin resistance and type 2 diabetes.
• Metabolic syndrome. This disorder, which includes high blood pressure, abnormal cholesterol levels, high blood sugar and an increased waist circumference, is linked to a higher risk of heart disease.

Complications with medicines and surgery. Obstructive sleep apnea is also a concern with certain medicines and general anesthesia. People with sleep apnea might be more likely to have complications after major surgery because they're prone to breathing problems, especially when sedated and lying on their backs.

Before you have surgery, tell your doctor about your sleep apnea and how it's being treated.

• Liver problems. People with sleep apnea are more likely to have irregular results on liver function tests, and their livers are more likely to show signs of scarring, known as nonalcoholic fatty liver disease.
• Sleep-deprived partners. Loud snoring can keep anyone who sleeps nearby from getting good rest. It's common for a partner to have to go to another room, or even to another floor of the house, to be able to sleep.

Complications of CSA can include:

Fatigue. The repeated awakening associated with sleep apnea makes typical, restorative sleep impossible. People with central sleep apnea often have severe fatigue, daytime drowsiness and irritability.

You might have difficulty concentrating and find yourself falling asleep at work, while watching television or even while driving.

Cardiovascular problems. Sudden drops in blood oxygen levels that occur during central sleep apnea can adversely affect heart health.

If there's underlying heart disease, these repeated multiple episodes of low blood oxygen — known as hypoxia or hypoxemia — worsen prognosis and increase the risk of irregular heart rhythms.

• Kline LR. Clinical presentation and diagnosis of obstructive sleep apnea in adults. https://www.uptodate.com/contents/search. Accessed June 28, 2022.
• Selim BJ, et al. The association of nocturnal cardiac arrhythmias and sleep-disordered breathing: The DREAM study. Journal of Clinical Sleep Medicine. 2016; doi:10.5664/jcsm.5880.
• Jameson JL, et al., eds. Sleep apnea. In: Harrison's Principles of Internal Medicine. 21st ed. McGraw-Hill; 2022. https://accessmedicine.mhmedical.com. Accessed June 28, 2022.
• Sleep apnea. National Heart, Lung, and Blood Institute. http://www.nhlbi.nih.gov/health/health-topics/topics/sleepapnea/. Accessed June 28, 2022.
• Badr MS. Central sleep apnea: Risk factors, clinical presentation, and diagnosis. https://www.uptodate.com/contents/search. Accessed June 28, 2022.
• Kryger MH, et al. Management of obstructive sleep apnea in adults. https://www.uptodate.com/contents/search. Accessed June 28, 2022.
• Aurora RN, et al. Practice parameters for the surgical modification of the upper airway for obstructive sleep apnea in adults. Sleep. 2010; doi:10.1093/sleep/33.10.1408.
• Amali A, et al. A comparison of uvulopalatopharyngoplasty and modified radiofrequency tissue ablation in mild to moderate obstructive sleep apnea: A randomized clinical trial. Journal of Clinical Sleep Medicine. 2017; doi:10.5664/jcsm.6730.
• Parthasarathy S. Treatment-emergent central sleep apnea. https://www.uptodate.com/contents/search. Accessed June 29, 2022.
• Mehra R. Sleep apnea and the heart. Cleveland Clinic Journal of Medicine. 2019; doi:10.3949/ccjm.86.s1.03.
• Badr MS. Central sleep apnea: Treatment. https://www.uptodate.com/contents/search. Accessed July 1, 2022.
• Olson EJ (expert opinion). Mayo Clinic. June 30, 2022.
• Continuous positive airway pressure (CPAP)
• CPAP machines: Tips for avoiding 10 common problems
• CPAP: How it works
• Which CPAP masks are best for you?

Associated Procedures

• Endoscopic sleeve gastroplasty
• Polysomnography (sleep study)
• Septoplasty
• Tonsillectomy
• Tracheostomy
• Symptoms & causes
• Diagnosis & treatment
• Doctors & departments

Mayo Clinic does not endorse companies or products. Advertising revenue supports our not-for-profit mission.

• Opportunities

Mayo Clinic Press

Check out these best-sellers and special offers on books and newsletters from Mayo Clinic Press .

• Mayo Clinic on Incontinence - Mayo Clinic Press Mayo Clinic on Incontinence
• The Essential Diabetes Book - Mayo Clinic Press The Essential Diabetes Book
• Mayo Clinic on Hearing and Balance - Mayo Clinic Press Mayo Clinic on Hearing and Balance
• FREE Mayo Clinic Diet Assessment - Mayo Clinic Press FREE Mayo Clinic Diet Assessment
• Mayo Clinic Health Letter - FREE book - Mayo Clinic Press Mayo Clinic Health Letter - FREE book

Your gift holds great power – donate today!

Make your tax-deductible gift and be a part of the cutting-edge research and care that's changing medicine.

( Getty Images (6) )

The 14 U.S. Metros Where the Number of Homes for Sale Is Bouncing Back to Pre-Pandemic Levels

The number of homes for sale has scraped bottom for nearly five years—but in some key U.S. markets, inventory is rising to pre-pandemic levels.

And these 14 markets share common traits. Most are in the Sun Belt, and some experienced a population boom during COVID-19 as homebuyers sought out bigger homes in warmer climates.

The Realtor.com® economics team noted the U.S. housing shortage before the pandemic , and then the COVID-19-era market conditions turbocharged the inventory squeeze. The ultralow mortgage rates, newfound mobility to work remotely, and desire for more living space pushed inventory to unseen lows.

Now, the rising inventory levels in these markets could signal a broader loosening of housing supply. And it could be a shot in the arm for homebuyers locked out of the market by still-rising home prices, elevated mortgage rates, and limited choices.

“Our inventory conditions are, for the most part, improving across the U.S.,” says Hannah Jones , a senior economic analyst at Realtor.com.

The laws of supply and demand could kick in, Jones says, leading to slowing price growth, or even falling prices in some places.

“This summer and fall could be better in terms of more inventory, which could take some pressure off of prices, or at the very least, it could dampen price growth,” says Jones.

What's notable, Jones notes, is that prices have not fallen yet, aside from a small correction in the months after mortgage interest rates first rose in 2022 .

Realtor.com dove deeper into the data to see what else there is to learn from the latest inventory numbers.

From among the 100 largest metros in the U.S., we compared the active inventory per 10,000 total housing units, which can provide a more accurate metric for comparisons between metros, over several years of data.

Not surprisingly, we found a few common traits among the places where the number of homes for sale is back above pre-pandemic levels.

Of the 14 metros we found, six are in the Sunshine State. The promise of a balmy, beach lifestyle and relatively affordable real estate made Florida particularly attractive during the pandemic. This influx of new residents likely set the stage for the inventory rebound we’re seeing now.

Many of the other metros also experienced population growth during the pandemic. Homebuyers were driven to some of these areas by the appeal of more space, lower costs of living, and the flexibility to work remotely.

Austin, TX , known for its vibrant tech scene and cultural appeal, saw the surge in population push inventory levels down as demand swelled. Now, with nearly 11,000 active listings, the Austin metro area has 13% more inventory than at the same time in 2019, and more than five times the number of homes for sale in early 2022, when active inventory dropped below 2,100 listings.

This has local experts saying Austin is finally " going back to normal ," with some areas of the metro seeing significant price drops as well .

Similarly, Myrtle Beach, SC , and New Orleans  saw increases in inventory. With the broader housing market moving more slowly over the past two years, many of the "pandemic boomtowns" have seen signs of slowing or even reversing , fitting with what we're seeing now with growing inventory.

Beyond some of these pandemic hot spots, the Sun Belt is well represented here. States such as Florida, Texas, and Arizona are now at the forefront of this inventory rebound.

Their warm climates, lower taxes, and affordable living costs drew many new residents during the pandemic.

"These areas were hot during the pandemic, attracting many new residents," says Jones. "But with prices climbing, demand has now tempered, leading to higher inventory."

Here's our look at where home inventory levels are back at or above pre-pandemic levels. We ranked metros based on how much inventory is up compared with five years ago.

(Sean Pavone/Getty Images)

1. Lakeland, FL

May median list price: $349,425 May active listings: 4,266 May active listings per 10,000 housing units: 140 Change from May 2019 to May 2024: 33%

(Getty Images)

2. Memphis, TN

May median list price: $349,900 May active listings: 3,565 May active listings per 10,000 housing units: 069 Percentage change from May 2019 to May 2024: 22%

3. Colorado Springs, CO

May median list price: $517,500 May active listings: 2,869 May active listings per 10,000 housing units: 098 Percentage change from May 2019 to May 2024: 30%

(TriggerPhoto/iStock)

4. Cape Coral, FL

May median list price: $461,750 May active listings: 11,110 May active listings per 10,000 housing units: 314 Percentage change from May 2019 to May 2024: 18%

(Creative Commons)

5. McAllen, TX

May median list price: $279,000 May active listings: 2,429 May active listings per 10,000 housing units: 089 Percentage change from May 2019 to May 2024: 16%

6. New Orleans, LA

May median list price: $338,500 May active listings: 5,054 May active listings per 10,000 housing units: 100 Percentage change from May 2019 to May 2024: 12%

( Kruck20/Getty Images)

7. Myrtle Beach, SC

May median list price: $345,544 May active listings: 6,726 May active listings per 10,000 housing units: 273 Percentage change from May 2019 to May 2024: 12%

8. Austin, TX

May median list price: $565,000 May active listings: 10,798 May active listings per 10,000 housing units: 110 Percentage change from May 2019 to May 2024: 13%

9. Sarasota, FL

May median list price: $499,900 May active listings: 8,412 May active listings per 10,000 housing units: 208 Percentage change from May 2019 to May 2024: 7%

(realtor.com)

10. Palm Bay, FL

May median list price: $399,000 May active listings: 4,007 May active listings per 10,000 housing units: 147 Percentage change from May 2019 to May 2024: 10%

(San Antonio: f11photo/iStock)

11. San Antonio, TX

May median list price: $348,262 May active listings: 12,221 May active listings per 10,000 housing units: 123 Percentage change from May 2019 to May 2024: 2%

(Thomas Roche/Getty Images)

12. Phoenix, AZ

May median list price: $545,000 May active listings: 17,436 May active listings per 10,000 housing units: 092 Percentage change from May 2019 to May 2024: 11%

(rgaydos/iStock)

13. Daytona Beach, FL

May median list price: $399,900 May active listings: 7,234 May active listings per 10,000 housing units: 239 Percentage change from May 2019 to May 2024: 6%

14. Tampa, FL

May median list price: $425,000 May active listings: 15,983 May active listings per 10,000 housing units: 116 Percentage change from May 2019 to May 2024: 2%

Evan Wyloge is a data journalist at Realtor.com. He covers trends in real estate.

• Related Articles

Share this Article

These are the top states for plug-in vehicle registrations

Plug-ins' overall market share is small but growing.

Electric vehicle sales numbers might not be growing as quickly as many had hoped, but they are growing. Plug-in hybrids are also increasing in popularity, though both types of vehicles are far more popular in some states than others. Energy.gov’s Fact of the Week (FOTW) for this week highlights the states where drivers register the most plug-in vehicles, and there are few surprises.

California is leading the way. Plug-ins (both all-electric and hybrid) hold more than a 4% share of vehicle registrations there, followed by the District of Columbia, where just over 3.5 percent were electrified. The top five states with the most EV or PHEV registrations as of December 2023 include:

• California: EVs/PHEVs are 4.3% of overall vehicle registrations
• District of Columbia: 3.52%
• Hawaii: 2.8%
• Washington: 2.64%
• Oregon: 2.27%

While it’s easy to get caught up in the news that the sky is falling in EV land, the data tell a slightly different story. Nine states recorded plug-in vehicle registrations of 2% or more, and half of the states had more than 1%. Those numbers obviously pale in comparison to ICE vehicle registrations, but they are growing. The Argonne National Laboratory, which provided data for the FOTW, noted that plug-in vehicle sales rose more than 50% between 2022 and 2023, with EVs accounting for 80% of plug-in sales.

On the opposite end of the spectrum were several states with less than half a percent of EV registrations. Mississippi was the worst, with just 0.19%, followed closely by North Dakota with 0.20%. Wyoming, West Virginia, and Louisiana rounded out the bottom five.

EV registrations in those states are likely lagging for a variety of reasons. Electrification is a uniquely political topic these days, with many states’ registration numbers closely aligning with the way their citizens vote. Charging infrastructure is another hurdle EVs face, as extremely rural states like North Dakota and Wyoming don’t have the same level of support that more densely populated areas like D.C. do.

• Government/Legal
• License License
• Facebook Share
• Twitter Share
• Tumblr Share
• Twitch Share
• Flipboard Share
• Instagram Share
• Newsletter Share
• Youtube Share
• Feeds Share

Popular Vehicles

Featured makes, product guides.

• The Best Electric Bikes
• The Best Car Covers
• The Best Portable Air Compressors
• The Best Car GPS Trackers

Choose a Display Name

Please enter a display name

Sign in to post

Please sign in to leave a comment.

• Share full article

Advertisement

Supported by

Guest Essay

The Long-Overlooked Molecule That Will Define a Generation of Science

By Thomas Cech

Dr. Cech is a biochemist and the author of the forthcoming book “The Catalyst: RNA and the Quest to Unlock Life’s Deepest Secrets,” from which this essay is adapted.

From E=mc² to splitting the atom to the invention of the transistor, the first half of the 20th century was dominated by breakthroughs in physics.

Then, in the early 1950s, biology began to nudge physics out of the scientific spotlight — and when I say “biology,” what I really mean is DNA. The momentous discovery of the DNA double helix in 1953 more or less ushered in a new era in science that culminated in the Human Genome Project, completed in 2003, which decoded all of our DNA into a biological blueprint of humankind.

DNA has received an immense amount of attention. And while the double helix was certainly groundbreaking in its time, the current generation of scientific history will be defined by a different (and, until recently, lesser-known) molecule — one that I believe will play an even bigger role in furthering our understanding of human life: RNA.

You may remember learning about RNA (ribonucleic acid) back in your high school biology class as the messenger that carries information stored in DNA to instruct the formation of proteins. Such messenger RNA, mRNA for short, recently entered the mainstream conversation thanks to the role they played in the Covid-19 vaccines. But RNA is much more than a messenger, as critical as that function may be.

Other types of RNA, called “noncoding” RNAs, are a tiny biological powerhouse that can help to treat and cure deadly diseases, unlock the potential of the human genome and solve one of the most enduring mysteries of science: explaining the origins of all life on our planet.

Though it is a linchpin of every living thing on Earth, RNA was misunderstood and underappreciated for decades — often dismissed as nothing more than a biochemical backup singer, slaving away in obscurity in the shadows of the diva, DNA. I know that firsthand: I was slaving away in obscurity on its behalf.

In the early 1980s, when I was much younger and most of the promise of RNA was still unimagined, I set up my lab at the University of Colorado, Boulder. After two years of false leads and frustration, my research group discovered that the RNA we’d been studying had catalytic power. This means that the RNA could cut and join biochemical bonds all by itself — the sort of activity that had been thought to be the sole purview of protein enzymes. This gave us a tantalizing glimpse at our deepest origins: If RNA could both hold information and orchestrate the assembly of molecules, it was very likely that the first living things to spring out of the primordial ooze were RNA-based organisms.

That breakthrough at my lab — along with independent observations of RNA catalysis by Sidney Altman at Yale — was recognized with a Nobel Prize in 1989. The attention generated by the prize helped lead to an efflorescence of research that continued to expand our idea of what RNA could do.

In recent years, our understanding of RNA has begun to advance even more rapidly. Since 2000, RNA-related breakthroughs have led to 11 Nobel Prizes. In the same period, the number of scientific journal articles and patents generated annually by RNA research has quadrupled. There are more than 400 RNA-based drugs in development, beyond the ones that are already in use. And in 2022 alone, more than $1 billion in private equity funds was invested in biotechnology start-ups to explore frontiers in RNA research.

What’s driving the RNA age is this molecule’s dazzling versatility. Yes, RNA can store genetic information, just like DNA. As a case in point, many of the viruses (from influenza to Ebola to SARS-CoV-2) that plague us don’t bother with DNA at all; their genes are made of RNA, which suits them perfectly well. But storing information is only the first chapter in RNA’s playbook.

Unlike DNA, RNA plays numerous active roles in living cells. It acts as an enzyme, splicing and dicing other RNA molecules or assembling proteins — the stuff of which all life is built — from amino acid building blocks. It keeps stem cells active and forestalls aging by building out the DNA at the ends of our chromosomes.

RNA discoveries have led to new therapies, such as the use of antisense RNA to help treat children afflicted with the devastating disease spinal muscular atrophy. The mRNA vaccines, which saved millions of lives during the Covid pandemic, are being reformulated to attack other diseases, including some cancers . RNA research may also be helping us rewrite the future; the genetic scissors that give CRISPR its breathtaking power to edit genes are guided to their sites of action by RNAs.

Although most scientists now agree on RNA's bright promise, we are still only beginning to unlock its potential. Consider, for instance, that some 75 percent of the human genome consists of dark matter that is copied into RNAs of unknown function. While some researchers have dismissed this dark matter as junk or noise, I expect it will be the source of even more exciting breakthroughs.

We don’t know yet how many of these possibilities will prove true. But if the past 40 years of research have taught me anything, it is never to underestimate this little molecule. The age of RNA is just getting started.

Thomas Cech is a biochemist at the University of Colorado, Boulder; a recipient of the Nobel Prize in Chemistry in 1989 for his work with RNA; and the author of “The Catalyst: RNA and the Quest to Unlock Life’s Deepest Secrets,” from which this essay is adapted.

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

Follow the New York Times Opinion section on Facebook , Instagram , TikTok , WhatsApp , X and Threads .

Food is Medicine: A Project to Unify and Advance Collective Action

The White House Conference on Hunger, Nutrition, and Health — held in September 2022 — renewed national attention and issued a call to action to end hunger and reduce the prevalence of chronic disease in the United States by 2030.  

Food is Medicine approaches that focus on integrating consistent access to diet- and nutrition- related resources are a critical component to achieve this goal. The approaches are increasingly present across many communities and systems. There’s also increasing federal investment and action to support Food is Medicine approaches in a variety of settings. 

Building on this collective energy, the Department of Health and Human Services (HHS) developed a Food is Medicine initiative in response to a congressionally funded initiative in fiscal year 2023. This congressional action directed the Secretary of HHS, in consultation with other federal agencies, to develop and implement a federal strategy to reduce nutrition-related chronic diseases and food insecurity to improve health and racial equity in the United States. This includes diet-related research and programmatic efforts that will increase access to Food is Medicine initiatives.

Learn about Food is Medicine framing language and principles.

Understanding the Connection Between Food and Health

Access to nutritious food is critical to health and resilience. Food is Medicine is a concept that reaffirms this connection, recognizing that access to high-quality nourishment is essential for well-being. By supporting the production of and facilitating access to nutritious food across a health continuum and range of settings, approaches to Food is Medicine support immediate and long-term resources for people, communities, and systems.

Nutrition Security and Health: By the Numbers

• Approximately 33.8 million people live in food-insecure households. 1
• Household food insecurity affected 12.5 percent of households with children in 2021. 1
• About half of all American adults — or 117 million individuals — have 1 or more preventable chronic disease, many of which are related to poor-quality eating patterns and physical inactivity. These include cardiovascular disease, high blood pressure, type 2 diabetes, some cancers, and poor bone health. 2
• Lower food security is associated with higher probability of chronic disease diagnosis — including hypertension, coronary heart disease, hepatitis, stroke, cancer, asthma, diabetes, arthritis, COPD (chronic obstructive pulmonary disease), and kidney disease. 3
• Nearly $173 billion a year is spent on health care for obesity alone. 4

HHS Approach

HHS will work collaboratively with federal partners and external organizations and communities to develop resources that can be used to advance Food is Medicine approaches across the country.

Our plans include the following steps:

1. Listen to Communities and Implementation Partners

• HHS will engage with a variety of external partners across the nation to better understand challenges and opportunities to advance Food is Medicine models.
• HHS will conduct an environmental scan of existing Food is Medicine models, initiatives, and federal, state, and local regulations and policies.

2. Cultivate Partnerships with Cross-Sector Leaders

• HHS and federal colleagues will build a public-private learning collaborative to support collectively measuring and demonstrating data-driven insights.

3. Develop Resources to Support Broad Uptake

• HHS will develop an evidence-based implementation resource that can serve as a practical guide to help communities understand how to design and implement effective Food is Medicine pilots and policy-sustaining programs.
• HHS will create a unified, applied-measures registry to support and create a transparent catalog of high-value, reliable measures.
• HHS will facilitate knowledge exchange and continue to identify opportunities for federal action to advance a robust Food is Medicine landscape.

Please note that this webpage will be updated with tools and resources as the initiative progresses.

Examples of Current HHS Food is Medicine Activities and Practice Resources

• Innovations in Medicaid programs: Section 1115 of the Social Security Act gives the Secretary of Health and Human Services authority to approve experimental, pilot, or demonstration projects, offering states an avenue to test new approaches in Medicaid that differ from what is required by federal statute. The Biden-Harris Administration has encouraged states to propose innovative Section 1115 waivers that expand coverage, reduce health disparities, and/or advance whole-person care (including addressing health-related social needs). HHS recently approved groundbreaking Medicaid initiatives in Massachusetts and Oregon and Arkansas that gave the states new authority to test coverage for evidence-based nutritional assistance and medically tailored meals. 
• Indian Health Service (IHS) Produce Prescription Pilot Program: American Indian and Alaska Native (AI/AN) people are disproportionately impacted by food insecurity when compared to non-Native people. They’re also more likely to live in areas with low or no access to fresh foods than any other racial or ethnic group. Produce prescription programs have been shown to increase access to nutritious foods in communities at risk for food insecurity. In 2023, IHS awarded a total of $2.5 million in funding to help decrease food insecurity in Native communities. Of that funding, 5 tribes and tribal organizations received $500,000 each in 2023 to implement a produce prescription program in their communities.
• Stimulating Research: In June 2022, the National Institutes of Health (NIH) released a Notice of Special Interest (NOSI): Stimulating Research to Understand and Address Hunger, Food and Nutrition Insecurity to encourage research on the efficacy of interventions that address nutrition security and the mechanisms of food insecurity on a variety of health outcomes. In April 2023, as part of a government-wide collaboration that includes 12 federal agencies and offices, NIH released a Request for Information on Food is Medicine Research Opportunities to gather input on the following topic areas: 1) Research, 2) Community Outreach and Engagement, 3) Education and Training, 4) Provision of Food is Medicine Services and Activities, and 5) Coverage for Services.
• Supporting Best Practices: The Administration for Community Living developed a public-facing resource with information to support partners’ design and implementation of Food is Medicine approaches, such as medically tailored meals, for older adults.

1 https://www.ers.usda.gov/topics/food-nutrition-assistance/food-security-in-the-u-s/key-statistics-graphics

2 https://www.usda.gov/media/blog/2016/03/16/healthy-eating-index-how-america-doing#:~:text=About%20half%20of%20all%20American,cancers%2C%20and%20poor%20bone%20health

3 https://www.ers.usda.gov/webdocs/publications/84467/err-235.pdf?v=9081.9

4 https://www.cdc.gov/chronicdisease/resources/publications/factsheets/nutrition.htm

The Office of Disease Prevention and Health Promotion (ODPHP) cannot attest to the accuracy of a non-federal website.

Linking to a non-federal website does not constitute an endorsement by ODPHP or any of its employees of the sponsors or the information and products presented on the website.

You will be subject to the destination website's privacy policy when you follow the link.

StarTribune

Declining muskie numbers don't support angler interest in these trophy fish.

Muskie fishing opens in Minnesota this weekend, but excitement for the season's first days isn't what it once was. That's because the state's lakes and rivers hold fewer muskies than they did a decade and more ago.

In fact, muskies — one of the world's most exciting sport fish — are at a crossroads in Minnesota. Perhaps in response, the Department of Natural Resources has announced development of a new muskie management plan , which likely won't be completed until next year. (I'll focus on the agency's perspective in an upcoming column.)

Whether the plan boosts muskie numbers in the approximately 100 Minnesota lakes and rivers that hold them (about half due to stocking) remains to be seen.

Below are snapshot interviews with three muskie-fishing experts — Bob Turgeon, 64, of the Twin Cities; Josh Stevenson , 48, a guide and owner of Blue Ribbon Bait in Oakdale; and muskie guide Josh Borovsky , 51, who fishes metro and northern lakes.

The status of Minnesota muskies

Stevenson: It's poor to below average and needs help. But the DNR appears to be doing very little to return muskies to their former populations. My understanding is that membership in Muskies Inc. is down as a result of declining interest. A guy who comes into my shop fishes aggressively and he caught only four muskies all last year in White Bear Lake.

Turgeon: The hours you put in vs. the reward is way down. A big issue is that stocking has not kept up with fishing pressure. Muskies took off when they were first stocked. But they've since settled back, in part I think due to too little stocking of larger muskies.

Borovsky: Muskies have taken a huge decline from where they were. Mille Lacs is the most obvious example of the DNR letting a fishery go downhill. Muskie stocking in Mille Lacs was stopped for four years in the early 2000s. Since then, they've stocked only 3,000 fish every other year. When Mille Lacs went downhill, its large army of muskie anglers dispersed, quadrupling pressure on many other lakes.

Is the DNR bowing to pressure from muskie opponents?

Turgeon: Whenever muskie stocking is proposed, opposition seems to spring up. Endless studies have determined what muskies eat, and it's not walleyes. But opponents keep saying the same thing, that muskies eat walleyes. In fact, some of the state's best walleye lakes are good muskie lakes.

Borovsky: Opposition to muskie stocking has been an issue in the past, in the Legislature particularly. In general, there's just been a lack of urgency at the DNR to address the problem. Mille Lacs, Minnetonka and Vermilion all took huge hits when they backed off stocking. Eventually, those declining fisheries caused all muskie fisheries to be overrun with pressure. Even with catch and release, there's delayed mortality.

Stevenson: Multiple studies have shown walleyes are not a primary prey fish of muskies. You would like to think the DNR would be guided by facts, not incorrect opinions. But the truth is that muskies have been down for quite a few years, and nothing significant at the DNR has happened in response.

Are muskie stocking adjustments needed?

Stevenson: Fingerling muskies have low survival when they're stocked, whereas larger muskies, meaning, generally, yearlings, have greater survival rates. The larger muskies cost more to produce. But if muskie anglers were informed about these expenses, I'm sure they would chip in more money than they already do to help. Unfortunately, the DNR distributes no reliable information about muskies' status, and what they plan to do about it, if anything.

Turgeon: Growing a muskie to 20 inches before stocking is obviously more expensive than stocking fingerlings. Those of us who are very involved in muskie fishing understand that. What we don't understand is why more of these fish aren't stocked. Is it cost? Politics? Whatever it is, let's acknowledge it and move forward.

Borovsky: If the DNR doesn't do something soon about stocking, I'll be too old to enjoy it. On Vermilion they did increase stocking in 2015, and we're starting to see some of those fish. But they're still sitting on their hands on Mille Lacs. Bringing it back would take a lot of pressure off other muskie lakes.

How have you adjusted to the downturn?

Stevenson: No guide can guarantee a muskie every trip. But in the past I produced enough to keep clients coming back. After all, catching a muskie is a life goal for many anglers. But the lower muskie populations have forced me more often to take clients bass or walleye fishing. Also, with my new jet boat, I explore new locations. If clients insist on muskies, I interview them extensively before taking the booking to make sure their stamina and expectations are in line with muskie numbers we're likely to encounter.

Turgeon: I still fish some metro lakes for muskies, but my expectations are way down. I've also been renting a Canadian lake cabin for a month in summer to find fishing similar to what I once had in Minnesota. In its prime, Mille Lacs was one of the best muskie lakes in the world. That's not even close to true now.

Borovsky: Higher fishing pressure for fewer fish has forced me and most other guides to adapt. Changing tactics and the lakes we fish and when we fish them are part of our adaptations. There's newfound pressure not only to refine our patterns but to be the first to create new patterns by designing new lures.

Outdoors columnist Dennis Anderson joined the Star Tribune in 1993 after serving in the same position at the St. Paul Pioneer Press for 13 years. His column topics vary widely, and include canoeing, fishing, hunting, adventure travel and conservation of the environment.

• What I've observed about Mary Moriarty, my 2022 opponent
• Scoggins: Why the Vikings have the best kind of owners
• Warring words, ejections and lawsuits: Why can't local governments work together?
• State's e-bike online rebate application crashes
• Scanners on staffed cash registers at Cub grocery stores down

Anderson: Where these guides fish, you should, too

Anderson: More women are out fishing, but where are more kids?

Anderson: Weather more than walleye was top of mind as we launched the opener

Live: Updates and photos from the Minnesota fishing opener

• Anderson: Where these guides fish, you should, too May. 24
• Electronic licensing, carp deterrent are outdoors wins at Capitol • Outdoors
• 'Canoe King of Ely' Bill Rom dies • Outdoors
• Trophy Tales: See the photos and send us your best catches • Outdoors
• Anderson: Before saunas got cool, Ely Steam Sauna was already hot • Outdoors
• Grand Marais author takes on danger and death in the BWCAW • Outdoors

© 2024 StarTribune. All rights reserved.

Claim your offer here