What Are Numbers? Types and Examples

Numbers describe amounts, measurements, positions and probabilities. We use them to count three books, record a temperature of $-8^\circ\text{C}$ , measure half a kilometre or express a probability of $0.2$ .

Different questions call for different kinds of numbers. Counting numbers tell us how many objects there are. Negative numbers describe values below a reference point, fractions describe division into equal parts, and irrational numbers describe lengths that cannot be written as ratios of integers.

On this page (12 sections)

Numbers, numerals and digits

A number is a mathematical value; a numeral is a written representation of that value. The digit 7, the word seven and the Roman numeral VII can all represent the same quantity. A collection of seven dots models that quantity.

Place and notation determine how a numeral is read. The decimal numeral $507$ has three digits with different place values. The binary numeral $111_2$ represents seven. The decimal and binary numerals have the same value.

The digits in base ten are $0$ through $9$ . Written representations of numbers may also use decimal points, fraction bars, negative signs, roots and other notation. The notation tells us how to read the value.

Four ways to represent the number seven

7Digit

sevenWord

VIIRoman numeral

Quantity

The numeral, word and Roman numeral represent seven. The seven dots model the same quantity.

Counting, ordering and measuring

The meaning of a number depends partly on how it is being used. A cardinal number answers “how many?” If there are 12 students in a room, 12 describes the size of the set. An ordinal number describes order or position, such as 1st, 2nd, or 10th. A measure attaches a number to a unit, such as $3.5\text{ m}$ or $20\text{ min}$ . A number can also serve as a label, such as a bus route number, where the numeral identifies something rather than measuring its quantity.

Number sense means understanding values, relationships, estimates and the effects of operations. In its guide to elementary mathematics intervention, the Institute of Education Sciences recommends clear mathematical language, representations and number lines to support this understanding.

Counting numbers and integers

Natural numbers

The natural numbers are the numbers used for counting. A common convention is

\mathbb{N} = \{1,2,3,4,5,\ldots\}.

This guide starts the natural numbers at $1$ . Some textbooks include $0$ , so check the convention when using another source.

Natural numbers answer questions such as “How many books are there?” and support the earliest forms of arithmetic. They continue without end. For every natural number $n$ , another natural number $n+1$ follows it.

Natural numbers are discrete. Between 4 and 5 there is no other natural number. That changes when we expand the number system to include fractions and decimals.

Whole numbers and zero

The whole numbers are

\{0,1,2,3,4,5,\ldots\}.

Zero is the additive identity: adding it leaves a number unchanged.

a+0=a.

Zero is also a placeholder in positional notation. In $5{,}072$ , the zero tells us that there are no hundreds. Removing it would change the value completely.

Zero is neither positive nor negative. Division by zero is undefined: if $8\div0=x$ , we would need $0\times x=8$ , but $0\times x=0$ for every $x$ . The expression 0 ÷ 0 also has no defined quotient, because every number would satisfy the corresponding multiplication, so there is no unique answer.

Integers and absolute value

The integers include whole numbers and their negative counterparts:

\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.

Negative numbers let us represent quantities relative to a reference point. Temperatures below zero, floors below ground level, losses, changes in elevation, and directional movement can all use negative values.

On a horizontal number line, numbers increase as you move right and decrease as you move left. For example,

-2>-5.

The number $-5$ is farther from zero, but it lies to the left of −2 and is therefore smaller. Absolute value describes distance from zero, separately from order on the line.

The absolute value of a real number is its distance from zero:

|-8|=8, \qquad |5|=5.

Distance is non-negative, so absolute value cannot be negative.

Positions of −5 and −2 on a number line

On this equally spaced number line, −2 is to the right of −5. Therefore −2 is greater than −5. Absolute value measures each number’s distance from zero.

How place value works

Our ordinary written number system is a base-ten positional system. Each position represents a power of ten. Consider

4{,}582.37.

Its value can be expanded as

4{,}582.37 =4(10^3)+5(10^2)+8(10^1)+2(10^0)+3(10^{-1})+7(10^{-2}).

Equivalently,

4{,}582.37=4000+500+80+2+0.3+0.07.

The digit 5 has a value of 500 because it occupies the hundreds place. The digit 3 has a value of three tenths because it is one place to the right of the decimal point.

Place value makes decimals a direct extension of whole-number notation rather than an unrelated topic. Moving one position to the left multiplies the place value by 10. Moving one position to the right divides it by 10.

Reference: place value in 4,582.37

Place value in 4,582.37
Place	Digit	Contribution
Thousands	4	4,000
Hundreds	5	500
Tens	8	80
Ones	2	2
Tenths	3	0.3
Hundredths	7	0.07

Fractions and other number types

Fractions between integers

Integers are not enough when a whole is divided into equal parts. The fraction

\frac{3}{4}

represents three parts when a whole has been partitioned into four equal parts. It also represents a number on the number line and the quotient $3\div4$ .

Fractions fill many of the gaps between integers. In fact, there are infinitely many fractions between any two distinct integers. Between 0 and 1 we can find

\frac12,\quad \frac13,\quad \frac23,\quad \frac{7}{10},\quad \frac{999}{1000},\quad \ldots

A fraction represents a single number, even though its notation has a numerator and denominator. Fractions in Maths develops this idea through equal parts, division and number-line positions.

Rational numbers

A rational number is any number that can be written in the form

\frac{p}{q}, \qquad p,q\in\mathbb{Z},\quad q\ne0.

The set of rational numbers is written $\mathbb{Q}$ . Every integer is rational because an integer $n$ can be written as

n=\frac{n}{1}.

Terminating decimals are rational. For example,

0.75=\frac{75}{100}=\frac34.

Repeating decimals are rational as well. For example,

0.333\ldots=\frac13.

This gives an important connection among representations: fractions and decimals are often different ways to write the same rational number. A comparison such as $0.4$ and $\frac38$ becomes easier when we translate one form into the other:

0.4=\frac{2}{5}=0.400, \qquad \frac38=0.375,

0.4>\frac38.

Irrational numbers

Not every point on the number line is rational. An irrational number cannot be expressed as $p/q$ for integers $p$ and non-zero $q$ .

Familiar examples include

\sqrt2,\quad \pi,\quad e.

Their decimal expansions do not terminate and do not enter a repeating cycle. For example,

\pi=3.1415926535\ldots

The ellipsis means more digits follow. It does not indicate a repeating block.

A square with side length 1 has a diagonal of length $\sqrt2$ . No ratio of integers gives that length exactly, so rational numbers alone cannot describe every geometric measurement.

Real numbers

The real numbers, written $\mathbb{R}$ , include all rational and irrational numbers. Informally, they are all the numbers that can be located on the ordinary continuous number line.

The number sets are nested as follows:

\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}.

If we use “whole numbers” as a separate set, then natural numbers sit inside whole numbers, whole numbers sit inside integers, integers sit inside rational numbers, and rational numbers sit inside real numbers.

A number can belong to several sets at once. For example, $8$ is natural, whole, integer, rational, and real. The number $-7$ is integer, rational, and real. The number $\sqrt2$ is irrational and real.

The number set determines which solutions are possible. For example, the equation

x^2=2

has no rational solution, but it has two real solutions:

x=\pm\sqrt2.

Complex numbers

The real numbers still do not solve every algebraic equation. Consider

x^2=-1.

No real number squared equals $-1$ . Mathematicians therefore define the imaginary unit $i$ by

i^2=-1.

A complex number has the form

a+bi,

where $a$ and $b$ are real numbers. The set of complex numbers is written $\mathbb{C}$ .

Every real number is also complex: a real number $a$ can be written as $a+0i$ , with an imaginary part of zero. Complex numbers also include values with a non-zero imaginary part, which cannot be placed on the real number line.

Number structure and powers

Factors, multiples and primes

For integers $a$ and $b$ , $a$ is a factor of $b$ when an integer $k$ satisfies

b=ak.

For example, 6 is a factor of 42 because

42=6\times7.

A prime number is a whole number greater than 1 with exactly two positive factors: 1 and itself. The number 29 is prime. A composite number is a whole number greater than 1 with more than two positive factors. For example,

30=2\times3\times5.

The number 1 is neither prime nor composite. Prime factorization becomes important in simplifying fractions, finding common denominators, working with ratios, and studying deeper properties of integers.

Powers and scientific notation

Repeated multiplication can be written using exponents. Instead of writing

10\times10\times10\times10,

we write

10^4.

This notation is the foundation of exponents and scientific notation. For example,

45{,}600=4.56\times10^4.

Scientific notation is especially useful when values are extremely large or extremely small. It separates a number into a coefficient and a power of ten, making scale easier to see and calculations easier to organize.

Comparing and representing numbers

Comparing numerical values

Written length is not a reliable guide to numerical size. For example, $0.57$ has more non-zero digits than $0.6$ , but

0.6=0.60>0.57.

Likewise, a larger denominator does not automatically make a fraction larger. The meaning depends on both numerator and denominator. Comparing $\frac58$ and $\frac23$ can be done using a common denominator:

\frac58=\frac{15}{24},\qquad \frac23=\frac{16}{24},

\frac23>\frac58.

Reliable comparison strategies include number lines, benchmark values such as $0$ , $\frac12$ , and $1$ , equivalent fractions, aligned place values, and conversion between representations when appropriate.

Order and distance on a number line

A number line shows order through position and difference through distance. Fractions and decimals have positions between integers, and negative values lie to the left of zero.

For two real numbers $a$ and $b$ , the distance between them is

|a-b|.

If $a=-3$ and $b=5$ , then

|-3-5|=|-8|=8.

The What Works Clearinghouse elementary intervention guide rates the evidence for its number-line recommendation as strong. In the examples here, the line provides a shared scale for comparing integers, fractions and decimals.

Equivalent fractions, decimals and percentages

Check a value by writing it in another form. For three quarters,

\frac34=0.75=75\%.

These forms emphasize different ideas. The fraction emphasizes a ratio or division. The decimal emphasizes base-ten place value. The percentage emphasizes a comparison per hundred.

The fraction, decimal and percentage represent exactly the same value.

The National Research Council’s Adding It Up describes mathematical proficiency as several connected abilities: understanding concepts, using procedures, choosing strategies, reasoning and seeing mathematics as useful. Explaining why equivalent forms agree brings these abilities together.

Try representing one value in words, on a number line and with an equation. CTS’s guide to learning how to learn offers further ways to choose and check a learning strategy.

Common errors when comparing numbers

A few misconceptions recur because notation can be mistaken for meaning. One is that a longer decimal must be larger. Another is that negative numbers with larger digits must be larger, as in incorrectly claiming $-9>-3$ . A third is that fractions are not “real numbers” in the ordinary mathematical sense. Fractions are rational numbers, and rational numbers are part of the real-number system.

Another common error is assuming that every square root is irrational. Some are rational because the number under the radical is a perfect square:

\sqrt{49}=7.

Finally, learners sometimes treat the equals sign as meaning “the answer comes next.” Equality actually states that two expressions have the same value. That broader meaning becomes critical in algebra.

How the number sets fit together

Number sets and their relationships

Complex numbersIncludes real and non-real complex numbers

Real numbersEvery point on the real number line

Rational numbersRatios of integers, such as 3/4

Integers…, −2, −1, 0, 1, 2, …

Whole numbers0, 1, 2, 3, …

Natural numbers1, 2, 3, …

Irrational numbers√2, π, …Real, but not rational

A non-real complex number: 2 + 3i

Natural numbers begin at 1 in this article. Each nested region belongs to the set around it.

Diagram explained

Natural numbers are contained in the whole numbers, which are contained in the integers, which are contained in the rational numbers. Rational and irrational numbers are disjoint and together make up the real numbers. The real numbers are contained in the complex numbers. A complex number with a non-zero imaginary part, such as 2 + 3i, is outside the real-number region.

Reference: number sets at a glance

Number sets at a glance
Number type	Example	Core idea
Natural	$6$	Counting numbers, conventionally starting at 1 in this guide
Whole	$0$	Natural numbers together with zero
Integer	$-12$	Whole numbers and their negatives
Rational	$\frac{7}{8}$	Can be written as a ratio of integers with non-zero denominator
Irrational	$\sqrt2$	Cannot be written as such a ratio
Real	$-4.2$	Any rational or irrational number on the real number line
Complex	$2+3i$	Has real and imaginary parts; includes all real numbers

Each larger set includes the earlier numbers. Rational and irrational numbers form separate parts of the real numbers.

Start with arithmetic operations to see how numbers combine. Then explore fractions, decimals, exponents and percentages as connected ideas.

For a real number, ask: what does it represent, which number sets contain it, where does it lie on the number line, and can it be written in another form?

For help with a particular concept, the CTS guide to personalized mathematics instruction explains how instruction can respond to the point of difficulty.

Practice questions

Try the questions, explain your reasoning, then check the answers.

Classify $-5$ : is it a whole number, an integer, a rational number, or a real number?
Write $0.25$ as a fraction in simplest form.
Which is larger: $-2$ or $-8$ ? Explain using a number line.

Show answers

−5 is an integer, a rational number and a real number. It is not a whole number.
$0.25=25/100=1/4$ . Divide 25 and 100 by their common factor 25.
−2 is greater than −8 because it lies farther to the right on the number line.

Sources and further reading

U.S. Institute of Education Sciences, What Works Clearinghouse. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. The guide supports systematic instruction, precise mathematical language, well-chosen representations, and number lines.
National Research Council. Adding It Up: Helping Children Learn Mathematics. The report describes mathematical proficiency as an integration of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
OpenStax. Prealgebra 2e. Open educational reference for number systems, fractions, decimals, percents, arithmetic, and related prealgebra concepts.