What are real numbers?

Every day we group or sort things; putting things in their proper place.

What are Real Numbers?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

I’m sure your kitchen has a designated place for silverware, cups, bowls, plates, etc., and in your bedroom I’m confident there is a closet and/or dresser were you sort your clothes and accessories.

  • Well guess what, we do the same thing in mathematics when it comes to numbers: Number Sets!
  • A set is a specifically defined collection of distinct elements, as Think Zone nicely states.
  • For example, the Alphabet is a set of letters, and your class contains a specific set of students.
  • Number Sets are sets of numbers that have the same characteristics, and this lesson is going to show you how to sort or categorize numbers into their appropriate sets.

Mathematical Number Sets

  • Natural Numbers are nothing more than your counting numbers: 1, 2, 3, …
  • Whole Numbers are your counting numbers but it also includes zero: 0, 1, 2, 3, …
  • Integers are the Natural Numbers and their opposites, or negatives: …-3, -2, -1, 0, 1, 2, 3…
  • Rational Numbers are Integers that can be expressed as terminating or repeating decimal (i.e, simple fraction).
  • Irrational Numbers are numbers that cannot be written as a simple fraction because their decimals never terminate or repeat.
  • Real Numbers are all the numbers on the Number Line and include all the Rational and Irrational Numbers
  • Complex Numbers are the set of Real Numbers and Imaginary Numbers.

Number Lines

Once we are able to classify numbers into their appropriate Number Sets, it is important to be able to place them on the Number Line.

What are Real Numbers?

Understanding the Real Number Line

  1. Why?
  2. Being able to visually see where a number is in relation to other numbers that are similar or different is an important tool in estimating and also when finding opposites or comparing numbers.
  3. Moreover, as Math is Fun so accurately points out,
    Number Lines can help us in adding or subtracting numbers and also in finding Absolute Value.
  4. Together we will walk through countless examples of how to classify numbers into Number Sets, place numbers on a Number Line, find Opposites and Absolute Value of numbers and how to Compare Numbers.

Real Numbers Explained – Video

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What are Real Numbers?

Real Numbers- Definition, Properties, Set of Real Numerals

Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also.

At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. The concepts related to real numerals are explained here in detail along with examples and practice questions.

The key concept in the number system is included in this article.

23 -12 6.99 5/2 π(3.14)

Table of contents:

What are Real Numbers?

Real Numbers Definition

Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals, and fractions come under this category. See the figure, given below, which shows the classification of real numerals.

Read More:

Set of Real Numbers

The set of real numbers consist of different categories, such as natural and whole numbers, rational and irrational numbers and integers. In the table given here, all these numbers are defined with examples.

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Natural Numbers Contain all counting numbers which start from 1.
N = {1,2,3,4,……}
All numbers such as 1, 2, 3, 4,5,6,…..…
Whole Numbers Collection of zero and natural number.
W = {0,1,2,3,…..}
All numbers including 0 such as 0, 1, 2, 3, 4,5,6,…..…
Integers The collective result of whole numbers and negative of all natural numbers. Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)
Rational Numbers Numbers that can be written in the form of p/q, where q≠0. Examples of rational numbers are ½, 5/4 and 12/6 etc.
Irrational Numbers All the numbers which are not rational and cannot be written in the form of p/q.

Algebra/Real Numbers

Real numbers consist of zero (0), the positive and negative integers (-3, -1, 2, 4), and all the fractional and decimal values in between (0.4, 3.1415927, 1/2). Real numbers are divided into rational and irrational numbers.

Rational numbers are numbers that can be expressed as a ratio (that is, a division) of two integers ( 2 3 {displaystyle {2 over 3}}
, 0.6 {displaystyle 0.6}
, 3 {displaystyle 3}
, − 4.7 {displaystyle -4.7}
, 0.11111… {displaystyle 0.11111…}
). If a number has a terminating decimal, or a decimal that ends ( 3.6 {displaystyle 3.6}
, 5.263 {displaystyle 5.263}
) or repeats ( 1.33333…. {displaystyle 1.33333….}
), it is rational.

Irrational numbers have decimal parts that do not terminate or repeat ( 2.71828… {displaystyle 2.71828…}
, 3.14159… {displaystyle 3.14159…}
) and cannot be expressed as a fractional equivalent. For example, the number 2 = 1.41421356… {displaystyle {sqrt {2}}=1.41421356…}
does not have an equivalent ratio or division of two numbers. There are several other different “sets” of rational numbers.

Natural numbers, also known as “counting numbers”, are the first numbers you learn. The natural numbers include all of the positive whole numbers (1, 24, 6, 2, 357). Note that zero is not included, and fractions or decimals are not included.

Whole numbers are the natural numbers, plus zero.

Integers are all positive and negative numbers without a decimal part (3, -1, 15, -42).

Properties Of Real Numbers[edit]

We begin this section with a review of the fundamental properties of arithmetic. It may seem unusual to give so much emphasis to the few properties listed below, but there is a good reason.

Roughly speaking, all of algebra follows from the 5 properties listed in the table below. In the table below, a, b and c can be any number unless stated otherwise.

So let's take a look:


a + b = b + a {displaystyle a+b=b+a}

This doesn't work: a − b ≠ b − a {displaystyle a-b
eq b-a}

This does: a + ( − b ) = ( − b ) + a {displaystyle a+(-b)=(-b)+a}

a ∗ b = b ∗ a {displaystyle a*b=b*a}

This doesn't work: a / b ≠ b / a {displaystyle a/b
eq b/a}

This does: a ∗ 1 / b = 1 / b ∗ a {displaystyle a*1/b=1/b*a}


( a + b ) + c = a + ( b + c ) {displaystyle (a+b)+c=a+(b+c)}

This doesn't work: ( a − b ) − c ≠ a − ( b − c ) {displaystyle (a-b)-c
eq a-(b-c)}

This does: ( a − b ) − c = a − ( b + c ) = a + ( − b − c ) {displaystyle (a-b)-c=a-(b+c)=a+(-b-c)}

( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) {displaystyle (a*b)*c=a*(b*c)}

This doesn't work: ( a / b ) / c ≠ a / ( b / c ) {displaystyle (a/b)/c
eq a/(b/c)}

This does: ( a / b ) / c = a ∗ 1 / b ∗ 1 / c = a / ( b ∗ c ) {displaystyle (a/b)/c=a*1/b*1/c=a/(b*c)}


a + 0 = a {displaystyle a+0=a}

a − 0 = a {displaystyle a-0=a}

a ∗ 1 = a {displaystyle a*1=a}

a / 1 = a {displaystyle a/1=a}


a + − a = 0 {displaystyle a+-a=0}

a − a = 0 {displaystyle a-a=0}

a ∗ ( 1 / a ) = 1 {displaystyle a*(1/a)=1}
  as long as a ≠ 0.
a / a = 1 {displaystyle a/a=1}
  as long as a ≠ 0.


a ∗ ( b + c ) = a ∗ b + a ∗ c {displaystyle a*(b+c)=a*b+a*c}

a ∗ ( b − c ) = a ∗ b − a ∗ c {displaystyle a*(b-c)=a*b-a*c}

( a + b ) / c = a / c + b / c {displaystyle (a+b)/c=a/c+b/c}

But wait: a / ( b + c ) ≠ a / b + a / c {displaystyle a/(b+c)
eq a/b+a/c}

But what does all this mean?
The commutative property is that you can exchange two numbers and still get the same answer.
The associative property is that you can change the grouping (i.e.

, change the position of the parenthesis) and still get the same answer.
The identity property is that there is a certain number that when operated with a number doesn't change it.

The inverse property is something that results to the identity number.
The distributive property

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Real Number

A real number is any positive or negative number. This includes all integers and all rational and irrational numbers.

Rational numbers may be expressed as a fraction (such as 7/8) and irrational numbers may be expressed by an infinite decimal representation (3.1415926535…).

Real numbers that include decimal points are also called floating point numbers, since the decimal “floats” between the digits.

Real numbers are relevant to computing because computer calculations involve both integer and floating point calculations.

Since integer calculations are generally more simple than floating point calculations, a computer's processor may use a different type of logic for performing integer operations than it does for floating point operations.

The floating point operations may be performed by a separate part of the CPU called the floating point unit, or FPU.

While computers can process all types of real numbers, irrational numbers (those with infinite decimal points) are generally estimated. For example, a program may limit all real numbers to a fixed number of decimal places. This helps save extra processing time, which would be required to calculate numbers with greater, but unnecessary accuracy.

Updated: May 14, 2010

Cite this definition:


This page contains a technical definition of Real Number. It explains in computing terminology what Real Number means and is one of many technical terms in the TechTerms dictionary.

All definitions on the TechTerms website are written to be technically accurate but also easy to understand. If you find this Real Number definition to be helpful, you can reference it using the citation links above. If you think a term should be updated or added to the TechTerms dictionary, please email TechTerms!

What are Real Numbers?

Updated April 24, 2018

By Bert Markgraf

The real numbers are all the numbers on a number line extending from negative infinity through zero to positive infinity. This construction of the set of real numbers is not arbitrary but rather the result of an evolution from the natural numbers used for counting.

The system of natural numbers has several inconsistencies, and as calculations became more complex, the number system expanded to address its limitations.

With real numbers, calculations give consistent results, and there are few exceptions or limitations such as were present with the more primitive versions of the number system.

The set of real numbers consists of all the numbers on a number line. This includes natural numbers, whole numbers, integers, rational numbers and irrational numbers. It does not include imaginary numbers or complex numbers.

Closure is the property of a set of numbers that means if allowed calculations are performed on numbers that are members of the set, the answers will also be numbers that are members of the set. The set is said to be closed.

Natural numbers are the counting numbers, 1, 2, 3…, and the set of natural numbers is not closed. As natural numbers were used in commerce, two problems immediately arose.

While the natural numbers counted real objects, for example cows, if a farmer had five cows and sold five cows, there was no natural number for the result. Early number systems very quickly developed a term for zero to address this problem.

The result was the system of whole numbers, which is the natural numbers plus zero.

The second problem was also associated with subtraction. As long as numbers counted real objects such as cows, the farmer could not sell more cows than he had.

But when numbers became abstract, subtracting larger numbers from smaller ones gave answers outside the system of whole numbers. As a result, integers, which are the whole numbers plus negative natural numbers were introduced.

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The number system now included a complete number line but only with integers.

Calculations in a closed number system should give answers from within the number system for operations such as addition and multiplication but also for their inverse operations, subtraction and division. The system of integers is closed for addition, subtraction and multiplication but not for division. If an integer is divided by another integer, the result is not always an integer.

Dividing a small integer by a larger one gives a fraction. Such fractions were added to the number system as rational numbers. Rational numbers are defined as any number that can be expressed as a ratio of two integers. Any arbitrary decimal number can be expressed as a rational number. For example 2.864 is 2864/1000 and 0.89632 is 89632/100,000. The number line now seemed to be complete.

There are numbers on the number line that cannot be expressed as a fraction of integers. One is the ratio of the sides of a right-angled triangle to the hypotenuse. If two of the sides of a right-angled triangle are 1 and 1, the hypotenuse is the square root of 2.

The square root of two is an infinite decimal that does not repeat. Such numbers are called irrational, and they include all real numbers that are not rational.

With this definition, the number line of all real numbers is complete because any other real number that is not rational is included in the definition of irrational.

Although the real number line is said to extend from negative to positive infinity, infinity itself is not a real number but rather a concept of the number system that defines it as being a quantity larger than any number.

Mathematically infinity is the answer to 1/x as x reaches zero, but division by zero is not defined. If infinity were a number, it would lead to contradictions because infinity does not follow the laws of arithmetic.

For example, infinity plus 1 is still infinity.

The set of real numbers is closed for addition, subtraction, multiplication and division except for division by zero, which is not defined. The set is not closed for at least one other operation.

The rules of multiplication in the set of real numbers specify that the multiplication of a negative and a positive number gives a negative number while the multiplication of positive or negative numbers gives positive answers.

This means that the special case of multiplying a number by itself yields a positive number for both positive and negative numbers. The inverse of this special case is the square root of a positive number, giving both a positive and a negative answer.

For the square root of a negative number, there is no answer in the set of real numbers.

The concept of the set of imaginary numbers addresses the issue of negative square roots in the real numbers. The square root of minus 1 is defined as i and all imaginary numbers are multiples of i.

To complete number theory, the set of complex numbers is defined as including all real and all imaginary numbers. Real numbers can continue to be visualized on a horizontal number line while imaginary numbers are a vertical number line, with the two intersecting at zero.

Complex numbers are points in the plane of the two number lines, each with a real and an imaginary component.

About the Author

Bert Markgraf is a freelance writer with a strong science and engineering background. He has written for scientific publications such as the HVDC Newsletter and the Energy and Automation Journal.

Online he has written extensively on science-related topics in math, physics, chemistry and biology and has been published on sites such as Digital Landing and Reference.

com He holds a Bachelor of Science degree from McGill University.

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