Real Numbers

Real Numbers: Definition, Types, Set, Chart & Examples

Real Numbers: The number system, often known as the numeral system, is a method of expressing numbers. There are two categories of the number system, i.e., real and imaginary numbers. Real numbers are the sum of rational and irrational numbers. All arithmetic operations may be done on these numbers in general. Furthermore, they can also be represented on a number line.

In comparison, imaginary numbers are unreal numbers that cannot be stated on a number line and are typically employed to represent complex numbers. Examples of real numbers are 23, -12, 6.99, 5/2, and so on. In this article, students will learn about the real numbers definition, their properties, chart, sets, and solved examples. Read on to find out more.

What are Real Numbers?

A combination of rational numbers and irrational numbers are known as real numbers. Real numbers can be both positive and negative, which is denoted as ‘R′.$‘R^{\prime }.$ natural numbers, fractions, and decimals all come under this category.

Types of Real Numbers

Different types along with real numbers examples are as given below:

1. Natural Numbers (N)$\left(N\right)$: Natural numbers are the counting numbers from 1$1$. For example, they are 1,2,3,4,5,…$1,2,3,4,5,\dots$ Natural numbers are a subset of real numbers. It is denoted by N$N$.
2. Whole Numbers (W)$\left(W\right)$ : Numbers 0,1,2,3…$0,1,2,3\dots$ are called whole numbers. These are natural numbers, including 0$0$. Whole numbers are also a subset of real numbers. It is denoted by W$W$.
3. Integers (Z)$\left(Z\right)$: The numbers …, –3,–2,–1,0,1,2,3,4,5,$–3,–2,–1,0,1,2,3,4,5,$ … are called integers. Integers are also a subset of real numbers. It is denoted by Z$Z$.
4. Rational Numbers (Q)$\left(Q\right)$ : A rational number is defined as a number that can be expressed in the form of pq$\frac{p}{q}$, where p$p$ and q$q$ are co-prime integers and q≠0.$q\ne 0.$. Rational numbers are also a subset of real numbers. It is denoted by Q$Q$. Examples: –23,0,5,310$–\frac{2}{3},0,5,\frac{3}{10}$, …. etc.
5. Irrational Numbers (P)$\left(P\right)$: An irrational number is defined as the number that cannot be expressed in the form of pq$\frac{p}{q}$, where p$p$ and q$q$ are co-prime integers and q≠0.$q\ne 0.$ Irrational numbers are also a subset of real numbers. It is denoted by P$P$. Examples: 2–√,5–√,π,e,$\sqrt{2},\sqrt{5},\pi ,e,$, …. etc.

What Is the Set of Real Numbers?

Now that you have understood the real numbers definition and examples, let’s see what constitute real numbers. The set of real numbers include natural numbers, whole numbers, integers, irrational numbers, and rational numbers, as defined and explained above.

What are the Properties of Real Numbers?

There are four main properties of real numbers. They are (i)$\left(\mathrm{i}\right)$ Commutative Property for addition and multiplication, (ii)$\left(\mathrm{i}\mathrm{i}\right)$ Associative Property for addition and multiplication, (iii)$\left(\mathrm{i}\mathrm{i}\mathrm{i}\right)$ Distributive Property of multiplication over addition and (iv)$\left(\mathrm{i}\mathrm{v}\right)$ Identity Property.

Commutative Property for Addition and Multiplication

The word commutative comes from “commute” or “move around”. Commutative properties mean that if the numbers we operate are changed or swapped from their position, the answer remains the same. This property is applicable only in the case of addition and multiplication but not for subtraction or division. If m$m$ and n$n$ are two real numbers, then,

For addition: m+n=n+m$m+n=n+m$, examples: 6+3=9=3+6$6+3=9=3+6$ or 2+8=10=8+2$2+8=10=8+2$

For multiplication: m×n=n×m$m\times n=n\times m$, example: 2×4=8=4×2$2\times 4=8=4\times 2$ or 5×10=50=10×5$5\times 10=50=10\times 5$

Associative Property for Addition and Multiplication

Associative Property states that you can add or multiply the numbers regardless of how they are grouped. If m,n$m,n$ and r$r$ are three real numbers, then,

For Addition: m+(n+r)=(m+n)+r$m+(n+r)=(m+n)+r$, example: 3+(4+5)=(3+4)+5$3+(4+5)=(3+4)+5$

For Multiplication: (m×n)×r=m×(n×r)$(m\times n)\times r=m\times (n\times r)$, example: (6×2)×3=6×(2×3)