What are Real Numbers and their Properties?

“Real numbers”, is a separate chapter of Mathematics that deals with both rational and irrational numbers. So, what exactly are real numbers? Real numbers are nothing but a quantity that includes all numbers right from integers to fractions, numbers containing roots, oi and so on. Real numbers are said to be numbers that can have an unending decimal expansion.

Real numbers are used in many measuring applications such as to measure size also time and are different from counting numbers which start from 1,2,3,4 without any decimal expansion. The real numbers include all counting numbers, i.e., natural numbers, rational for portraying fractions like ½ and irrational for representing square roots.

The real numbers are different from complex numbers or imaginary numbers and they consist of integers both negative and positive, rational numbers or fractions and irrational numbers. The rational numbers are different from irrational numbers in the sense that rational numbers have decimal expansions which repeat themselves, example like 2/7=0.285714285714….and so on. The irrational numbers on the other hand include decimal expansions that do not repeat themselves like for example 0.1115353553241.

In Mathematics most of the numbers are pure i.e., if one says 0.5 it exactly means ½ or half. However, it is not the same when it comes to the real world the ½ or the 0.5 might not be the exact value for example try to cut an apple into half you might not get the exact half. Therefore, in mathematics the values are pure and real, hence they are called real numbers.

In Algebra, the irrational numbers are very familiar where they are seen as solutions or roots of a given algebraic equation and are also sometimes called as algebraic irrational  numbers.

For example: Find the solution of a given algebraic equation g(y)=y²-2=0

Solution: In the equation,

• y²-2=0
• y=√2

This is known as the irrational number or the algebraic irrational number. The value Π however, is not a solution of any equation in Algebra and such irrational numbers are known as transcendental irrational values.

So, any number which comes to mind can be considered real as long as it does not include complex numbers or imaginary. The set of real numbers is quite big and is often denoted by R and is the union rational numbers’ set i.e., Q and the irrational numbers’ set represented by ¯QQ¯. Hence, one can write the set R is equal to Q U ¯QQ¯.

The question then arises which numbers are considered not real?

The simple answer is the numbers which does not fall in the category of rational or irrational  are often considered not real. For example, 2+i, -i and √-1. These numbers fall in the set of complex numbers of Mathematics.

Symbols that have been given to Real numbers:

By definition, real numbers include the natural numbers, all the rational and irrational numbers, integers both positive and negative, whole numbers here are some symbols that represent them. The set of all real numbers is usually represented by R. Here in this article the symbols of the individuals are given:

• R :  Set of real numbers
• W: Whole numbers
• N : Natural numbers
• Z: Integers
• Q : Rational numbers
• ¯QQ¯  : Irrational numbers

Real Numbers: Properties

The real numbers just like any number in mathematics say natural, numbers or integers also satisfies different properties like associative, distributive, commutative and closure properties.

Some of which are highlighted below:

Associative Property: The associative property states that the value is same even when grouping of three real numbers is change for calculating the sum or product of these real numbers. For example,

• a + (b + c) = (a + b) + c
• a × (b × c) = (a × b) × c

Closure Property:The closure property says that the value is always a real number from the sum or the product of any two real numbers.

Commutative Property:The value of the sum or product of two real numbers after interchanging the order remain the same. Example:

• a + b = b + a
• a × b = b × a

Distributive property:The property gives the following:

• a × (b + c) = (a × b) + (a × c)
• a × (b – c) = (a × b) – (a × c)