Rational Number

Rational numbers are a cornerstone of mathematics with broad-reaching applications in our daily lives. Understanding rational numbers is essential for performing basic arithmetic operations. This chapter explores what they are, how they differ from other number types and their relevance in everyday life.

Natural Numbers

Within the number system, natural numbers are all counting numbers that begin with 1 and extend to infinite. Some examples of natural numbers are:

Whole Numbers

Whole numbers are natural numbers including zero. Some examples of whole numbers are:

Integers

Integers are an extension of the concept of natural and whole numbers. They comprise positive counting numbers, zero and the negative counterparts of counting numbers. Some examples of integers are:

Rational Numbers

Rational numbers are denoted by ‘r’ that can be expressed in the form of p/q where both ‘p’ and ‘q’ are integers and ‘q’ is not equal to zero.
A fraction 2/5 is a rational number where ‘p’ is 2, ‘q’ is 5 and q ≠ 0.

Some examples of rational numbers are shown on the number line:

Irrational Numbers

Irrational numbers are any numbers that don’t represent in the form of p/q where ‘p’ and ‘q’ are integers with 'q' ≠ 0.

Some examples of irrational numbers are:

1. Root of the numbers like 2 , 3 , 5 , 32 , 33 ,  etc.
2. Mathematical constants like ‘π’.
3. Non-repeating decimals like 0.01324563……….,1.792894…………….., etc.

Additive Inverse of rational number

Additive inverse of a numerical value refers to the number when combined with the original number which results in a sum of zero.
This concept is alternatively referred to as the opposite number, sign reversal or negation. The additive inverse of ‘X’ is denoted as ‘−X’. The sum of a number and its additive inverse is:

X + (−X) = 0

Example: −2/7 stands as the additive inverse of 2/7.

Hence, 2/7 + (−2/7) = 2/7 −2/7 = 0

Reciprocal of Rational Numbers

Reciprocal of a rational number is a multiplicative inverse. It is denoted as 1/x or x?¹ which is a number that when multiplied by another number ‘x’ results in the multiplicative identity which is 1.

x × ¹⁄_x  = 1

Example: Reciprocal of 2/7 is 7/2.

Hence, ²⁄₇ × ⁷⁄₂  = 1