Rational and Irrational number

Hi, Welcome back! In this post let's learn about rational and irrational numbers. Before that, it's necessary to know about Integers. If you know about it then let's begin,

Rational Number

A number that can be written in the form of a fraction like `\frac{a}{b}` where a (numerator) and b (denominator) are integers and b (denominator) must not be zero are called Rational Numbers. Rational numbers can be denoted by Q.

For example : 5, 0, `\frac{2}{3}`, 3.5,  1.333......., `\sqrt{4}` etc are rational numbers. Let's see, how?

• 5 = `\frac{5}{1}` is in the form of fraction. Where both numerator (5) and the denominator (1) are integers. And, denominator i.e. 1 ≠ 0. So, 5 is a rational number.

• 0 = `\frac{0}{1}` is in the form of fraction. Where both numerator (0) and the denominator (1) are integers. And, denominator i.e. 1 ≠ 0. So, 0 is a rational number.

In the same way integers, the whole number, counting numbers and natural numbers are rational numbers.

• `\frac{2}{3}` is a rational number.

`\frac{2}{3}` is in the form of a fraction where 2(numerator) and 3(denominator) are integers and 3(denominator) ≠ 0. 

• 2`\frac{2}{3}` = `frac\{8}{3}` is also a rational number.

As, its numerator and denominator are integers and denominator ≠ 0.

• 3.5 =  `\frac{35}{10}` = `\frac{7}{2}`

3.5 is a terminating decimal, and all the terminating decimals can be expressed in the form of fractions. So, it is a rational number.

• 1.333..... = 1.33`\overline{3}` = `\frac{4}{3}` 

1.333..... is a Non-Terminating Recurring decimal. And, it can be expressed as a fraction. So, it is also a rational number.

• `\sqrt{64}` = `\sqrt{8 . 8}` = 8

Here, `\sqrt{64}` = 8 which is an integer. So, `\sqrt{64}` is a rational number.

• `\sqrt[3]{8}`= `\sqrt[3]{2 . 2 . 2}` = 2
  
  

Here, `\sqrt[3]{8}` = 2, which is an integer. So, it is a rational number.

Irrational Number

The numbers which are not rational are called irrational numbers. It cannot be expressed in the form of a fraction like `\frac{a}{b}`. In short, when we decimalized the irrational numbers, we get non-terminating non-recurring decimals. Irrational numbers can be denoted by Q'. 

For example:

`\sqrt{8}` = `\sqrt{2 . 2 . 2} = 2 `\sqrt{2}` = 2.8284271247461900976033774484194