Explain the basic mathematical operations on irrational numbers.

Addition of the Irrational Numbers :

Irrational Number $+$ Irrational Number $=$ May or may not be an Irrational Number

Example: $√2 = 1.414… , √3 = 1.732…$

$√2 + √3$

$= 1.414… + 1.732…$

$= 3.146……$

3.146…… is non-repeating and non terminating. Therefore, it is an irrational number. 

$( 1 – √2 )  + √2$

$= 1 – √2  + √2 = 1$

1 is a rational number.

The addition of two irrational numbers may or may not be an irrational number.

Subtraction of the Irrational Numbers

Irrational Number $–$ Irrational Number $=$ May or may not be an Irrational Number

$√2 = 1.414… , √3 = 1.732… ,$ 

$√3 –  √2$

$=  1.732… –  1.414… = 0.318…$

0.318… is non-repeating and non terminating. Therefore, 0.318.... is an irrational number.

$( 1 + √2 )  – √2$

$ = 1 + √2  – √2 = 1$

1 is a rational number.

The subtraction of two irrational numbers may or may not be an irrational number.

Multiplication of the Irrational Numbers

Irrational Number $\times$ Irrational Number $=$ May or may not be an Irrational Number

$√2 = 1.414… , √3 = 1.732…$ 

$√2 \times √3$

$= 1.414… \times 1.732… = 2.449….$

2.449..... is non-repeating and non terminating. Therefore, 2.449..... is an irrational number.

$( 2 √3 ) \times √3$

$= 2 \times √3 \times √3 = 2 \times 3 = 6$

6 is a rational number.

The multiplication of two irrational numbers may or may not be an irrational number.

Division of the Irrational Numbers

$\frac{Irrational Number}{Irrational Number} = May or may not be an Irrational Number$

$√2 = 1.414… , √3 = 1.732… , √5 = 2.236…$

$\frac{\sqrt{2}}{\sqrt{3}}=\frac{1.414..}{1.732..}=0.816..$

When we divide two irrational numbers we may or may not get an irrational number.