Question

Using suitable examples, show that the

1. sum of two irrational numbers may be rational.
2. difference of two irrational numbers may be rational.
3. product of two irrational numbers may be rational.
4. quotient of two irrational numbers may be rational.

Answer 1

i. Sum of two irrational numbers may be rational

Given: `m = 3 + 5sqrt(2), n = 6 - 5sqrt(2)` both m and n are irrational. 

Calculation: `m + n = (3 + 5sqrt(2)) + (6 - 5sqrt(2))`

= `3 + 6 + 5sqrt(2) - 5sqrt(2)`

= 9

The sum m + n = 9 is rational.

So, the sum of two irrational numbers may be rational.

ii. Difference of two irrational numbers may be rational

Given: `m = 3 + 2sqrt(5), n = 6 + 2sqrt(5)` both m and n are irrational.

Calculation: `m - n = (3 + 2sqrt(5)) - (6 + 2sqrt(5))` 

= `3 - 6 + 2sqrt(5) - 2sqrt(5)`

= –3

The difference m – n = –3 is rational.

So, the difference of two irrational numbers may be rational.

iii. Product of two irrational numbers may be rational

Given: `m = 5 + sqrt(2), n = 5 - sqrt(2)` both m and n are irrational.

Calculation: `m xx n = (5 + sqrt(2))(5 - sqrt(2))` 

= `5^2 - (sqrt(2))^2`

= 25 – 2

= 23

The product m × n = 23 is rational.

So, the product of two irrational numbers may be rational.

iv. Quotient of two irrational numbers may be rational

Given: `m = 5sqrt(3), n = 2sqrt(3)` both m and n are irrational.

Calculation: `m/n = (5sqrt(3))/(2sqrt(3)) = 5/2`

The quotient `m/n = 5/2` is rational.

So, the quotient of two irrational numbers may be rational.

Hence, all four properties are shown with examples.