Question

Show that 1/√2 is rational if√2 is given an irrational number

Answer 1

Answer:

(i)

2

1

Let us assume

2

1

is rational.

So we can write this number as

2

1

=

b

a

---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) multiply by

2

both sides, we get

1=

b

a

2

Now, divide by b, we get

b=a

2

or

a

b

=

2

Here, a and b are integers so,

a

b

is a rational number,

so

2

should be a rational number.

But

2

is a irrational number, so it is contradictory.

Therefore,

2

1

is irrational number.

(ii) 7

5

Let us assume 7

5

is rational.

So, we can write this number as

7

5

=

b

a

---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) divide by 7 both sides, we get

5

=

7b

a

Here, a and b are integers, so

7b

a

is a rational

number, so

5

should be a rational number.

But

5

is a irrational number, so it is contradictory.

Therefore, 7

5

is irrational number.

(iii) 6+

2

Let us assume 6+

2

is rational.

So we can write this number as

6+

2

=

b

a

---- (1)

Here, a and b are two co-prime number and b is not equal to zero.

Simplify the equation (1) subtract 6 on both sides, we get

2

=

b

a

−6

2

=

b

a−6b

Here, a and b are integers so,

b

a−6b

is a rational

number, so

2

should be a rational number.

But

2

is a irrational number, so it is contradictory.

Therefore, 6+

2

is irrational number.