Question

Prove that the following are irrationals:
(I)1/√2
(ii) 7/√5
(iii) 6 +√ 2​

Answer 1

Answer:

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.

Answer 2

Step-by-step explanation:

)

2

1

2

1

×

2

2

=

2

2

Let a=(

2

1

)

2

be a rational number.

⇒2a=

2

2a is a rational number since product of two rational number is a rational number .

Which will imply that

2

is a rational number.But it is a contradiction since

2

is an irrational number

Therefore 2a is irrational or a is irrational.

Therefore

2

1

is irrational .Hence proved.

(2) 7

5

Let a=7

5

be a rational number

⇒

7

a

=

5

Now ,

7

a

is a rational number since quotient of two rational number is a rational number.

The above will imply that

5

is a rational number. But

5

is an irrational number.

This contradicts our assumption.Therefore we can conclude that 7

5

is an irrational number and hence the result.

(3) 6+

2

If possible let a=6+

2

be a rational number.

Squaring both side

a

2

=(6+

2

)

2

a

2

=38+12

2

2

=

12

a

2

−38

--(1)

Since a is a rational number the expression

12

a

2

−38

is also rational number.

⇒

2

is rational number.

This is a contradiction .Hence 6+

2

is irrational

Hence proved.

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