Question

Prove that $4- 5{\sqrt{2}}$ is an irrational number

Answer 1

SOLUTION :  

Let us assume , to the contrary ,that 4 - 5√2 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

4 - 5√2 = a/b

4 - a/b = 5√2

[(4b - a)/b]/5 = √2

(4b - a)/5b  = √2

since, a & b is an integer so, (4b - a)/5b

 is a rational number.  

∴ √2 is rational  

But this contradicts the fact that √2 is an irrational number .

Hence, 4 - 5√2 is an irrational .

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Answer 2

Answer:

Step-by-step explanation:

Let a/b = 4 - 5√2 be rational  number.

where a and b are integers .

⇒ - 5√2 = a/b - 4

⇒ 5√2 = 4 - a/b

⇒ 5√2 = 4b/b - a/b

⇒ √2 = (4b - a)/5b

Therefore, √2 will be rational.

But we know that √2 is irrational, there is a contradiction

Hence, 4 - 5√2 is an irrational number.

Hence, Proved.