Question

prove that 4-5√2 is an irrational number, given that √2 is an irrational number ​

Answer 1

Let us assume that (4−5
2
​
) is rational .

Subtract given number from 4, considering 4 is rational number. as we know difference of two rational numbers is rational.

4−(4−5
2
​
)is rational

⇒5
2
​
is rational

which is only possible if 5 is rational and root2 is rational.

As we know prouduct of two rational number s rational
But the fact is root2 is an irrational.

Which is contradictory to our assumption.

Hence, 4−5 root2 is irrational . hence proved.

Answer 2

Step-by-step explanation:

let,

$4 - 5 \sqrt{2} \: \: be \: a \: rational \: number \: \\ \\ \: \therefore4 - 5 \sqrt{2} = \frac{a}{b} (where \: a \: and \: b \: are \: co - prime \: and \: b \: ≠0) \\ \\ \implies - 5 \sqrt{2} = \frac{a}{b} - 4 \\ \\ \implies - 5 \sqrt{2} = \frac{a - 4b}{b} \\ \\ = \sqrt{2 } = \frac{a - 4b}{ - 5b} \\ \\ \\ hence \: we \: get \: that \: the \: two \: numbers \: are \: equl. \\ but \: this \: is \: not \: possibe \: because \: an \: irrational \: number \: cannot \: be \: equal \: to \: a \: rational \: number. \\ \\ \therefore4 - 5 \sqrt{2} \: is \: not \: rational. \\ hence \: it \: is \: irrational.$