Question

prove that root 5 minus root 3 is irrational

Answer 1

Answer:

Let 5-√2 be a rational number.

5-√2 =a/b (where a and b are integer b is not equal to 0, a and b are co prime no.)

-√2 =a/b -5

-√2 = a-5b/b

√2= 5b-a/b

5b-a/b is a rational no.

Where as √2 is irrational

And 5-√2 must be irrational so as to satisfy the above statement

Hence our assumption is wrong

5-√2 is an irrational number....

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Answer 2

Let us assume that 5-√3 is rational.

That is, we can find coprimes a and b (b≠0) such that 5-√3 = a/b.

Therefore,

$5 - \frac{a}{b} = \sqrt{3}$

we get,

$\sqrt{3} = 5 - \frac{a}{b}$

$since \: a \: and \: b \: are \: integers \: \: 5 - \frac{a}{b} \: is \: rational \: and \: \sqrt{3} \: is \: also \: rational.$

$but \: this \: contadicts \: the \: fact \: that \: \sqrt{3} is \:irrational.$

$this \: contradiction \: has \: arisen \: of \: our \: assumption \: 5 - \sqrt{3} \: is \: rational.$

$so \: we \: can \: conclude \: that \: 5 - \sqrt{3} \: is \: irrational.$