Question

Show that 5 + 3√2 is an irrational number?​

Answer 1

First of all the definition of an irrational number is,

A number wich is non-terminating. That is, non ending.

So here, √2 is an irrational number.

The value of √2 will be like 1.41421356237309.........

So here the answer will be a never ending number.

That's why, this number 5+3√2 is an irrational number.

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Answer 2

Till now, we have seen many proofs but I will show you a proof in a totally different way ! So, we have to show that 5+3√2 is an irrational number.

First of all, we have to prove three things :

(i) √2 is an irrational number.

(ii) The sum of a rational and an irrational number is irrational

(iii) The product of a non-zero rational number and an irrational number is an irrational number. (i)

The proofs of (i), (ii) and (iii) are shown in the pictures attached with this answer. So, nowz after proving these three statements, we will come to our original proof :

PROOF :-

√2 is an irrational number (Already proved)

So, 3√2 is also an irrational number (Because, The product of a non-zero rational number and an irrational number is an irrational number.)

Therefore, 5+3√2 is a irrational number (Because, The sum of a rational and an irrational number is irrational)

Hence, 5+3√2 is a irrational number [proved]