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Question

Prove that 5 is irrational number


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Solution

Given: the number 5

We need to prove that 5 is irrational

Let us assume that 5 is a rational number.

So it can be expressed in the form p/q where p,q are co-prime integers and q0
5=pq

On squaring both the sides we get,
5=p²/q²5q²=p²(i)p²/5=q²So5dividesp2,pisamultipleof5p=5mp²=25m²-(ii)Fromequations(i)and(ii),weget,5q²=25m²q²=5m²q²isamultipleof5qisamultipleof5
Thus,p,q have a common factor5. This contradicts our assumption that they are co-primes. Therefore,pq is not a rational number

Hence, 5 is an irrational number.


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