Question

prove that√18 is irrational​

Answer 1

Answer:

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Step-by-step explanation:

ANSWER ::

Assume that √18 is rational.

is rational.Then √18 = p/q where p and q are coprime intege

(√18 )^2 = 18 = p²/q²

= p²/q²p² = 18q²

q²therefore p² is an even number since an even number multiplied by any other integer is also an even number. If p² is even then p must also be even since if p were odd, an odd number multiplied by an odd number would also be odd.

q²therefore p² is an even number since an even number multiplied by any other integer is also an even number. If p² is even then p must also be even since if p were odd, an odd number multiplied by an odd number would also be odd.So we can replace p with 2k where k is an integer.

q²therefore p² is an even number since an even number multiplied by any other integer is also an even number. If p² is even then p must also be even since if p were odd, an odd number multiplied by an odd number would also be odd.So we can replace p with 2k where k is an integer.(2k)² = 18q²

q²4k² = 18q²

q²2k² = 9q²

q²Now we see that 9q² is even. For 9q² to be even, q² must be even since 9 is odd and an odd times an even number is even. And by the same argument above, if q² is even then q is even.

is odd and an odd times an even number is even. And by the same argument above, if q² is even then q is even.So both p and q are even which means both are divisible by 2. But that means they are not coprime,

------ this is a contradiction

contradicting our assumption so √18 is not rational.