prove sqrt 7 is irrational

Massatfy

Massatfy

Answered question

2022-08-03

prove sqrt 7 is irrational

Answer & Explanation

agergadas3b

agergadas3b

Beginner2022-08-04Added 16 answers

Let x = ?7 and assume it is rational. Then it can be written x = p/q where p and q are integers. Further assume that the fraction p/q is in simplest form. That is, GCF(p,q) = 1. Now:
x^2 = (?7)^2
x^2 = 7
(a/b)^2 = 7
a^2 = 7b^2
So a must be a multiple of 7. Let a = 7k for some integer k. Then:
(7k)^2 = 7b^2
49k^2 = 7b^2
7k^2 = b^2
Now, b must also be a multiple of 7. But this contradicts the fact that a/b is in simplest form. So our assumptions that ?7 is rational must be false.
QED
The same line of reasoning can be applied to the square root of 2, 3, 5, or any other prime number.

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