Question

Prove that √37 is irrational

Answer 1

Answer:

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Answer 2

Answer:

Step-by-step explanation: As usual, assume  $\sqrt{37}$= $a/b$ , with the fraction  $a/b$  fully reduced.

If  $a/b$  is fully reduced, so is  $a$²/$b$² . This can be made clear if you pass to prime factorizations; hopefully it’s clear enough.

Thus,  37= $a$²/$b$²  where the right side is fully reduced.

A fully reduced fraction which equals an integer must have its denominator equal 1. Hopefully this too is clear enough.

This means  $b$$=1$ , so  $\sqrt{37}$$=a$ . But there is no integer which squares to 37, since  62=36  and  72=49 , and the squaring function is monotonically increasing over the positive integers. So we have our contradiction.

Note that 37 can be replaced with any integer that's not a perfect square