Is 7 pie a rational number why or why not?

Answer 1

Assuming that you mean pi, and not pie, it is not a rational number.

The set of rational numbers is a field and this means that for every non-zero rational number, there exists a multiplicative inverse in the setand also, due to closure, the product of any two rational numbers is a rational number.

Now suppose 7*pi were rational.

7 is rational and so there is its multiplicative inverse, which is (1/7).

(1/7) is also rational so (1/7)*(7*pi) is rational

But by the associative property, this is (1/7*7)*pi = 1*pi = pi.

But it has been proven that pi is irrational. Therefore the supposition must be wrong ie 7*pi is not rational.