Question

The formula for the circumference of a circle is c=pi d, where d is the length of the diameter. If d is a rational number, what can you conclude about the circumference?

Answer 1

The circumference is an irrational number.

Explanation:

`\pi` is an irrational number, multiplying
`\pi` by a rational number different from 0 will give as a result an irrational number. This is because if that is not the case, then there would exist rational numbers a and b, with a different from 0, such that
`\pi *a = b`, then
`\pi = (b)/(a)` .

Since the quotient of two rational numbers is a rational number, if b and a were rational, then
`\pi` should also be rational, which is not the case.

We conclude that, since the diameter is positive and rational, that the circumference cant be rational, therefore it is irrational.