Question

If c is rational, c does not = 0, and x is irrational, then cx and c+x are irrational.

Answer 1

sum:
Assume that it is rational. That is, if c = a/b, then

a/b + x = m/n
for some integers m and n.

That means that

x = m/n - a/b
But m/n-a/b = (bm-an)/(bn), clearly a rational number, and x is irrational.

So, if x is irrational, c+x cannot be rational.

Clearly, cx must also be irrational, since multiplication is just a shorthand for addition. You can prove it in the same way as the sum. Assume cx is rational and show that it cannot be.

Answer 2

Thank you!