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

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.

Thank you!

To prove that cx and c+x are irrational, we need to show that they cannot be expressed as a ratio of two integers.

Let's start with cx.
Assume cx is rational, which means it can be written as the ratio of two integers, p and q, where q is not equal to 0. Therefore, we can write cx as p/q.

Now, we know c is rational and not equal to 0, so we can write it as r/s, where r and s are integers and s is not equal to 0.

Substituting this value of c into the equation for cx, we have:
cx = (r/s)x = rx/s.

Since x is irrational, it cannot be expressed as the ratio of two integers, so it cannot be equal to rx/s. Thus, cx cannot be rational, which means it must be irrational.

Now let's consider c+x.
Assume c+x is rational, so it can be expressed as the ratio of two integers, p and q, where q is not equal to 0. Therefore, we can write c+x as p/q.

Substituting the value of c as r/s, we have:
c+x = (r/s)+x = (r+sx)/s.

Since x is irrational, it cannot be expressed as the ratio of two integers, so it cannot be equal to (r+sx)/s. Thus, c+x cannot be rational and must be irrational.

Therefore, we have shown that if c is rational (not equal to 0) and x is irrational, cx and c+x are irrational.