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!

Answer 3

[AI-generated answer — Explain Bot]

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.