proving transcendental numbers are irrational

Question

I don't understand how every transcendental number is irrational, is there a way to prove that? I know it just means it's a non-algebraic number, but how does that correlate to irrationality?

Answer 1

If $x$ is transcendental but not irrational, then $x = a/b$, with $a,b$ integers, and so $x$ solves the rational equation $b t - a = 0$, but then $x$ is algebraic and hence not transcendental.

Answer 2

Summing up the comments...,there are two types of numbers in $\Bbb{R}$ in the sense , one type is algebraic and the other one is transcendental.

In particular every rational $x=\frac{p}{q}$ is algebraic, since $x$ satisfies $qx-p$, which is a non zero integer polynomial. Therefore if any $x$ is not algebraic ,it cannot be a rational!

So every transcendental number is irrational not every irrational is transcendental!