# Math Help - number theory

1. ## number theory

Give a proof that shows where =0,1,2,3... is irrational.

The idea here is to show that the decimal expansion of x does not terminate or eventually repeat.

2. Originally Posted by bigb
Give a proof that shows where =0,1,2,3... is irrational.

The idea here is to show that the decimal expansion of x does not terminate or eventually repeat.
Assume that this number, call it $x$, is rational, then $x=\frac{a}{b}$. And thus by this it means we can write $\frac{a}{b} = \sum_{k=0}^{\infty} b_k 10^{-k}$ where $\{ b_k\}$ is eventually repeating. Therefore, $x - \frac{a}{b} = 0 = \sum_{k=0}^{\infty} (a_k - b_k)10^{-k}$. But the only way to express zero in a decimal expansion is trivially i.e. $a_k - b_k = 0 \implies a_k = b_k$. But then this implies that $\{a_k\}$ must also be eventually repeating which is a contradiction because it is an increasing sequence. Contradiction.