Irrational numbers are non-terminating/non-repeating decimals Why is it true that all irrational numbers are non-terminating/non-repeating decimals?
By definition, an irrational number is one that can't be expressed as a ratio of integers.

The definition: a number is irrational if and only if it's not rational, i.e. it can't be expressed as a ratio of two integers. This answers one part of your question.
The other part: I'll prove the contrapositive. If $x$ has a repeating decimal expansion (this includes terminating decimal expansions), then $x$ is rational.  
Proof: If $x$ has a repeating decimal expansion, then it can always be written in the following form: 
Let $c,b$ be non-negative integers and $a_i\in\{0,1,2,\ldots,9\}$ and $t$ is the number of digits of $b$.
$$x=\overline{c.ba_1a_2\ldots a_ka_1a_2\ldots a_ka_1a_2\ldots}$$
$$10^tx=\overline{cb.a_1a_2\ldots a_ka_2a_2\ldots a_ka_1a_2\ldots}$$
$$10^{kt}x=\overline{cba_1a_2\ldots a_k.a_1a_2\ldots a_ka_1a_2\ldots}$$
$$10^{kt}x-10^{t}x=\overline{cba_1a_2\ldots a_k}-\overline{cb}$$
$$x=\frac{\overline{cba_1a_2\ldots a_k}-\overline{cb}}{10^{kt}-10^t}$$
