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.

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*If the decimal expansion of a number $x$ is terminating, with $n$ digits after the decimal point, say, then $10^nx$ is an integer $m$ (the decimal expansion is shifted by $n$ places to the left, hence has nothing after the decimal point) so that $x =\frac{m}{10^n}$ is a fraction of integers, aka. rational number.

*If the decimal expansion of a number $x$ is eventually repeating, with a epriod of length $n\ge1$, say, then $10^nx$ has  a decimal expansion that matches that of $x$ beyond some point, so that in computing $10^nx-x$ everything cancels beyond some point, i.e., $10^nx-x$ is a rational $\frac ab$ per first bullet point. Solving for $x$ we find $x=\frac{a}{(10^n-1)b}$, which is again a fraction of integers.

*If $x$ is the ratio/fraction of integers, then by definition it is rational.


Hence for an irrational number (by definition a number that is not a rational number)  none of the three options above can occur.
