## Irrational Numbers as Decimals

Introduction

In Rational and Irrational Numbers post, we have discussed that  $\sqrt{2}$ is irrational.  Aside from its radical form,  using a calculator or a computer, we can approximate its value; for example, $\sqrt{2} \approx 1.414213562$. As we can see, irrational numbers can also be represented as decimals. The more powerful the computer, the more accurate we can approximate.

Some Definitions

Decimal numbers with finite number of digits are called terminating decimals, while decimals with infinite number of digits are called non-terminating decimals. The number $0.345$ is a terminating decimal, while $0.999 \cdots$ is a non-terminating decimal. The $\cdots$ symbol means that the $9$s extend indefinitely.

Decimals with repeating digits; that is, the digits that repeat infinitely are called repeating decimals. The numbers $0.999 \cdots$$0.4545 \cdots$,  and $5.144144144 \cdots$ are repeating decimals. » Read more