A Non-Terminating Decimal (see previous entry in the quant mountain), will do one of two things:
- Repeat in some endless pattern
- We call these Repeating Decimals
- Go on endlessly with no pattern at all
- We call these Non-Repeating Decimals
Repeating Decimals are always from fractions, where the numerator and denominator are both integers. Here are two examples below:
$$\frac{4}{9} = 0.444444...$$
$$\frac{5}{11} = 0.45454545...$$
The first endlessly repeats the digit $4$ and the second endlessly repeats the digits $45$.
Non-Repeating Decimals are always irrational numbers and never fractions. Here are two examples below:
$$e = 2.718281828459...$$
$$\sqrt{3} = 1.732050807569$$
Is There a Shortcut to Write Repeating Decimals?
Why yes! I'm so glad you asked. Instead of writing $\frac{1}{3}$ as $0.3333333333333...$, we can simply write it with a little line over the repeating part. Take a look at the examples below:
$$\frac{1}{3} = 0.\overline3$$
$$\frac{23}{99} = 0.\overline{23}$$
$$\frac{3}{7} = 0.\overline{428571}$$