A **decimal expansion** is the most popular way to represent rational and irrational real numbers. It formalizes an idea you’re already probably aware of: fractions can be written as decimals.

A simple example: the fraction ½ could be represented by the decimal 0.5; So 0.5 is a *decimal expansion *of ½

More examples:

Fraction |
Decimal Expansion |

1/3 | .3 |

1/4 | .25 |

1/0 | .1 |

1/20 | .05 |

You can find a full list of decimal expansions for 1/*b* for values between 2 and 10 here. Another alternative: use a calculator. For example, typing in 1/7 into the Google calculator gives 0.14285714285.

Instead of writing out those numbers over and over again, we can write 0.142857, where the overline indicates a series of indefinitely repeating numbers.

## Period of a Decimal Expansion

The **period** of a decimal expansion is the smallest block of repeating digits (if there are no repeating digits, the decimal is **terminating**).

**Examples**:

- The solution above repeats a block of 6 numbers 142857. This is called a
**repeating decimal**because of the infinitely repeating digits. The period is**6.** - The number ½ can be expressed as a repeating decimal 0.3, which has a period of 1.

## More Formal Definition

Any decimal expansion can be expressed as a base-10 geometric series. A geometric series can be written as:

a + ar + ar^{2} + ar^{3} + …

Where:

- a is the initial term,
- r is the common ratio.

**Example question:** Express the decimal expansion 0.121212… as a geometric series.

Step 1: **Split the decimal into its repeating parts**:

0.121212… = 0.12 + 0.0012 + 0.000012…

Step 2: **Rewrite the parts from Step 1 as fractions**:

Step 3: **Find the common ratio**. Each of the above terms can be found by multiplying by the common ratio of 1/100:

Which means this geometric series has:

The sum of this geometric series tells us that the fraction (4/33) has a decimal expansion of 0.121212…

## References

Comez, D. (2016). Decimal Expansion Representation of Real Numbers. Retrieved April 21, 2021 from: https://www.ndsu.edu/pubweb/~comez/Decimal%20expansion.pdf