---
tag: 06XY4X9430
kind: definition
name: Decimal expansion
msc: [11A63]
defines: 0VM8B2SB4P
uses: [08J2M1JYAG, 0AXWET80YW]
sources:
  - url: https://en.wikipedia.org/wiki/Decimal_representation
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Decimal_representation
    locator: Integer and fractional parts
---

## Definition

Let $x$ be a real number with $x \ge 0$.
A decimal expansion of $x$ is a natural number $a_0$ and a sequence $(d_k)_{k \ge 1}$ of digits $d_k \in \{0, 1, \dots, 9\}$ with
$$x = a_0 + \sum_{k=1}^{\infty} \frac{d_k}{10^k}.$$
The notation is $x = a_0.d_1 d_2 d_3 \dots$, with $a_0$ in base $10$.

## Remarks

The series always converges, because its partial sums are increasing and bounded above by $a_0 + 1$.
A number can have two expansions. This happens exactly when one expansion ends in infinitely many $0$ and the other in infinitely many $9$, as for [[0.999... and 1|0NN5JEMTVE]].
For a negative number $x$, one writes $-$ followed by an expansion of $-x$.

## Explanations

### L1 human

The digits after the point say how many tenths, hundredths, thousandths, and so on, the number has.
For example, $3.14$ means $3 + 1/10 + 4/100$.
For a number such as $\pi$, the digits go on forever.