# Definition:Decimal Expansion

## Definition

Let $x \in \R$ be a real number.

The decimal expansion of $x$ is the expansion of $x$ in base $10$.

$x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {10^j}$:

$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$

where:

$s = \floor x$, the floor of $x$
it is not the case that there exists $m \in \N$ such that $d_M = 9$ for all $M \ge m$.

(That is, the sequence of digits does not end with an infinite sequence of $9$s.)

### Decimal Point

The dot that separates the integer part from the fractional part of $x$ is called the decimal point.

That is, it is the radix point when used specifically for a base $10$ representation.

### Decimal Place

Let the decimal expansion of $x$ be:

$x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_{10}$

Then $d_k$ is defined as being the digit in the $k$th decimal place.

## Examples

### Decimal Expansion of $17 / 10$

$\dfrac {17} {10}$ has a decimal expansion of $1 \cdotp 7$.

### Decimal Expansion of $9 / 100$

$\dfrac 9 {100}$ has a decimal expansion of $0 \cdotp 09$.

### Decimal Expansion of $1 / 6$

$\dfrac 1 6$ has a decimal expansion of $0 \cdotp 1666 \ldots$.

### $1 \cdotp 23999 \ldots$ is not a Decimal Expansion

$1 \cdotp 23999 \ldots$

is not a decimal expansion as defined on $\mathsf{Pr} \infty \mathsf{fWiki}$.

This is because it ends in an infinite sequence of $9$s.

The number $1 \cdotp 23999 \ldots$ is equal to, and is best expressed as, $1 \cdotp 24$.

## Historical Note

The idea of representing fractional values by extending the decimal notation to the right appears to have been invented by Simon Stevin, who published the influential book De Thiende.

The idea was borrowed from the Babylonian number system, but streamlined to base $10$ from the cumbersome sexagesimal.