# Definition:Decimal Expansion

## Definition

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

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

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

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

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.

### Size Less than 1

A number $x$ such that $\size x < 0$ has a units digit which is zero.

Such a number may be expressed either with or without the zero, for example:

- $0 \cdotp 568$

or:

- $\cdotp 568$

While both are commonplace, the form with the zero is less prone to the mistake where decimal point is missed when reading it.

### 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 Number $234 \cdotp 568$

The number:

- $234 \cdotp 568$

is effectively shorthand for:

plus:

- $5$ tenths
- $6$ hundredths
- $8$ thousandths

### Decimal Number $0.207$

The number:

- $0 \cdotp 207$

can be expressed as a fraction as:

\(\ds 0 \cdotp 207\) | \(=\) | \(\ds \dfrac 2 {10} + \dfrac 0 {100} + \dfrac 7 {1000}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {207} {1000}\) |

### Decimal Number $23.23$

The number:

- $23 \cdotp 23$

can be expressed as a mixed number as:

\(\ds 23 \cdotp 23\) | \(=\) | \(\ds 23 + \dfrac 2 {10} + \dfrac 3 {100}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 23 \tfrac {23} {100}\) |

### 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, $1 \cdotp 24$.

## Also see

## 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.

However, his notation was cumbersome: he would write, for example, $25 \bigcirc \! \! \! \! \! \! 0 \ \, 3 \bigcirc \! \! \! \! \! \! 1 \ \, 7 \bigcirc \! \! \! \! \! \! 2 \ \, 9 \bigcirc \! \! \! \! \! \! 3$ for what we now give as $25 \cdotp 379$.

John Napier, in the early $17$th century, appears to have been the first into print with the contemporary notation, although Walter William Rouse Ball suggests that credit for this ought to be due to Henry Briggs.

It was not until a century later, however, that the decimal point came into general use.

## Sources

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-5}$: Decimal representation of real numbers - 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.38$. Decimals - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm - 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Decimal Representation of Real Numbers - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers: The Dark Ages?