Definition:Decimal Expansion

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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 $\size x < 1$ 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:

$2$ hundreds
$3$ tens
$4$ units

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 fraction as:

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


Decimal Number $6735 \cdotp 249$

The number:

$6735 \cdotp 249$

is shorthand for:

$\paren {6 \times 10^3} + \paren {7 \times 10^2} + \paren {3 \times 10^1} + \paren {5 \times 10^0} + \paren {2 \times 10^{-1} } + \paren {4 \times 10^{-2} } + \paren {9 \times 10^{-3} }$


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

  • Results about decimal expansions can be found here.


Historical Note

The use of a positional number system to represent fractional values was introduced around $1579$ by François Viète.

The idea was also implemented, by extending the decimal notation to the right, by Simon Stevin, who published the influential book De Thiende in $1585$.

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

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