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 $10$.


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

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

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.


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, $1 \cdotp 24$.


Also see


Sources