Definition:Decimal

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Definition

Decimal is a word used to denote $10$-ness.

It is usually used to mean the decimal number system and related concepts.


Decimal System

A decimal system is a system of measurement in which the standard multiples and fractions of the units of measurement are powers of $10$.


Decimal Notation

Decimal notation is the quotidian technique of expressing numbers in base $10$.

Every number $x \in \R$ is expressed in the form:

$\ds x = \sum_{j \mathop \in \Z} r_j 10^j$

where:

$\forall j \in \Z: r_j \in \set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$


Decimal Expansion

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.


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.


Decimal Part

Erroneously used to mean fractional part:


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

Let $\floor x$ be the floor function of $x$.


The fractional part of $x$ is the difference:

$\fractpart x := x - \floor x$


Also known as

Some sources use the term denary for decimal.


Also see

  • Results about decimal can be found here.


Sources

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