Definition:Decimal
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): decimal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): decimal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): denary