Editing Truncation

{{Short description|In mathematics, limiting the number of digits right of the decimal point}}
{{Other uses}}

In [[mathematics]] and [[computer science]], '''truncation''' is limiting the number of [[numerical digit|digit]]s right of the [[decimal point]].

== Truncation and floor function ==
{{main|Floor and ceiling functions}}
Truncation of positive real numbers can be done using the [[floor function]]. Given a number <math>x \in \mathbb{R}_+</math> to be truncated and <math>n \in \mathbb{N}_0</math>, the number of elements to be kept behind the decimal point, the truncated value of x is
:<math>\operatorname{trunc}(x,n) = \frac{\lfloor 10^n \cdot x \rfloor}{10^n}.</math>

However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the <math> \operatorname{floor} </math> function rounds towards negative infinity. For a given number <math>x \in \mathbb{R}_-</math>, the function <math> \operatorname{ceil} </math> is used instead
:<math>\operatorname{trunc}(x,n) = \frac{\lceil 10^n \cdot x \rceil}{10^n}</math>.

== Causes of truncation ==
With computers, truncation can occur when a decimal number is [[type conversion|typecast]] as an [[integer]]; it is truncated to zero decimal digits because integers cannot store non-integer [[real numbers]].

== In algebra ==
An analogue of truncation can be applied to [[polynomial]]s. In this case, the truncation of a polynomial ''P'' to degree ''n'' can be defined as the sum of all terms of ''P'' of degree ''n'' or less. Polynomial truncations arise in the study of [[Taylor polynomial]]s, for example.<ref>{{cite book|first=Michael|last=Spivak|title=Calculus|edition=4th|year=2008|isbn=978-0-914098-91-1|page=[https://archive.org/details/calculus4thediti00mich/page/434 434]|publisher=Publish or Perish |url-access=registration|url=https://archive.org/details/calculus4thediti00mich/page/434}}</ref>

== See also ==
* [[Arithmetic precision]]
* [[Quantization (signal processing)]]
* [[Precision (computer science)]]
* [[Truncation (statistics)]]

== References ==
{{Reflist}}

== External links ==
* [http://to-campos.planetaclix.pt/fractal/walle.html Wall paper applet] that visualizes errors due to finite precision

[[Category:Numerical analysis]]

[[ja:端数処理]]