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In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.

Truncation and floor function

Truncation of positive real numbers can be done using the floor function. Given a number $x\in \mathbb {R} _{+}$ to be truncated and $n\in \mathbb {N} _{0}$ , the number of elements to be kept behind the decimal point, the truncated value of x is

\operatorname {trunc} (x,n)={\frac {\lfloor 10^{n}\cdot x\rfloor }{10^{n}}}.

However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the $\operatorname {floor}$ function rounds towards negative infinity. For a given number $x\in \mathbb {R} _{-}$ , the function $\operatorname {ceil}$ is used instead

\operatorname {trunc} (x,n)={\frac {\lceil 10^{n}\cdot x\rceil }{10^{n}}}

.

Causes of truncation

With computers, truncation can occur when a decimal number is 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 polynomials. 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 polynomials, for example.^[1]

References

^ Spivak, Michael (2008). Calculus (4th ed.). Publish or Perish. p. 434. ISBN 978-0-914098-91-1.

External links

Wall paper applet that visualizes errors due to finite precision

[1] Spivak, Michael (2008). Calculus (4th ed.). Publish or Perish. p. 434. ISBN 978-0-914098-91-1.

[1]

@@ Line 2: / Line 2: @@
 {{Other uses}}
-In [[mathematics]] and [[computer Science]], '''truncation''' is limiting the number of [[numerical digit|digit]]s right of the [[decimal point]].
+In [[mathematics]] and [[computer science]], '''truncation''' is limiting the number of [[numerical digit|digit]]s right of the [[decimal point]].
 == Truncation and floor function ==
@@ Line 9: / Line 9: @@
 :<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 floor function rounds towards negative infinity. For a given number <math>x \in \mathbb{R}_-</math>, the function ceil is used instead.
+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>
+:<math>\operatorname{trunc}(x,n) = \frac{\lceil 10^n \cdot x \rceil}{10^n}</math>.
-In some cases {{math|trunc(''x'',0)}} is written as {{math|[''x'']}}.{{cn|date=July 2021}} See [[Floor_and_ceiling_functions#Notation|Notation of floor and ceiling functions]].
 == Causes of truncation ==
@@ Line 18: / Line 16: @@
 == 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]|url-access=registration|url=https://archive.org/details/calculus4thediti00mich/page/434}}</ref>
+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]]
-* [[Floor function]]
 * [[Quantization (signal processing)]]
 * [[Precision (computer science)]]