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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 1: / Line 1: @@
+{{Short description|In mathematics, limiting the number of digits right of the decimal point}}
 {{Other uses}}
-In [[mathematics]] and [[computer science]], '''truncation''' is the term for limiting the number of [[numerical digit|digit]]s right of the [[decimal point]], by discarding the least significant ones.
+In [[mathematics]] and [[computer science]], '''truncation''' is limiting the number of [[numerical digit|digit]]s right of the [[decimal point]].
-For example, consider the [[real number]]s
-The term was first used in television by Creighton Sweeney.
-:5.6341432543653654
-:32.438191288
-:−6.3444444444444
-To ''truncate'' these numbers to 4 decimal digits, we only consider the 4 digits to the right of the decimal point.
-The result would be:
-:5.6341
-:32.4381
-:−6.3444
-Truncation is equivalent to [[rounding]] towards zero (or [[rounding]] down the absolute value of the number while maintaining the sign).
-The truncation [[Approximation error|error]] can be twice the maximum error in rounding.{{Citation needed|date=February 2014}}
 == Truncation and floor function ==
@@ Line 28: / 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.
+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 [[real numbers]] (that are not themselves integers).
+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=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)]]