Truncation

For other uses, see Truncation (disambiguation).

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

Truncation and floor function

Main article: Floor and ceiling functions

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

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

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

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

In some cases trunc(x,0) is written as [x].^{[citation needed]} See Notation of floor and ceiling functions.

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]

See also

• Arithmetic precision
• Quantization (signal processing)
• Precision (computer science)
• Truncation (statistics)

References

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

External links

• Wall paper applet that visualizes errors due to finite precision