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 ${\displaystyle \operatorname {floor} }$ function rounds towards negative infinity. For a given number ${\displaystyle x\in \mathbb {R} _{-}}$, the function ${\displaystyle \operatorname {ceil} }$ is used instead

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

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]

See also

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

References

1. 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