Truncation

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 floor function rounds towards negative infinity. For a given number $x\in \mathbb {R} _{-}$ , the function

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

is used instead.

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 Linux, the truncate command allows emptying or shrinking a file to a specified size. The command also supports expanding files to a specified size by appending null bytes.^[1]

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.^[2]

References

^ "How to Truncate (Empty) Files in Linux". linuxize.com. 12 September 2019.
^ 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

[1] "How to Truncate (Empty) Files in Linux". linuxize.com. 12 September 2019.

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

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