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 $x\in \mathbb {R} _{+}$ 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

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

Where the part in the dividend is read as "the Integer part of" by using the operand "[ ]" in this case with the typesetting that symbolizes the usage of the floor function (see Notation of floor and ceiling functions)

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 ceil is used instead.

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

Where the part in the dividend is read as "the Integer part of" by using the operand "[ ]" in this case with the typesetting that symbolizes the usage of the ceil function (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.

See also

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

External links

• Wall paper applet that visualizes errors due to finite precision