Truncation

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]. See Notation of floor and ceiling functions.

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.

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]

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

• Wall paper applet that visualizes errors due to finite precision