# Truncation

For other uses, see Truncation (disambiguation).

In mathematics and computer science, truncation is limiting the number of digits right of the decimal point, by discarding the least significant ones.

For example, consider the real numbers

5.6341432543653654
32.438191288
−6.3444444444444

To truncate these numbers to 4 decimal digits, we only consider the 4 digits to the right of the decimal point.

The result would be:

5.6341
32.4381
−6.3444

Truncation is equivalent to rounding towards zero (or rounding down the absolute value of the number while maintaining the sign).

The truncation error can be twice the maximum error in rounding.[citation needed]

## 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.

## 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 real numbers (that are not themselves integers).

## 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]