# Truncation

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In mathematics and computer science, truncation is the term for 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

Note that in some cases, truncating would yield the same result as rounding, but truncation does not round up or round down the digits; it merely cuts off at the specified digit. The truncation error can be twice the maximum error in rounding.

## 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]

## References

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