Truncation (math)

In mathematics and computer science, truncation describes an arithmetic operation in which a sequence or a decimal number is shortened to a certain length. For decimal numbers this means that decimal places are deleted.

Difference to the rounding function

The following applies to a number that is to be truncated to decimal places ${\displaystyle x\in \mathbb {R} _{+}}$ ${\displaystyle n\in \mathbb {N} }$

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

For negative real numbers , however, the following applies ${\displaystyle x}$

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

where and stand for the rounding function and the rounding function . ${\displaystyle \lfloor x\rfloor }$ ${\displaystyle \lceil x\rceil }$

For positive numbers, the truncation with therefore corresponds to the rounding function. In the case of negative numbers, however, the rounding function is equivalent in this case. ${\displaystyle n=0}$

Typecast

Truncation with occurs in C / C ++ when converting a floating point number to an integer . ${\displaystyle n=0}$

Polynomials

Analogous to truncation with decimal numbers, the truncation of a polynomial can be defined. That is the sum of the polynomial terms up to the degree . ${\displaystyle [P]_{n}}$ ${\displaystyle n}$

example

Be . The different truncations are then ${\displaystyle P(x)=x^{4}-5x^{3}+{\frac {2}{3}}x^{2}-5}$

${\displaystyle [P]_{4}=x^{4}}$, , . ${\displaystyle [P]_{3}=x^{4}-5x^{3}}$ ${\displaystyle [P]_{2}=x^{4}-5x^{3}+{\frac {2}{3}}x^{2}=[P]_{1}}$

Individual evidence

1. ↑ Ronald B. Guenther, Abdelwahab Kharab: An Introduction to Numerical Methods: a MATLAB approach . 4th edition. CRC Press, Boca Raton 2019, ISBN 978-1-138-09307-2 , pp. 32 .
2. ↑ Type conversions. In: cplusplus.com. Retrieved July 5, 2019 . 
3. ↑ Michael Spivak: Calculus . 4th edition. Publish or Perish, Houston 2008, ISBN 978-0-914098-91-1 , pp. 434 .