truncation error

Truncation error is the difference between a truncated value and the actual value. A truncated quantity is represented by a numeral with a fixed number of allowed digits, with any excess digits "chopped off" (hence the expression "truncated").

As an example of truncation error, consider the speed of light in a vacuum. The official value is 299,792,458 meters per second. In scientific (power-of-10) notation, that quantity is expressed as 2.99792458 x 10⁸. Truncating it to two decimal places yields 2.99 x 10⁸. The truncation error is the difference between the actual value and the truncated value, or 0.00792458 x 10⁸. Expressed properly in scientific notation, it is 7.92458 x 10⁵.

In computing applications, truncation error is the discrepancy that arises from executing a finite number of steps to approximate an infinite process. For example, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ... adds up to exactly 1. However, if we truncate the series to only the first four terms, we get 1/2 + 1/4 + 1/8 + 1/16 = 15/16, producing a truncation error of 1 - 15/16, or 1/16.