Rounding

Rounding is the process of replacing a number with a nearby value that has fewer digits or a simpler form, intended to approximate the original value within a defined precision. It is used in measurement, finance, and computation to simplify results and to align with reporting conventions.

Common methods include rounding to a fixed number of decimal places: examine the first dropped digit. If it is less than 5, keep the retained digits unchanged; if greater than 5, increase the last retained digit by one; if exactly 5, apply a tie-breaking rule such as round half up (away from zero for the magnitude) or round half to even (banker’s rounding). Other approaches include truncation (round toward zero), floor (round toward negative infinity), and ceiling (round toward positive infinity). Rounding to significant figures uses the same principle relative to the number’s magnitude.

Examples: 3.14159 rounded to two decimals is 3.14. 2.5 rounds to 3 under round-half-up, but to 2 under round-half-to-even. -3.6 rounded with round-away-from-zero becomes -4, while floor would also yield -4 in this case.

In computing, rounding is formalized by rounding modes, such as those in the IEEE 754 floating-point standard. Floating-point rounding can introduce small errors that propagate in calculations, especially in summations or iterative processes.

Rounding has practical implications in finance, statistics, and measurements, where precision limits and reporting requirements constrain how numbers are presented. Awareness of bias and accumulated error is important when applying rounding in data analysis.