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Concept
Rounding
Summary
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing , the fraction 312/937 with 1/3, or the expression with .
Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as but is known to be accurate only to within a few hundred units is usually better stated as "about ".
On the other hand, rounding of exact numbers will introduce some round-off error in the reported result. Rounding is almost unavoidable when reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain ill-conditioned cases they may make the result meaningless.
Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma".
Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals.
A wavy equals sign (≈: approximately equal to) is sometimes used to indicate rounding of exact numbers, e.g., 9.98 ≈ 10. This sign was introduced by Alfred George Greenhill in 1892.
Ideal characteristics of rounding methods include:
Rounding should be done by a function. This way, when the same input is rounded in different instances, the output is unchanged.
Calculations done with rounding should be close to those done without rounding.
As a result of (1) and (2), the output from rounding should be close to its input, often as close as possible by some metric.
Official source
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