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Rounding to a specified power is very different from rounding to a specified multiple; for example, it is common in computing to need to round a number to a whole power of 2. The steps, in general, to round a positive number x to a power of some positive number b other than 1, are:
Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski-Young theorem. The mathematical theory of apportionment identifies what properties can be expected from an apportionment ...
Nanson's method can be adapted to handle incomplete ballots (including "plumping") and equal rankings ("bracketing"), though he describes two different methods to handle these cases: a theoretically correct method involving fractions of a vote, and a practical method involving whole numbers (which has the side effect of diminishing the voting ...
Subnormal numbers are not supported, but can be emulated with the mpfr_subnormalize function. MPFR is not able to track the accuracy of numbers in a whole program or expression; this is not its goal. Interval arithmetic packages like Arb, [4] MPFI, [5] or Real RAM implementations like iRRAM, [6] which may be based on MPFR, can do that for the user.
where P is the population of the state, and n is the number of seats it currently holds before the possible allocation of the next seat. Consider the reapportionment following the 2010 U.S. census: after every state is given one seat: The largest value of A 1 corresponds to the largest state, California, which is allocated seat 51.
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.