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Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. [150]
A typical solution is to represent the number in a small base, b, such that, for example, 8b is a representable machine integer. Several additions can then be performed before an overflow occurs. When the number becomes too large, we add part of it to the result, or we carry and map the remaining part back to a number that is less than b.
Adding and subtracting numbers below 20 using Mayan numerals is very simple. Addition is performed by combining the numeric symbols at each level: If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed and a dot is added to the next higher row.
Empty numbers are sometimes made up, with obvious meaning: "squillions" is obviously an empty, but very large, number; a "squintillionth" would be a very small number. Some empty numbers may be modified by actual numbers, such as "four zillion", and are used for jest, exaggeration, or to relate abstractly to actual numbers.
He used it for building a table from successive differences, and for adding large numbers of harmonic terms". "Tables of functions are constructed from their differences with great efficiency, either as printed tables or as a file of punched cards ."
A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative. [77]
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