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If the digit 9 is ignored when summing the digits, the effect is to "cast out" one more 9 to give the result 12. More generally, when casting out nines by summing digits, any set of digits which add up to 9, or a multiple of 9, can be ignored. In the number 3264, for example, the digits 3 and 6 sum to 9.
Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers in decimal, a method known as long ago as the 12th century. [3] If an odd perfect number exists, it will have at least nine distinct prime factors. [4] Non-intersecting chords between four points on a circle
For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations. Digit sums are also a common ingredient in checksum algorithms to check the arithmetic operations of early computers. [ 3 ]
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For example, 90% would be described as "one nine"; 99% as "two nines"; 99.9% as "three nines"; and so forth. However, there are different conventions for representing inexact multiples of 9. For example, a percentage of 99.5% could be expressed as "two nines five" (2N5, or N2.5) [ 2 ] or as 2.3 nines, [ citation needed ] following from the ...
Also b n −1 is not a multiple of p for any value of n less than ℓ, because otherwise the repeating period of a/p in base b would be less than ℓ. Now suppose that ℓ = hk. Then b ℓ − 1 is a multiple of b k − 1. (To see this, substitute x for b k; then b ℓ = x h and x − 1 is a factor of x h − 1.
A sequence of six consecutive nines occurs in the decimal representation of the number pi (π), starting at the 762nd decimal place. [ 1 ] [ 2 ] It has become famous because of the mathematical coincidence , and because of the idea that one could memorize the digits of π up to that point, and then suggest that π is rational .
Let be a Banach space, let ′ be the dual space of , let : ′ be a linear map, and let ′.A vector is a solution of the equation = if and only if for all , () = ().A particular choice of is called a test vector (in general) or a test function (if is a function space).
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