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To "cast out nines" from a single number, its decimal digits can be simply added together to obtain its so-called digit sum. The digit sum of 2946, for example is 2 + 9 + 4 + 6 = 21. The digit sum of 2946, for example is 2 + 9 + 4 + 6 = 21.
Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0. If the resulting sum has two or more digits, sum those digits as in step one; repeat this step until the resulting sum has only one digit. Repeat steps one and two with the second operand.
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 ]
Casting out nines makes use of the fact that if + =, then (() + ()) = (). In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal, the original addition must have been faulty.
The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).
The topic of Chapter 9 is modular arithmetic, divisibility, and their connections to positional notation, including the use of casting out nines to determine divisibility by nine. [ 4 ] [ 5 ] [ 10 ] In Chapter e {\displaystyle e} , From Zero to Infinity shifts from the integers to irrational numbers , complex numbers , logarithms , and Euler's ...
The next number in the sequence (the smallest number of additive persistence 5) is 2 × 10 2×(10 22 − 1)/9 − 1 (that is, 1 followed by 2 222 222 222 222 222 222 222 nines). For any fixed base, the sum of the digits of a number is proportional to its logarithm ; therefore, the additive persistence is proportional to the iterated logarithm .