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Casting out nines is any of three arithmetical procedures: [1] Adding the decimal digits of a positive whole number , while optionally ignoring any 9s or digits which sum to 9 or a multiple of 9.
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 .
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.
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 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
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 smaller numbers, for use when subtracting, are the nines' complement of the larger numbers, which are used when adding. In mathematics and computing , the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism ) for addition throughout ...
The Archimedean property: any point x before the finish line lies between two of the points P n (inclusive).. It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits.