Mental calculation

Mental calculation

Mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation (for example, conventional methods as taught in educational institutions), or in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems.

Many of these techniques take advantage of or rely on the decimalnumeral system. Usually, the choice of radix determines what methods to use and also which calculations are easier to perform mentally. For example, multiplying or dividing by ten is an easy task when working in decimal (just move the decimal point), whereas multiplying or dividing by sixteen is not; however, the opposite is true when working in hexadecimal.

Methods and techniques

Casting out nines

Main article: Casting out nines

After applying an arithmetic operation to two operands and getting a result, you can use this procedure to improve your confidence that the result is correct.

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. You now have two one-digit numbers, one condensed from the first operand and the other condensed from the second operand. (These one-digit numbers are also the remainders you would end up with if you divided the original operands by 9; mathematically speaking, they're the original operands modulo 9.)Apply the originally specified operation to the two condensed operands, and then apply the summing-of-digits procedure to the result of the operation.Sum the digits of the result you originally obtained for the original calculation..If the result of step 4 does not equal the result of step 5, then the original answer is wrong. If the two results match, then the original answer may be right, though it isn't guaranteed to be.

Example

Say we've calculated that 6338 × 79 equals 500702Sum the digits of 6338: (6 + 3 = 9, so count that as 0) + 3 + 8 = 11Iterate as needed: 1 + 1 = 2Sum the digits of 79: 7 + (9 counted as 0) = 7Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14; 1 + 4 = 5Sum the digits of 500702: 5 + 0 + 0 + (7 + 0 + 2 = 9, which counts as 0) = 55 = 5, so there's a good chance that we were right that 6338 × 79 equals 500702.

You can use the same procedure with multiple operations just repeat steps 1 and 2 for each operation.