Games - Mental calculation

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	Games - Mental calculation

Mental calculation is the practice of doing mathematical calculations using only the human brain, with no help from any computing devices. It is practiced as a sport in the Mind Sports Olympiad. Mental calculation is said to improve mental capability, speed of response, memory power and concentration.

There are many different techniques for performing mental calculations, many of which are specific to a type of problem.

Calculating differences: a − b

Direct calculation

When the digits of b are all smaller than the digits of a, the calculation can be done digit by digit. For example, evaluate 872 − 41 simply by subtracting 1 from 2 in the units place, and 4 from 7 in the tens place: 831.

Indirect calculation

When the above situation does not apply, the problem can sometimes be modified:

If only one digit in b is larger than its corresponding digit in a, diminish the offending digit in b until it is equal to its corresponding digit in a. Then subtract further the amount b was diminished by from a. For example, to calculate 872 − 92, turn the problem into 872 − 72 = 800. Then subtract 20 from 800: 780.

If more than one digit in b is larger than its corresponding digit in a, it may be easier to find how much must be added to b to get a. For example, to calculate 8192 − 732, we can add 8 to 732 (resulting in 740), then add 60 (to get 800), then 200 (for 1000). Next, add 192 to arrive at 1192, and, finally, add 7000 to get 8192. Our final answer is 7460.

Calculating products: a × b

Multiplying by 2

In this case, the product can be essentially calculated digit by digit. This is not exactly the case because it is possible to have a remainder, but if there is a remainder, it is always 1, which simplifies things greatly. Still, the product must be calculated from right to left: 2 × 167 is by 4 with a remainder, then a 2 (so 3) with another remainder, then a 2 (so 3). Thus, we get 334.

Multiplying by 5

To multiply by 5, first multiply by 10, then divide by 2. Adjoin a 0 to the right end of the number. Then read the number from left to right, dividing the digits by 2, and eventually adding 5 to the next digit if the digit that was divided was odd (after having been divided). For example, 176 × 5 = 1760 ÷ 2. Digit by digit we get 0 (in the thousands digit), 5 + 3, 5 + 3, and 0. This gives 880.

Multiplying by 9

Since 9 = 10 − 1, to multiply by 9, multiply the number by 10 and then subtract the original number from this result. For example, 9 × 27 = 270 − 27 = 243.

Multiplying by 10

To multiply an integer by 10, simply add an extra 0 to the end of the number. To multiply a non-integer by 10, move the decimal point to the right one character.

Multiplying by 11

For single digit numbers simply duplicate the number into the tens digit, for example: 1 × 11 = 11, 2 × 11 = 22, up to 9 × 11 = 99.

The product for any larger non-zero integer can be found by a series of additions to each of its digits from right to left, two at a time.

First take the ones digit and copy that to the temporary result. Next, starting with the ones digit of the multiplier, add each digit to the digit to its left. Each sum is then added to the left of the result, in front of all others. If a number sums to 10 or higher take the tens digit, which will always be 1, and carry it over to the next addition. Finally copy the multipliers left-most (highest valued) digit to the front of the result, adding in the carried 1 if necessary, to get the final product.

In the case of a negative 11, multiplier, or both apply the sign to the final product as per normal multiplication of the two numbers.

A step-by-step example of 759 × 11: # The ones digit of the multiplier, 9, is copied to the temporary result. #* result: 9 # Add 5 + 9 = 14 so 4 is placed on the left side of the result and carry the 1. #* result: 49 # Similarly add 7 + 5 = 12, then add the carried 1 to get 13. Place 3 to the result and carry the 1. #* result: 349 # Add the carried 1 to the highest valued digit in the multiplier, 7+1=8, and copy to the result to finish. #* Final product of 759 × 11: 8349

Further examples:

−54 × −11 = 5 5+4(9) 4 = 594
999 × 11 = 9+1(10) 9+9+1(9) 9+9(8) 9 = 10989
* Note the handling of 9+1 as the highest valued digit.
−3478 × 11 = 3 3+4+1(8) 4+7+1(2) 7+8(5) 8 = −38258
62473 × 11 = 6 6+2(8) 2+4+1(7) 4+7+1(2) 7+3(0) 3 = 687203

Using hands to multiply numbers

This technique allows a number from 6 to 10 to be multiplied by another number from 6 to 10.

This method uses the fingers of both hands, face to face:

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