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Mental calculation
This article should be Mental calculation is doing mathematical calculations using only the human brain, with no help from any computing devices. Mental calculation is practiced as a sport in the Mind Sports Olympiad. Mental calculation is said to improve mental capability, increases speed of response, memory power and concentration power.
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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 a to get b. 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 10
To multiply a number by 10, simply add an extra 0 to the end of the number.
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 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
Note that 9 = 10 - 1. Thus, 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.
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:
-10-- -10-- --9-- --9-- --8-- --8-- --7-- --7-- --6-- --6--
Here are two examples:
- 9 × 6
above:
-10-- --9-- --8-- -10-- --7--
below:
--9-- --6-- --8-- --7-- --6--
- 5 fingers below make 5 tens - 4 fingers above to the right - 1 finger above to the left
the result: 9 × 6 = 50 + 4 × 1 = 54
- 6 × 8
above:
-10-- --9-- --8-- -10-- --7-- --9--
below:
--6-- --8-- --7-- --6--
- 4 fingers below make 4 tens - 2 fingers above to the right - 4 fingers above to the left
result: 6 × 8 = 40 + 2 × 4 = 48
How it works: each finger represents a number (between 6 and 10). Join the fingers representing the numbers you wish to multiply (x and y). The fingers below give the number of tens, that is (x - 5) + (y - 5). The digits to the upper left give (10 - x) and those to the upper right give (10 - y), leading to [(x - 5) + (y - 5)] × 10 + (10 -x) × (10 - y) = x × y.
Multiplying a two-digit number by 11
Add the two digits together, write down the answer, then append the tens' digit on the left and the units' digit on the right. Thus, for example, 17 × 11 = 187, 35 × 11 = 385.
Using square numbers
The products of small numbers may be calculated by using the squares of integers; for example, to calculate 13 × 17, you can note that 15 is the mean of the two factors, and thus think of it as (15 - 2) ×(15 + 2), i.e. 15^{2} - 2^{2}. Knowing that 15^{2} is 225 and 2^{2} is 4, simple subtraction shows that 225 - 4 = 221, which is the desired product.
This method requires knowing by heart a certain number of squares:
- 1^{2} = 1
- 2^{2} = 4
- 3^{2} = 9
- 4^{2} = 16
- 5^{2} = 25
- 6^{2} = 36
- 7^{2} = 49
- 8^{2} = 64
- 9^{2} = 81
- 10^{2} = 100
- 11^{2} = 121
- 12^{2} = 144
- 13^{2} = 169
- 14^{2} = 196
- 15^{2} = 225
- 16^{2} = 256
- 17^{2} = 289
- 18^{2} = 324
- 19^{2} = 361
It should be noted that if one cannot memorize all of the squares on this list, any square number may be easily calculated by finding the sum of the previous square number, its positive square root, and the number whose square you wish to know. For example, the square of 13 is 144 + 12 + 13 = 169.
Other systems
There are other methods of mental mathematics