Mental calculation
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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.
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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 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
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.
Multiplying by 10
To multiply a 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.
Note that 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:
-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.
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. 152 − 22. Knowing that 152 is 225 and 22 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:
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 62 = 36
- 72 = 49
- 82 = 64
- 92 = 81
- 102 = 100
- 112 = 121
- 122 = 144
- 132 = 169
- 142 = 196
- 152 = 225
- 162 = 256
- 172 = 289
- 182 = 324
- 192 = 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.
Checking
Estimation
When checking the mental calculation, it is useful to think of it in terms of scaling. For example, when dealing with large numbers, say 1531 × 19625, it, be aware of the number of digits expected for the final value. A useful way of checking is to estimate. 1531 is around 1500, and 19625 is around 20000, so therefore a result of around 20000X1500 (30000000) would be a good estimate. So if the answer has too many zeros, you know you've made a mistake.
9 and 3 rules
If you multiply numbers which have factors of 3, you must end up with a value with a factor of 3 (provided you are dealing with integers). To check this, if you add up all the digits you should end up with a sum that is a multiple of 3. Also, if you know the product to be a multiple of 9, you should end up with the sum of its digits being a multiple of 9.
Approximating square roots
Say we want to find out the square root of a non-square number. Using the formula (a − b)2 = a2 − 2ab + b2. If you choose a 'b' value small enough you can get an accurate estimate. For example, if we are asked to find the square root of 15, we could start with the knowledge that the root of 16 is 4. Now we need a 'b' to put into the equation (4 − b)2 = 15, or thereabouts. Since (4 − b)2 = 16 − 2 × 4 × b roughly, we get b = (16 − 15) ÷ (2 × 4), or roughly 0.125. So then an estimation for the square root is 3.875. If you're after a more accurate value, then restart with an estimate of around 3.9. 3.9)2 we can work out as 15.21, so we do the same working as before; but end up with (3.9 − b)2 = 15, getting b = (15 − 3.92) ÷ (2 × 3.9) = (15 − 15.21) ÷ (7.8) = roughly -0,02 ÷ 8 or around -0,0025. The square of 15 is now estimated as 3.9 − 0.0025 or 3.8875 .
Squaring numbers near 50
Suppose we need to square a number x near 50. This number may be expressed as (50-n), which is 502 - 100n + n2. We know that 502 is 2500. So we subtract n a hundred times from 2500, and then add the difference between x and 50 squared. Example, say we want to square 48, which is 50 - 2. We subtract 200 from 2500 and add 4, and get x2 = 2304. For numbers larger than 50, add n a hundred times instead of subtracting it.
Other systems
There are other methods of mental mathematics
See also
External links
- Fast Arithmetic Tips (http://www.cut-the-knot.org/arithmetic/rapid/index.shtml)
- Divisibility Criteria (http://www.cut-the-knot.org/blue/divisibility.shtml)de:Kopfrechnen
es:Cálculo mental fr:Techniques_de calcul_mental sv:Huvudräkning