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.^{[citation needed]}

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

[edit] Calculating differences: a − b

[edit] 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.

[edit] 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.

[edit] Look-ahead borrow method

This method can be used to subtract numbers left to right, and if all that is required is to read the result aloud, it requires little of the user's memory even to subtract numbers of arbitrary size.

One place at a time is handled, left to right.

Example:

          4075
        - 1844
        ------


Thousands: 4-1=3, look to right, 075<844, need to borrow.
           3-1=2, say "Two thousand"

 Hundreds: 0-8=negative numbers not allowed here,
           10-8=2, 75>44 so no need to borrow,
           say "two hundred"

     Tens: 7-4=3, 5>4 so no need to borrow, say "thirty"

     Ones: 5-4=1, say "one"

[edit] Calculating products: a × b

[edit] 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.

[edit] Multiplying by 5

To multiply a number by 5, first multiply that number by 10, then divide it by 2. The following algorithm is a quick way to produce this result: First, append a zero to right side of the desired number. Next, starting from the leftmost numeral, divide by 2 and append each result in the respective order to form a new number; fraction answers should be rounded down to the nearest whole number. For example, if you intended to multiply 176 by 5, you would first append a zero to 176 to make 1760. Next, divide 1 by 2 to get .5, rounded down to zero. Divide 7 by 2 to get 3.5, rounded down to 3. Divide 6 by 2 to get 3. Zero divided by two is simply zero. The resulting number is 0330. The final step involves adding 5 to the number that follows any single numeral in this new number that was odd before dividing by two; this is better understood through the example. In the original number, 176, the first place is 1, which is odd. Therefore, we add 5 to the numeral after the first place in our newly constructed number (0330), which is 3; 3+5=8. The numeral in the second place of 176, 7, is also odd. Therefore the number-place after the corresponding numeral in the constructed number (0830) is increased by 5 as well; 3+5=8. The numeral in the third place of 176, 6, is even, therefore the final number, zero, in our answer is not changed. That final answer is 0880. The leftmost zero can be omitted, leaving 880. So 176 times 5 equals 880.

[edit] 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.

[edit] 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 digit.

[edit] 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

[edit] Multiplying two 2 digit numbers between 11 and 19

To easily multiply 2 digit numbers together between 11 and 19 a simple algorithm is as follows:

1a x 1b

100 + 10 * (a+b) + a*b
which can be visualized as:

1
xx
 yy

for example:

17 * 16

1
13
 42

272

[edit] Multiplying Any 2 digit Numbers Together

To easily multiply any 2 digit numbers together a simple algorithm is as follows:


ab * cd

100*(a*c) + 10*(b*c) +  10*(a*d)+ b*d

for example

23
47

 800
 120
 140 
  21

1081

[edit] 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.

[edit] 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 remark 15 is the mean of the two factors, and 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

Any square number may be easily calculated by adding 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.

[edit] Checking

[edit] 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 digits, you know you've made a mistake.

[edit] 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.

[edit] 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 root of 15 is now estimated as 3.9 − 0.0025 or 3.8975. (the real square root of 15 is 3.8729833...)

[edit] Squaring numbers near 50

Suppose we need to square a number x near 50. This number may be expressed as x=50-n, and hence the answer x² is (50−n)², which is 50² − 100n + n². We know that 50² is 2500. So we subtract 100n from 2500, and then add n². 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 (x=50+n), add n a hundred times instead of subtracting it.

[edit] Other systems

There are other methods of mental mathematics

[edit] Mental Calculation World Cup

The first World Mental Calculation Championships (Mental Calculation World Cup) took place 2004. They are repeated every second year. The event of 2006 took place on November 4 2006 in Giessen, Germany. It consists of four different tasks (addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation square roots and calculation weekdays for gives dates) plus two surprise tasks. It was won by Robert Fountain from England.

The next Championships are scheduled for 2008.

[edit] See also

[edit] External links

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