Mental calculation 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. 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. 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 X 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 zeroes, 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 your 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 acurate 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 Other systems There are other methods of mental mathematics Vedic mathematics Trachtenberg system Abacus system See also Mental calculator FAVORITE ARTICLES: History of the World    History of the United States    History of Europe    Ancient History    History    Military History    World War I    Bombing of Dresden    California    US Constitution    Frank Sinatra    Boston    Marbury v. Madison This article is licensed under the GNU Free Documentation License at http://www.gnu.org/copyleft/fdl.html You may copy and modify it as long as the entire work (including additions) remains under this license. 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