Mental calculation is the practice of doing mathematical calculations using only the human brain, with no help from any computing devices. A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change The human brain controls the Central nervous system (CNS by way of the Cranial nerves and Spinal cord, the Peripheral nervous system (PNS A computer is a Machine that manipulates data according to a list of instructions.
Practically, mental calculations are not only helpful when computing tools are not available, but they also can be helpful in situations where it is beneficial to calculate with speed. When a method is much faster than the conventional methods (as taught in school), it may be called a shortcut. Although used to aid or quicken tedious computation, many also practice or create such tricks to impress their peers with their quick calculating skills.
Almost all such methods make use of the decimal numeral system. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. It should be noted that the choice of radix determines what methods to use and also which calculations are easier to perform mentally. In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral For example, multiplying or dividing by ten is very easy in decimal (just move the decimal point), whereas multiplying or dividing by sixteen is not; but the opposite happens if one uses the hexadecimal base instead of decimal. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a
There are many different techniques for performing mental calculations, many of which are specific to a type of problem.
A quick test to further increase confidence that the correct answer to a calculation has been found. A sanity test or sanity check is a basic test to quickly evaluate the validity of a claim or calculation
After applying an arithmetic operation to two operands and getting a result, you can use this procedure to improve your confidence that the result is correct. Casting out nines is a Sanity check to ensure that hand computations of sums differences products and quotients of Integers are correct
You can use the same procedure with multiple operands; just repeat steps 1 and 2 for each operand.
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, estimation instructs you to 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 20000 × 1500 (30000000) would be a good estimate for the actual answer (30045875). So if the answer has too many digits, you know you've made a mistake.
When multiplying, a useful thing to remember is that the factors of the operands still remain. For example, to say that 14 × 15 was 211 would be unreasonable. Since 15 was a multiple of 5, so should the product. The correct answer is 210.
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.
When the above situation does not apply, the problem can sometimes be modified:
You may guess what is needed, and accumulate your guesses. Your guess is good as long as you haven't gone beyond the "target" number. 8192-732, mentally, you want to add 8000 but that would be too much, so we add 7000, then 700 to 1100, is 400 (so far we have 7400), and 32 to 92 can easily be recognized as 60. The result is 7460.
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"
Many of these methods work because of the distributive property. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law
Where one number being multiplied is sufficiently small to be multiplied with ease by any single digit, the product can be calculated easily digit by digit from right to left. This is particularly easy for multiplication by "2" since the carry digit cannot be more than 1.
For example, to calculate 2 × 167: 2x7=14, so the final digit is 4, with a "1" carried and added to the 2x6=12 to give 13, so the next digit is 3 with a "1" carried and added to the 2x1=2 to give 3. Thus, the product is 334.
To multiply a number by 5,
1. First multiply that number by 10, then divide it by 2.
The following algorithm is a quick way to produce this result:
2. Add a zero to right side of the desired number. (A. ) 3. Next, starting from the leftmost numeral, divide by 2 (B. )and append each result in the respective order to form a new number;(fraction answers should be rounded down to the nearest whole number).
EXAMPLE: Multiply 176 by 5. A. Add a zero to 176 to make 1760. B. Divide by 2 starting at the left. 1. Divide 1 by 2 to get . 5, rounded down to zero. 2. Divide 7 by 2 to get 3. 5, rounded down to 3. 3. Divide 6 by 2 to get 3. Zero divided by two is simply zero.
The resulting number is 0330. (This is not the final answer, but a first approximation which will be adjusted in the following step:)
C. Add 5 to the number that follows any single numeral in this new number that was odd before dividing by two;
EXAMPLE: 176 (IN FIRST, SECOND THIRD PLACES):
1. The FIRST place is 1, which is odd. ADD 5 to the numeral after the first place in our new number (0330)which is 3; 3+5=8. 2. The number in the second place of 176, 7, is also odd. The corresponding number (0 8 3 0) is increased by 5 as well; 3+5=8.
3. 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.
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.
Hold hands in front of you, palms facing you. Assign the left thumb to be 1, the left index to be 2, and so on all the way to right thumb is ten. Each | symbolizes a raised finger and a - represents a bent finger.
1 2 3 4 5 6 7 8 9 10| | | | | | | | | |left hand right hand
Bend the finger which represents the number to be multiplied by nine down.
Ex: 6 × 9 would be
| | | | | - | | | |
The right little finger is down. Take the number of fingers still raised to the left of the bent finger and append it to the number of fingers to the right.
Ex: There are five fingers left of the right little finger and four to the right of the right little finger. So 6 × 9 = 54.
5 4| | | | | - | | | |
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.
In general for base ten, to multiply by 10n (where n is an integer), move the decimal point n digits to the right. If n is negative, move the decimal |n| digits to the left.
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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French
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:
Another method is to simply multiply the number by 10, and add the original number to the result.
17 × 11
17 × 10 = 170 + 17 = 187
17 × 11 = 187
To easily multiply 2 digit numbers together between 11 and 19 a simple algorithm is as follows:
1a x 1b100 + 10 * (a+b) + a*bwhich can be visualized as:1xx yyfor example:17 * 16113 (7+6) 42 (7*6)272 (total)
To easily multiply any 2 digit numbers together a simple algorithm is as follows:
800 +120 +140 + 21----- 1081
Note that this is the same thing as the conventional sum of partial products, just restated with brevity. To minimize the number of elements being retained in one's memory, it may be convenient to perform the sum of the "cross" multiplication product first, and then add the other two elements:
i. e. , in this example
to which is it is easy to add 21: 281 and then 800: 1081
This technique allows a number from 6 to 10 to be multiplied by another number from 6 to 10. This is only a test.
Assign 6 to the little finger, 7 to the ring finger, 8 to the middle finger, 9 to the index finger, and 10 to the thumb. Touch the to desired numbers together. The point of contact and below is considered the "below" section and everything above the two fingers that are touching are part of the "above" section. For example, 6 × 9 would look like this:
-10-- --9-- --8-- (above)-10-- --7--====================--9-- --6-- left index and right little finger are touching--8-- (below)--7-- --6-- (9 × 6)
-10-- -10----9-- --9----8-- --8----7-- --7----6-- --6--
Here are two examples:
-10-- --9-- --8---10-- --7--
--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
-10----9-- --8-- -10----7-- --9--
--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.
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:
It may be useful to be aware that the difference between two successive square numbers is the sum of their respective square roots. Hence if you know that 12*12=144 and wish to know 13*13, calculate 144 + 12 + 13 = 169.
This is because (x+1)² - x² = x² + 2 x + 1 - x² = x + (x+1)
x² = (x-1)² + (2x - 1)
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.
This method requires the memorization of squares from 1 to 25.
The square of n (most easily calculated when n is between 26 and 75 inclusive) is
(50 − n)² + 100(n − 25)
In other words, the square of a number is the square of its difference from fifty added to one hundred times the difference of the number and twenty five. For example, to square 62, we have:
(−12)² + [(50-12) * 100]= 144 + 3,700= 3,844
This method requires the memorization of squares from 1 to 25.
The square of n (most easily calculated when n is between 76 and 99 inclusive) is
(100 − n)² + 100(100 − 2(100 − n))
In other words, the square of a number is the square of its difference from one hundred added to the product of one hundred and the difference of one hundred and the product of two and the difference of one hundred and the number. For example, to square 93, we have:
7² + 100(100 − 2(7))= 49 + 100 × 86= 49 + 8,600= 8,649
Another way to look at it would be like this:
93² = ? (is -7 from 100)93 - 7 = 86 (this gives us our first two digits)(-7)² = 49 (these are the second two digits)93² = 8649
82² = ? (is -18 from 100) 82-18 = 64 (subtract. First digits. ) (-18)² = 324 (second pair of digits. We'll need to carry the 3. ) 82² = 6724
This method requires memorization of the squared numerals 1 to 9.
The square of mn, mn being a two-digit integer, can be calculated as
10 × m(mn + n) + n²
Meaning the square of mn can be found by adding n to mn, multiplied by m, adding 0 to the end and finally adding the square of n.
For example, we have 23²:
23²= 10 × 2(23 + 3) + 3²= 10 × 2(26) + 9= 520 + 9= 529
So 23² = 529.
An easy way to approximate the square root of a number is to use the following equation:
The closer the known square is to the unknown, the more accurate the approximation. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Algebra, the square of a number is that number multiplied by itself For instance, to estimate the square root of 15, we could start with the knowledge that the nearest perfect square is 16 (4²).
So we've estimated the square root of 15 to be 3. 875. The actual square root of 15 is 3. 872983. . .
Say we want to find the square root of a number we'll call 'x'. By definition
We then redefine the root
where 'a' is a known root (4 from the above example) and 'b' is the difference between the known root and the answer we seek.
And here's the trick. If 'a' is close to your target, 'b' will be a small enough number to render the element of the equation negligible. So we drop out and rearrange the equation to
This is a surprisingly easy task for many higher powers, but not very useful except for impressing friends (practical uses of finding roots rarely use perfect powers). The task is not as hard as it sounds mainly because the basic method is to find the last digit using the last digit of the given power and then finding the other digits by using the magnitude of the given power. Such feats may seem obscure but are nevertheless recorded and practiced. See 13th root.
An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42³ = 74088.
Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10:
A neat trick here is that there is a pattern see if you can see it. Remember that the pattern is adding and subtracting. Starting from zero:
There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above.
Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit.
The second step is to determine the first digit of the two digit cube root by looking at the magnitude of the given cube. To do this, remove the last three digits of the given cube (29791 -> 29) and find the greatest cube it is greater than (this is where knowing the cubes of numbers 1-10 is needed). Here, 29 is greater than 1 cubed, greater than 2 cubed, greater than 3 cubed, but not greater than 4 cubed. The greatest cube it is greater than is 3, so the first digit of the two digit cube must be 3.
Therefore, the cube root of 29791 is 31.
To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. In Mathematics, an n th root of a Number a is a number b such that bn = a. One must know:
From this information, one can find the log of any number 1-9.
The first step in approximating the common log is to put the number given in scientific notation. For example, the number 45 in scientific notation is 4. 5 * 10^1, but we will call it a * 10^b. Next, find the log of a, which is between 1 and 10. Start by finding the log of 4, which is . 60, and then the log of 5, which is . 70 because 4. 5 is between these two. Next, and skill at this comes with practice, place a 5 on a logarithmic scale between . 6 and . 7, somewhere around . 653 (NOTE: the actual value of the extra places will always be greater than if it were placed on a regular scale. i. e. , you would expect it to go at . 650 because it is halfway, but instead it will be a little larger, in this case . 653) Once you have obtained the log of a, simply add b to it to get the approximation of the common log. In this case, a+b = . 653+1 = 1. 653. The actual value of log(45) = 1. 65321.
The same process applies for numbers between 0 and 1. For example, . 045 would be written as 4. 5*10^-2. The only difference is that b is now negative, so when adding you are really subtracting. This would yield the result . 653-2, or -1. 347.
There are many other methods of calculation in mental mathematics. The list below shows a few other methods of calculating, though they may not be entirely mental.
The first World Mental Calculation Championships (Mental Calculation World Cup) took place 2004. For the actual mathematics of the Vedic period, see the articles on Sulba Sūtras and Indian mathematics. The Trachtenberg System is a system of rapid Mental calculation, somewhat similar to Vedic mathematics. The abacus system of Mental calculation is a system where users mentally visualize an Abacus to do Calculations No physical abacus is used only the answers Chisanbop or chisenbop (from Korean chi (ji finger + sanpŏp (sanbeop calculation 지산법 is an Abacus -like Finger counting Mental Calculation World Cup is an international competition of Mental calculators Mental Calculation World Cup 2004 The first Mental Calculation World Cup was They are repeated every second year. The event of 2006 took place on November 4, 2006 in Giessen, Germany. Events 1333 - Flood of the Arno River, causing massive damage in Florence as recorded by the Florentine chronicler Giovanni Villani Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Gießen (ˈgiːsən is a town in the German federal state ( Bundesland) of Hessen, capital of both the district of Gießen and the administrative Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. It consists of four different tasks: addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation of square roots and calculation of weekdays for given dates, plus two surprise tasks. It was won by Robert Fountain from England. England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland
The next Championship is scheduled for 2008.