3606:
which is .60, and then the logarithm 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., one would expect it to go at .650 because it is halfway, but instead, it will be a little larger, in this case, .653) Once one has obtained the logarithm of a, simply add b to it to get the approximation of the common logarithm. In this case, a + b = .653 + 1 = 1.653. The actual value of log(45) ~ 1.65321.
79:
257:
place. Hundreds: 0 − 8 = negative numbers not allowed here. One is going to increase this place by using the number one borrowed from the column to the left. Therefore: 10 − 8 = 2. It is 10 rather than 0, because one borrowed from the
Thousands place. 75 > 44 so no need to borrow, say "two hundred" Tens: 7 − 4 = 3, 5 > 4, so 5 - 4 = 1
25:
3689:) took place in 2004. It is an in-person competition that occurs every other year in Germany. It consists of four different standard tasks --- addition of ten ten-digit numbers, multiplication of two eight-digit numbers, calculation of square roots, and calculation of weekdays for given dates --- in addition to a variety of "surprise" tasks.
3337:
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 42
3670:
The first Mental
Calculations World Championship took place in 1998. This event repeats every year and now occurs online. It consists of a range of different tasks such as addition, subtraction, multiplication, division, irrational and exact square roots, cube roots and deeper roots, factorizations,
3605:
The first step in approximating the common logarithm is to put the number given in scientific notation. For example, the number 45 in scientific notation is 4.5 × 10, but one will call it a × 10. Next, find the logarithm of a, which is between 1 and 10. Start by finding the logarithm of 4,
3502:
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).
2482:
205:
When multiplying, a useful thing to remember is that the factors of the operands still remain. For example, to say that 14 × 15 was 201 would be unreasonable. Since 15 is a multiple of 5, the product should be as well. Likewise, 14 is a multiple of 2, so the product should be even. Furthermore,
1295:
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
3641:
by
Ranjana Mehta includes a recent study that involved participants engaging in concurrent mental and physical tasks. This study investigated the effects of mental demands on physical performance at different levels of physical exertion and ultimately found a decrease in physical performance when
256:
Example: 4075 − 1844 ------ Thousands: 4 − 1 = 3, look to right, 075 < 844, need to borrow. 3 − 1 = 2, say "Two thousand". One is performing 3 - 1 rather than 4 - 1 because the column to the right is going to borrow from the thousands
844:
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:
2595:
3324:
is often practiced. The difficulty of the task does not depend on the number of digits of the perfect power but on the precision, i.e. the number of digits of the root. In addition, it also depends on the order of the root; finding perfect roots, where the order of the root is
2806:
3625:, like doing mental calculations, performed afterward. It has been shown that during high levels of physical activity there is a negative effect on mental task performance. This means that too much physical work can decrease accuracy and output of mental math calculations.
1564:
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 one knows that 12 × 12 = 144 and wish to know 13 × 13, calculate 144 + 12 + 13 = 169.
578:
To easily multiply any 2-digit numbers together a simple algorithm is as follows (where a is the tens digit of the first number, b is the ones digit of the first number, c is the tens digit of the second number and d is the ones digit of the second number):
4079:
Tzourio-Mazoyer, Nathalie; Pesenti, Mauro; Zago, Laure; Crivello, Fabrice; Mellet, Emmanuel; Samson, Dana; Duroux, Bruno; Seron, Xavier; Mazoyer, Bernard (2001). "Mental calculation in a prodigy is sustained by right prefrontal and medial temporal areas".
3609:
The same process applies for numbers between 0 and 1. For example, 0.045 would be written as 4.5 × 10. The only difference is that b is now negative, so when adding one is really subtracting. This would yield the result 0.653 − 2, or −1.347.
2304:
In other words, the square of a number is the square of its difference from 100 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:
206:
any number which is a multiple of both 5 and 2 is necessarily a multiple of 10, and in the decimal system would end with a 0. The correct answer is 210. It is a multiple of 10, 7 (the other prime factor of 14) and 3 (the other prime factor of 15).
3463:
There are two steps to extracting the cube root from the cube of a two-digit number. For example, extracting the cube root of 29791. Determine the one's place (units) of the two-digit number. Since the cube ends in 1, as seen above, it must be 1.
1444:
The products of small numbers may be calculated by using the squares of integers; for example, to calculate 13 × 17, one can remark 15 is the mean of the two factors, and think of it as (15 − 2) × (15 + 2),
2600:
So the estimated square root of 15 is 3.875. The actual square root of 15 is 3.872983... One thing to note is that, no matter what the original guess was, the estimated answer will always be larger than the actual answer due to the
1408:
Next, subtract one of the two variables from 100. Then subtract the difference from the other variable. That difference will be the first two digits of the final product, and the resulting 4 digit number will be the final product.
2421:
3709:(Mental addition, Mental Multiplication, Mental Square Roots (non-integer), Mental Calendar Dates calculation and Flash Anzan). The third Memoriad was held in Las Vegas, USA, from November 8, 2016 through November 10, 2016.
1415:
95 x 97 ---- Last two digits: 100-95=5 (subtract first number from 100) 100-97=3 (subtract second number from 100) 5*3=15 (multiply the two differences)
3637:. Using an EEG as a measure of mental workload after different levels of physical activity can help determine the level of physical exertion that will be the most beneficial to mental performance. Previous work done at
836:
2497:
3535:
These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work.
1156:
3232:
729:
1296:
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.
3304:
2752:
3503:
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.
1271:
Since 9 = 10 − 1, to multiply a number by nine, multiply it by 10 and then subtract the original number from the result. For example, 9 × 27 = 270 − 27 = 243.
250:
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.
1161:
this expression is analogous to any number in base 10 with a hundreds, tens and ones place. FOIL can also be looked at as a number with F being the hundreds, OI being the tens and L being the ones.
165:
If the result of step 4 does not equal the result of step 5, then the original answer is wrong. If the two results match, then the original answer may be right, though it is not guaranteed to be.
3756:
Note that these single-digit numbers are really the remainders one would end up with if one divided the original operands by 9, which is to say that each is the result of its associated operand
2502:
1634:
Multiply these numbers together to get 242,000 (This can be done efficiently by dividing 484 by 2 = 242 and multiplying by 1000). Finally, add the difference (8) squared (8 = 64) to the result:
104:. People may use mental calculation when computing tools are not available, when it is faster than other means of calculation (such as conventional educational institution methods), or even in
3154:
1847:
1274:
This method can be adjusted to multiply by eight instead of nine, by doubling the number being subtracted; 8 × 27 = 270 − (2×27) = 270 − 54 = 216.
492:
156:
Repeat steps one and two with the second operand. At this point there are two single-digit numbers, the first derived from the first operand and the second derived from the second operand.
1420:
First two digits: 100-95=5 (Subtract the first number of the equation from 100) 97-5=92 (Subtract that answer from the second number of the equation)
568:
896:
1756:
328:
108:. Mental calculation often involves the use of specific techniques devised for specific types of problems. People with unusually high ability to perform mental calculations are called
2491:
is to the unknown, the more accurate the approximation. For instance, to estimate the square root of 15, one could start with the knowledge that the nearest perfect square is 16 (4).
3705:, Turkey, on 24–25 November 2012. 89 competitors from 20 countries participated. Awards and money prizes were given for 10 categories in total; of which 5 categories had to do about
2741:
2247:
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:
2858:
1921:
234:, 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.
3976:
Mehta, Ranjana K.; Nussbaum, Maury A.; Agnew, Michael J. (2012). "Muscle- and task-dependent responses to concurrent physical and mental workload during intermittent static work".
3027:
2900:
966:
636:
1233:
2703:
2663:
2961:
1430:
5+3=8 (Add the two single digits derived when calculating "Last two digits" in previous step) 100-8=92 (Subtract that answer from 100)
1683:
928:
3096:
3063:
1051:
1013:
3544:
To approximate a common logarithm (to at least one decimal point accuracy), a few logarithm rules, and the memorization of a few logarithms is required. One must know:
413:
372:
1261:
1187:
2746:
It may simplify mental calculation to notice that this method is equivalent to the mean of the known square and the unknown square, divided by the known square root:
1405:
Subtract both variables from 100 which will result in 2 one-digit number. The product of the 2 one-digit numbers will be the last two digits of one's final product.
1277:
Similarly, by adding instead of subtracting, the same methods can be used to multiply by 11 and 12, respectively (although simpler methods to multiply by 11 exist).
1285:
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.
2477:{\displaystyle {\text{root }}\simeq {\text{ known square root}}-{\frac {{\text{known square}}-{\text{unknown square}}}{2\times {\text{known square root}}}}\,}
2602:
2342:
82 = ? (is −18 from 100) 82 − 18 = 64 (subtract. First digits.) (−18) = 324 (second pair of digits. One will need to carry the 3.) 82 = 6724
1388:
Second example: 87 x 11 = 957 because 8 + 7 = 15 so the 5 goes in between the 8 and the 7 and the 1 is carried to the 8. So it is basically 857 + 100 = 957.
3890:
Brisswalter, J.; Arcelin, R.; Audiffren, M.; Delignieres, D. (1997). "Influence of
Physical Exercise on Simple Reaction Time: Effect of Physical Fitness".
290:
For any 2-digit by 2-digit multiplication problem, if both numbers end in five, the following algorithm can be used to quickly multiply them together:
1449:
15 − 2. Knowing that 15 is 225 and 2 is 4, simple subtraction shows that 225 − 4 = 221, which is the desired product.
159:
Apply the originally specified operation to the two condensed operands, and then apply the summing-of-digits procedure to the result of the operation.
3847:
Lambourne, Kate; Tomporowski, Phillip (2010). "The effect of exercise-induced arousal on cognitive task performance: A meta-regression analysis".
2590:{\displaystyle {\begin{aligned}{\text{root}}&\simeq 4-{\frac {16-15}{2\times 4}}\\&\simeq 4-0.125\\&\simeq 3.875\\\end{aligned}}\,\!}
740:
1402:
The product of two variables ranging from 90-99 will result in a 4-digit number. The first step is to find the ones-digit and the tens digit.
3499:
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.
1081:
35:
153:
If the resulting sum has two or more digits, sum those digits as in step one; repeat this step until the resulting sum has only one digit.
3165:
2801:{\displaystyle {\text{root }}\simeq {\frac {{\text{mean}}({\text{known square}},{\text{unknown square}})}{\text{known square root}}}\,}
1399:
This technique allows easy multiplication of numbers close and below 100.(90-99) The variables will be the two numbers one multiplies.
1299:
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.
642:
3243:
3794:
3638:
1391:
Or if 43 x 11 is equal to first 4+3=7 (For the tens digit) Then 4 is for the hundreds and 3 is for the tens. And the answer is 473.
1382:
If one has a two-digit number, take it and add the two numbers together and put that sum in the middle, and one can get the answer.
4119:"Developmental Changes in Mental Arithmetic: Evidence for Increased Functional Specialization in the Left Inferior Parietal Cortex"
2278:
and 100. For example, students who have memorized their squares from 1 to 24 can apply this method to any integer from 76 to 124.
1330:
Add the carried 1 to the highest valued digit in the multiplier, 7 + 1 = 8, and copy to the result to finish.
3722:
4166:
333:
As a preliminary step simply round the smaller number down and the larger up to the nearest multiple of ten. In this case:
2196:
For example, say one wants to square 48, which is 50 − 2. One subtracts 200 from 2500 and add 4, and get
3686:
3680:
3104:
4161:
1762:
424:
1618:
Take a given number, and add and subtract a certain value to it that will make it easier to multiply. For example:
1626:
492 is close to 500, which is easy to multiply by. Add and subtract 8 (the difference between 500 and 492) to get
511:
146:
and getting a result, the following procedure can be used to improve confidence in the correctness of the result:
3382:
Observe that there is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero:
851:
3643:
1689:
296:
3647:
4019:
2708:
2826:
2350:
This method is a straightforward extension of the explanation given above for squaring an integer near 100.
1853:
2972:
3630:
2869:
2628:
is their difference, it is more convenient to express this approximation in the form of mixed fraction as
934:
585:
78:
3341:
Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10:
1194:
2668:
1322:
Similarly add 7 + 5 = 12, then add the carried 1 to get 13. Place 3 to the result and carry the 1.
276:
3784:
3532:
This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11.
3329:
with 10 are somewhat easier since the digits are scrambled in consistent ways, as in the next section.
2631:
197:
The same procedure can be used with multiple operations, repeating steps 1 and 2 for each operation.
2919:
187:
Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14; 1 + 4 = 5
1361:
Another method is to simply multiply the number by 10, and add the original number to the result.
193:
5 = 5, so there is a good chance that the prediction that 6,338 × 79 equals 500,702 is right.
150:
Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0.
4105:
4001:
3958:
3915:
3872:
3809:
3706:
1648:
902:
3068:
3035:
2376:
This method is a straightforward extension of the explanation given above for integers near 10.
1024:
992:
242:
When the above situation does not apply, there is another method known as indirect calculation.
3933:
Murata, Atsuo (2005). "An
Attempt to Evaluate Mental Workload Using Wavelet Transform of EEG".
1292:
can be found by a series of additions to each of its digits from right to left, two at a time.
380:
339:
4140:
4097:
3993:
3950:
3907:
3864:
3790:
3732:
3659:
3655:
2383:
trailing digits) 407 + 7 = 414 414 × 4 = 1656 (leading digits; note this multiplication by
1240:
1166:
137:
109:
105:
1385:
For example: 24 x 11 = 264 because 2 + 4 = 6 and the 6 is placed in between the 2 and the 4.
4130:
4089:
3985:
3942:
3899:
3856:
2488:
3622:
162:
Sum the digits of the result that were originally obtained for the original calculation.
3634:
3321:
2398:
trailing digits) 79991 - 9 79982 × 8 = 639856 (leading digits) 79991 = 6398560081
46:
4155:
3876:
3727:
3718:
986:. F meaning first, O meaning outer, I meaning inner and L meaning last. For example:
4109:
4051:
4005:
3962:
3919:
4068:
4033:
3860:
173:
Assume the calculation 6,338 × 79, manually done, yielded a result of 500,702:
1931:
This method requires memorization of the squares of the one-digit numbers 1 to 9.
3989:
100:, with no help from any supplies (such as pencil and paper) or devices such as a
2412:
983:
97:
93:
3946:
3903:
3650:
is a widely known task using mental arithmetic. This procedure, mostly used in
3032:
If 'a' is close to the target, 'b' will be a small enough number to render the
4052:"Mental Calculation World Cup - the World Championship for Mental Calculators"
4034:"Mental Calculation World Cup - the World Championship for Mental Calculators"
3822:
3626:
1314:
Add 5 + 9 = 14 so 4 is placed on the left side of the result and carry the 1.
101:
90:
1235:
is the addition of the product of the outer digits and the inner digits; OI.
4135:
4118:
3651:
3315:
190:
Sum the digits of 500702: 5 + 0 + 0 + (7 + 0 + 2 = 9, which counts as 0) = 5
4144:
4101:
3997:
3954:
3868:
831:{\displaystyle 23\cdot 47=100(2\cdot 4)+10(3\cdot 4)+10(2\cdot 7)+3\cdot 7}
3911:
3654:
experiments, suggests mental subtraction is useful in testing the effects
3698:
3618:
3693:
Memoriad – World Memory, Mental
Calculation & Speed Reading Olympics
1306:
The ones digit of the multiplier, 9, is copied to the temporary result.
3737:
3702:
3326:
2364:
trailing digits) 9997 - 3 = 9994 (leading digits) 9997 = 99940009
1289:
143:
120:
82:
Mental calculation has long been a component of mathematical education.
3935:
Human
Factors: The Journal of the Human Factors and Ergonomics Society
2394:
79991 = ? (79991 is -9 from 80000) (-9) = 0081 (
1151:{\displaystyle a\cdot c\cdot 100+(a\cdot d+b\cdot c)\cdot 10+b\cdot d}
3757:
3570:
From this information, one can find the logarithm of any number 1-9.
2357:
trailing digits) 1012 + 12 = 1024 (leading digits) 1012 = 1024144
2346:
Squaring any integer near 10 (e.g., 976 to 1024, 9976 to 10024, etc.)
178:
Sum the digits of 6,338: (6 + 3 = 9, so count that as 0) + 3 + 8 = 11
4020:"Mental Calculations World Championship – World Mental Calculation"
2913:
is the difference between the known root and the answer one seeks.
2360:
9997 = ? (9997 is -3 from 10000) (-3) = 0009 (
1452:
This method requires knowing by heart a certain number of squares:
4093:
77:
3495:
If the perfect cube ends in 9, the cube root of it must end in 9.
3492:
If the perfect cube ends in 8, the cube root of it must end in 2.
3489:
If the perfect cube ends in 7, the cube root of it must end in 3.
3486:
If the perfect cube ends in 6, the cube root of it must end in 6.
3483:
If the perfect cube ends in 5, the cube root of it must end in 5.
3480:
If the perfect cube ends in 4, the cube root of it must end in 4.
3477:
If the perfect cube ends in 3, the cube root of it must end in 7.
3474:
If the perfect cube ends in 2, the cube root of it must end in 8.
3471:
If the perfect cube ends in 1, the cube root of it must end in 1.
3468:
If the perfect cube ends in 0, the cube root of it must end in 0.
3227:{\displaystyle \mathrm {root} \simeq a-{\frac {a^{2}-x}{2a}}\,\!}
2353:
1012 = ? (1012 is +12 from 1000) (+12) = 144 (
1189:
is the product of the first digit of each of the two numbers; F.
724:{\displaystyle =100(a\cdot c)+10(b\cdot c)+10(a\cdot d)+b\cdot d}
3621:
of the proper level can lead to an increase in performance of a
1263:
is the product of the last digit of each of the two numbers; L.
3299:{\displaystyle \mathrm {root} \simeq {\frac {a^{2}+x}{2a}}\,\!}
2220:
This method requires the memorization of squares from 1 to 24.
3789:. United States: CreateSpace Independent Publishing Platform.
1354:−3478 × 11 = 3 3+4+1(8) 4+7+1(2) 7+8(5) 8 = −38258
18:
4073:
2605:. Thus, one should try rounding the estimated answer down.
2266:
This method requires the memorization of squares from 1 to
58:
2665:. Thus, in the previous example, the square root of 15 is
2379:
407 = ? (407 is +7 from 400) (+7) = 49 (
119:
Many of these techniques take advantage of or rely on the
50:
2387:
was not needed for integers from 76 to 124 because their
1357:
62473 × 11 = 6 6+2(8) 2+4+1(7) 4+7+1(2) 7+3(0) 3 = 687203
979:
to which is it is easy to add 21: 281 and then 800: 1081
501:
is the tens unit of the original larger number (75) and t
3595:
log(8) = log(2 × 2 × 2) = log(2) + log(2) + log(2) ~ .90
3519:
After the last three digits are taken away, 456 remains.
54:
3065:
element of the equation negligible. Thus, one can drop
1981:, adding 0 to the end and finally adding the square of
42:
4117:
Rivera, S.M.; Reiss, AL; Eckert, MA; Menon, V (2005).
3823:"How to Do Cube Roots of 9 Digit Numbers in Your Head"
3642:
mental tasks were completed concurrently, with a more
2716:
2676:
2639:
505:
is the tens unit of the original smaller number (35).
3701:, Turkey, in 2008. The second Memoriad took place in
3246:
3168:
3107:
3071:
3038:
2975:
2922:
2872:
2829:
2755:
2711:
2671:
2634:
2500:
2424:
1856:
1765:
1692:
1651:
1349:
Note the handling of 9+1 as the highest valued digit.
1243:
1197:
1169:
1084:
1027:
995:
937:
905:
854:
743:
645:
588:
514:
427:
383:
342:
299:
2612:
is the closest perfect square to the desired square
2329:
93 − 7 = 86 (this gives the first two digits)
16:
Arithmetical calculations using only the human brain
2372:
10 (e.g., 276 to 324, 4976 to 5024, 79976 to 80024)
2332:(−7) = 49 (these are the second two digits)
2262:
Squaring an integer near 100 (e.g., from 76 to 124)
3298:
3226:
3148:
3090:
3057:
3021:
2955:
2894:
2852:
2800:
2735:
2697:
2657:
2589:
2476:
1915:
1841:
1750:
1677:
1255:
1227:
1181:
1150:
1045:
1007:
960:
922:
890:
830:
723:
630:
562:
486:
407:
366:
322:
3295:
3223:
3149:{\displaystyle b\simeq {\frac {a^{2}-x}{2a}}\,\!}
3145:
3018:
2952:
2891:
2849:
2586:
2185:. One knows that 50 is 2500. So one subtracts 100
230:are all smaller than the corresponding digits of
3522:456 is greater than all the cubes up to 7 cubed.
1942:being a two-digit integer, can be calculated as
1432:Now, the difference will be the first two digits
1422:Now, the difference will be the first two digits
3786:On a New Method of Multiplication and Shortcuts
2909:is a known root (4 from the above example) and
1842:{\displaystyle n^{2}=(n^{2}-an+an-a^{2})+a^{2}}
982:An easy mnemonic to remember for this would be
253:One place at a time is handled, left to right.
3646:at the higher level of physical workload. The
2415:of a number is to use the following equation:
2322:Another way to look at it would be like this:
1395:Multiplying two numbers close to and below 100
487:{\displaystyle (X\times Y)+50(t_{1}-t_{2})+25}
184:Sum the digits of 79: 7 + (9 counted as 0) = 7
142:After applying an arithmetic operation to two
3697:The first international Memoriad was held in
3633:, have been shown to be useful in indicating
3586:log(5) = log(10 / 2) = log(10) − log(2) ~ .70
3540:Approximating common logarithms (log base 10)
1346:999 × 11 = 9+1(10) 9+9+1(9) 9+9(8) 9 = 10989
8:
2603:inequality of arithmetic and geometric means
563:{\displaystyle =30\times 80+50(7-3)+25=2625}
3598:log(9) = log(3 × 3) = log(3) + log(3) ~ .96
3589:log(6) = log(2 × 3) = log(2) + log(3) ~ .78
3583:log(4) = log(2 × 2) = log(2) + log(2) ~ .60
891:{\displaystyle (a\cdot d+b\cdot c)\cdot 10}
3614:Mental arithmetic as a psychological skill
275:Many of these methods work because of the
4134:
3506:Therefore, the cube root of 29791 is 31.
3294:
3271:
3264:
3247:
3245:
3222:
3199:
3192:
3169:
3167:
3144:
3121:
3114:
3106:
3087:
3081:
3072:
3070:
3054:
3048:
3039:
3037:
3017:
3005:
2980:
2974:
2951:
2939:
2921:
2890:
2873:
2871:
2848:
2836:
2831:
2828:
2797:
2783:
2775:
2767:
2764:
2756:
2754:
2715:
2710:
2705:As another example, square root of 41 is
2675:
2670:
2638:
2633:
2585:
2523:
2505:
2501:
2499:
2473:
2465:
2452:
2444:
2441:
2433:
2425:
2423:
2032:Multiply this number by itself plus one:
2017:Take the digit(s) that precede the five:
1907:
1861:
1855:
1833:
1817:
1786:
1770:
1764:
1751:{\displaystyle n^{2}=(n^{2}-a^{2})+a^{2}}
1742:
1726:
1713:
1697:
1691:
1669:
1656:
1650:
1242:
1196:
1168:
1083:
1026:
994:
938:
936:
906:
904:
853:
742:
644:
587:
513:
469:
456:
426:
382:
341:
323:{\displaystyle \mathrm {Ex} :35\times 75}
300:
298:
3810:multiplying two numbers close, below 100
3685:The first Mental Calculation World Cup (
3343:
1454:
1343:−54 × −11 = 5 5+4(9) 4 = 594
3775:
3749:
2736:{\displaystyle 6{\tfrac {5}{12}}=6.416}
2326:93 = ? (is −7 from 100)
36:instructions, advice, or how-to content
3783:Cheprasov, Artem (September 3, 2009).
3666:Mental Calculations World Championship
3525:The first digit of the cube root is 7.
2853:{\displaystyle \mathrm {r} ^{2}=x\,\!}
1916:{\displaystyle n^{2}=(n-a)(n+a)+a^{2}}
3022:{\displaystyle a^{2}-2ab+b^{2}=x\,\!}
2158:Suppose one needs to square a number
7:
2895:{\displaystyle \mathrm {r} =a-b\,\!}
2743:while the actual value is 6.4031...
2200:= 2304. For numbers larger than 50 (
1302:A step-by-step example of 759 × 11:
1288:The product for any larger non-zero
961:{\displaystyle {}+a\cdot c\cdot 100}
631:{\displaystyle (10a+b)\cdot (10c+d)}
2274:is the absolute difference between
2231:is between 26 and 74 inclusive) is
1228:{\displaystyle (a\cdot d+b\cdot c)}
975:(12 + 14) = 26, 26 × 10 = 260,
45:so that it is more encyclopedic or
3310:Extracting roots of perfect powers
3257:
3254:
3251:
3248:
3179:
3176:
3173:
3170:
3098:out and rearrange the equation to
2874:
2832:
2698:{\displaystyle 4{\tfrac {-1}{8}}.}
2169: = 50 −
1572: + 1) −
841:800 +120 +140 + 21 ----- 1081
304:
301:
14:
3639:Michigan Technological University
3548:log(a × b) = log(a) + log(b)
2658:{\displaystyle n{\tfrac {d}{2n}}}
2216:Squaring an integer from 26 to 74
2154:Squaring numbers very close to 50
1584: + 1 −
4074:Memoriad - World Mental Olympics
3671:fractions, and calendar dates.
418:The algorithm reads as follows:
23:
3723:calculating the day of the week
2956:{\displaystyle (a-b)^{2}=x\,\!}
2411:An easy way to approximate the
2165:The number may be expressed as
1333:Final product of 759 × 11: 8349
574:Multiplying any 2-digit numbers
3861:10.1016/j.brainres.2010.03.091
3528:The cube root of 456533 is 77.
2936:
2923:
2820:is the square root of x, then
2788:
2772:
1897:
1885:
1882:
1870:
1823:
1779:
1732:
1706:
1428:Alternate for first two digits
1222:
1198:
1127:
1103:
879:
855:
813:
801:
792:
780:
771:
759:
706:
694:
685:
673:
664:
652:
625:
610:
604:
589:
545:
533:
475:
449:
440:
428:
1:
3513:Find the cube root of 456533.
2227:(most easily calculated when
2043:Take above result and attach
2012:Squaring a number ending in 5
4069:Mental Calculation World Cup
3990:10.1080/00140139.2012.703695
3687:Mental Calculation World Cup
3681:Mental Calculation World Cup
3675:Mental Calculation World Cup
3551:log(a / b) = log(a) - log(b)
2863:One then redefines the root
2204: = 50 +
1927:Squaring any 2-digit integer
181:Iterate as needed: 1 + 1 = 2
3892:Perceptual and Motor Skills
2212:instead of subtracting it.
1678:{\displaystyle n^{2}=n^{2}}
923:{\displaystyle {}+b\cdot d}
260:Hence, the result is 2231.
4183:
3947:10.1518/001872005774860096
3904:10.2466/pms.1997.85.3.1019
3678:
3313:
3091:{\displaystyle {}+b^{2}\,}
3058:{\displaystyle {}+b^{2}\,}
2407:Approximating square roots
2368:Squaring any integer near
1606: − 1) + (2
1046:{\displaystyle ab\cdot cd}
1008:{\displaystyle 75\cdot 23}
286:The "Ends of Five" Formula
135:
2309:100(100 − 2(7)) + 7
1995:= 10 × 2(23 + 3) + 3
408:{\displaystyle 75+5=80=Y}
367:{\displaystyle 35-5=30=X}
210:Calculating differences:
3648:Brown–Peterson procedure
3601:log(10) = 1 + log(1) = 1
3516:The cube root ends in 7.
2189:from 2500, and then add
2078:Mathematical explanation
1256:{\displaystyle b\cdot d}
1182:{\displaystyle a\cdot c}
246:Look-ahead borrow method
3629:measures, specifically
3237:that can be reduced to
2435: known square root
1969:can be found by adding
3300:
3228:
3150:
3092:
3059:
3023:
2957:
2896:
2854:
2802:
2737:
2699:
2659:
2591:
2478:
2050:Example: 85 × 85
1965:Meaning the square of
1917:
1843:
1752:
1679:
1610: − 1)
1257:
1229:
1183:
1152:
1047:
1009:
971:i.e., in this example
962:
924:
892:
832:
725:
632:
564:
488:
409:
368:
324:
264:Calculating products:
127:Methods and techniques
83:
4167:Games of mental skill
4136:10.1093/cercor/bhi055
4038:www.recordholders.org
3658:can have on how long
3656:maintenance rehearsal
3554:log(0) does not exist
3333:Extracting cube roots
3301:
3229:
3151:
3093:
3060:
3024:
2958:
2897:
2855:
2803:
2738:
2700:
2660:
2592:
2487:The closer the known
2479:
2173:so its square is (50−
1998:= 10 × 2(26) + 9
1918:
1844:
1753:
1680:
1258:
1230:
1184:
1153:
1048:
1010:
963:
925:
893:
833:
726:
633:
565:
489:
410:
369:
325:
277:distributive property
106:a competitive context
81:
3320:Extracting roots of
3244:
3166:
3105:
3069:
3036:
2973:
2920:
2870:
2827:
2753:
2709:
2669:
2632:
2498:
2422:
1854:
1763:
1690:
1649:
1440:Using square numbers
1241:
1195:
1167:
1082:
1025:
993:
935:
903:
852:
741:
643:
586:
512:
425:
381:
340:
297:
238:Indirect calculation
114:lightning calculator
4082:Nature Neuroscience
2391:= 1) 407 = 165649
1642:The proof follows:
1614:Squaring any number
1435:Final Product- 9215
1425:Final Product- 9215
1418:Final Product- yx15
1379:One last easy way:
226:When the digits of
43:rewrite the content
4162:Mental calculation
3707:Mental Calculation
3644:significant effect
3296:
3224:
3146:
3088:
3055:
3019:
2953:
2892:
2850:
2816:By definition, if
2798:
2733:
2725:
2695:
2690:
2655:
2653:
2587:
2583:
2474:
2070:12 × 13 = 156
1913:
1839:
1748:
1675:
1630:492 -> 484, 500
1339:Further examples:
1253:
1225:
1179:
1148:
1043:
1005:
958:
920:
888:
828:
721:
628:
560:
484:
405:
364:
320:
222:Direct calculation
110:mental calculators
87:Mental calculation
84:
3821:Dorrell, Philip.
3733:Mental calculator
3660:short-term memory
3619:Physical exertion
3509:Another example:
3380:
3379:
3292:
3220:
3142:
2966:Expanding yields
2795:
2794:
2793:known square root
2786:
2778:
2770:
2759:
2724:
2689:
2652:
2547:
2508:
2471:
2468:
2467:known square root
2455:
2447:
2436:
2428:
2339:Another example:
2151:
2150:
1988:For example, 23:
1568:This is because (
1557:
1556:
1281:Multiplying by 11
138:Casting out nines
132:Casting out nines
123:numeral system.
76:
75:
4174:
4148:
4138:
4113:
4056:
4055:
4048:
4042:
4041:
4030:
4024:
4023:
4016:
4010:
4009:
3973:
3967:
3966:
3930:
3924:
3923:
3887:
3881:
3880:
3844:
3838:
3837:
3835:
3833:
3818:
3812:
3807:
3801:
3800:
3780:
3763:
3760:
3754:
3344:
3305:
3303:
3302:
3297:
3293:
3291:
3283:
3276:
3275:
3265:
3260:
3233:
3231:
3230:
3225:
3221:
3219:
3211:
3204:
3203:
3193:
3182:
3155:
3153:
3152:
3147:
3143:
3141:
3133:
3126:
3125:
3115:
3097:
3095:
3094:
3089:
3086:
3085:
3073:
3064:
3062:
3061:
3056:
3053:
3052:
3040:
3028:
3026:
3025:
3020:
3010:
3009:
2985:
2984:
2962:
2960:
2959:
2954:
2944:
2943:
2901:
2899:
2898:
2893:
2877:
2859:
2857:
2856:
2851:
2841:
2840:
2835:
2807:
2805:
2804:
2799:
2796:
2792:
2791:
2787:
2784:
2779:
2776:
2771:
2768:
2765:
2760:
2757:
2742:
2740:
2739:
2734:
2726:
2717:
2704:
2702:
2701:
2696:
2691:
2685:
2677:
2664:
2662:
2661:
2656:
2654:
2651:
2640:
2596:
2594:
2593:
2588:
2584:
2571:
2552:
2548:
2546:
2535:
2524:
2509:
2506:
2483:
2481:
2480:
2475:
2472:
2470:
2469:
2466:
2457:
2456:
2453:
2448:
2445:
2442:
2437:
2434:
2429:
2426:
2084:
2083:
2073:So, 125 = 15,625
1977:, multiplied by
1922:
1920:
1919:
1914:
1912:
1911:
1866:
1865:
1848:
1846:
1845:
1840:
1838:
1837:
1822:
1821:
1791:
1790:
1775:
1774:
1757:
1755:
1754:
1749:
1747:
1746:
1731:
1730:
1718:
1717:
1702:
1701:
1684:
1682:
1681:
1676:
1674:
1673:
1661:
1660:
1596: + 1)
1560:Squaring numbers
1455:
1267:Multiplying by 9
1262:
1260:
1259:
1254:
1234:
1232:
1231:
1226:
1188:
1186:
1185:
1180:
1157:
1155:
1154:
1149:
1052:
1050:
1049:
1044:
1014:
1012:
1011:
1006:
967:
965:
964:
959:
939:
929:
927:
926:
921:
907:
897:
895:
894:
889:
837:
835:
834:
829:
730:
728:
727:
722:
637:
635:
634:
629:
569:
567:
566:
561:
493:
491:
490:
485:
474:
473:
461:
460:
414:
412:
411:
406:
373:
371:
370:
365:
329:
327:
326:
321:
307:
71:
68:
62:
27:
26:
19:
4182:
4181:
4177:
4176:
4175:
4173:
4172:
4171:
4152:
4151:
4129:(11): 1779–90.
4123:Cerebral Cortex
4116:
4078:
4065:
4060:
4059:
4050:
4049:
4045:
4032:
4031:
4027:
4018:
4017:
4013:
3984:(10): 1166–79.
3975:
3974:
3970:
3932:
3931:
3927:
3889:
3888:
3884:
3846:
3845:
3841:
3831:
3829:
3820:
3819:
3815:
3808:
3804:
3797:
3782:
3781:
3777:
3772:
3767:
3766:
3758:
3755:
3751:
3746:
3715:
3695:
3683:
3677:
3668:
3635:mental workload
3616:
3542:
3335:
3318:
3312:
3284:
3267:
3266:
3242:
3241:
3212:
3195:
3194:
3164:
3163:
3134:
3117:
3116:
3103:
3102:
3077:
3067:
3066:
3044:
3034:
3033:
3001:
2976:
2971:
2970:
2935:
2918:
2917:
2868:
2867:
2830:
2825:
2824:
2814:
2766:
2751:
2750:
2707:
2706:
2678:
2667:
2666:
2644:
2630:
2629:
2582:
2581:
2569:
2568:
2550:
2549:
2536:
2525:
2510:
2496:
2495:
2458:
2443:
2420:
2419:
2409:
2404:
2399:
2392:
2374:
2365:
2358:
2348:
2343:
2312:= 100 × 86 + 49
2264:
2218:
2156:
2056:8 × 9 = 72
2014:
1929:
1903:
1857:
1852:
1851:
1829:
1813:
1782:
1766:
1761:
1760:
1738:
1722:
1709:
1693:
1688:
1687:
1665:
1652:
1647:
1646:
1616:
1562:
1442:
1437:
1397:
1373:170 + 17 = 187
1371:
1283:
1269:
1239:
1238:
1193:
1192:
1165:
1164:
1080:
1079:
1023:
1022:
991:
990:
933:
932:
901:
900:
850:
849:
842:
739:
738:
641:
640:
584:
583:
576:
510:
509:
504:
500:
465:
452:
423:
422:
379:
378:
338:
337:
295:
294:
288:
282:
273:
258:
248:
240:
224:
219:
203:
140:
134:
129:
96:using only the
72:
66:
63:
40:
28:
24:
17:
12:
11:
5:
4180:
4178:
4170:
4169:
4164:
4154:
4153:
4150:
4149:
4114:
4076:
4071:
4064:
4063:External links
4061:
4058:
4057:
4043:
4025:
4011:
3968:
3941:(3): 498–508.
3925:
3898:(3): 1019–27.
3882:
3849:Brain Research
3839:
3813:
3802:
3795:
3774:
3773:
3771:
3768:
3765:
3764:
3748:
3747:
3745:
3742:
3741:
3740:
3735:
3730:
3725:
3714:
3711:
3694:
3691:
3679:Main article:
3676:
3673:
3667:
3664:
3615:
3612:
3603:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3568:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3541:
3538:
3530:
3529:
3526:
3523:
3520:
3517:
3514:
3497:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3461:
3460:
3453:
3446:
3439:
3432:
3425:
3418:
3411:
3404:
3397:
3390:
3378:
3377:
3374:
3371:
3368:
3365:
3361:
3360:
3357:
3354:
3351:
3348:
3334:
3331:
3322:perfect powers
3311:
3308:
3307:
3306:
3290:
3287:
3282:
3279:
3274:
3270:
3263:
3259:
3256:
3253:
3250:
3235:
3234:
3218:
3215:
3210:
3207:
3202:
3198:
3191:
3188:
3185:
3181:
3178:
3175:
3172:
3159:and therefore
3157:
3156:
3140:
3137:
3132:
3129:
3124:
3120:
3113:
3110:
3084:
3080:
3076:
3051:
3047:
3043:
3030:
3029:
3016:
3013:
3008:
3004:
3000:
2997:
2994:
2991:
2988:
2983:
2979:
2964:
2963:
2950:
2947:
2942:
2938:
2934:
2931:
2928:
2925:
2903:
2902:
2889:
2886:
2883:
2880:
2876:
2861:
2860:
2847:
2844:
2839:
2834:
2813:
2810:
2809:
2808:
2790:
2785:unknown square
2782:
2774:
2763:
2732:
2729:
2723:
2720:
2714:
2694:
2688:
2684:
2681:
2674:
2650:
2647:
2643:
2637:
2598:
2597:
2580:
2577:
2574:
2572:
2570:
2567:
2564:
2561:
2558:
2555:
2553:
2551:
2545:
2542:
2539:
2534:
2531:
2528:
2522:
2519:
2516:
2513:
2511:
2504:
2503:
2485:
2484:
2464:
2461:
2454:unknown square
2451:
2440:
2432:
2408:
2405:
2403:
2400:
2393:
2378:
2373:
2366:
2359:
2352:
2347:
2344:
2341:
2337:
2336:
2333:
2330:
2327:
2320:
2319:
2316:
2313:
2310:
2302:
2301:
2281:The square of
2263:
2260:
2259:
2258:
2255:
2252:
2245:
2244:
2223:The square of
2217:
2214:
2155:
2152:
2149:
2148:
2137:
2134:
2133:
2122:
2119:
2118:
2107:
2104:
2103:
2092:
2082:
2081:
2080:
2079:
2076:
2075:
2074:
2071:
2068:
2062:
2061:
2060:
2059:So, 85 = 7,225
2057:
2054:
2041:
2030:
2013:
2010:
2006:
2005:
2002:
1999:
1996:
1993:
1963:
1962:
1934:The square of
1928:
1925:
1924:
1923:
1910:
1906:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1864:
1860:
1849:
1836:
1832:
1828:
1825:
1820:
1816:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1789:
1785:
1781:
1778:
1773:
1769:
1758:
1745:
1741:
1737:
1734:
1729:
1725:
1721:
1716:
1712:
1708:
1705:
1700:
1696:
1685:
1672:
1668:
1664:
1659:
1655:
1640:
1639:
1632:
1631:
1624:
1623:
1615:
1612:
1592: + (
1580: + 2
1561:
1558:
1555:
1554:
1551:
1548:
1545:
1542:
1539:
1535:
1534:
1531:
1528:
1525:
1522:
1519:
1515:
1514:
1511:
1508:
1505:
1502:
1499:
1495:
1494:
1491:
1488:
1485:
1482:
1479:
1475:
1474:
1471:
1468:
1465:
1462:
1459:
1441:
1438:
1414:
1396:
1393:
1376:17 × 11 = 187
1370:17 × 10 = 170
1369:
1359:
1358:
1355:
1352:
1351:
1350:
1344:
1337:
1336:
1335:
1334:
1328:
1327:
1326:
1320:
1319:
1318:
1312:
1311:
1310:
1282:
1279:
1268:
1265:
1252:
1249:
1246:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1178:
1175:
1172:
1159:
1158:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1054:
1053:
1042:
1039:
1036:
1033:
1030:
1016:
1015:
1004:
1001:
998:
977:
976:
969:
968:
957:
954:
951:
948:
945:
942:
930:
919:
916:
913:
910:
898:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
840:
839:
838:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
732:
731:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
638:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
575:
572:
571:
570:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
502:
498:
495:
494:
483:
480:
477:
472:
468:
464:
459:
455:
451:
448:
445:
442:
439:
436:
433:
430:
416:
415:
404:
401:
398:
395:
392:
389:
386:
375:
374:
363:
360:
357:
354:
351:
348:
345:
331:
330:
319:
316:
313:
310:
306:
303:
287:
284:
272:
262:
255:
247:
244:
239:
236:
223:
220:
218:
208:
202:
199:
195:
194:
191:
188:
185:
182:
179:
175:
174:
167:
166:
163:
160:
157:
154:
151:
136:Main article:
133:
130:
128:
125:
74:
73:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4179:
4168:
4165:
4163:
4160:
4159:
4157:
4146:
4142:
4137:
4132:
4128:
4124:
4120:
4115:
4111:
4107:
4103:
4099:
4095:
4094:10.1038/82831
4091:
4087:
4083:
4077:
4075:
4072:
4070:
4067:
4066:
4062:
4053:
4047:
4044:
4039:
4035:
4029:
4026:
4021:
4015:
4012:
4007:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3972:
3969:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3929:
3926:
3921:
3917:
3913:
3909:
3905:
3901:
3897:
3893:
3886:
3883:
3878:
3874:
3870:
3866:
3862:
3858:
3854:
3850:
3843:
3840:
3828:
3827:Thinking Hard
3824:
3817:
3814:
3811:
3806:
3803:
3798:
3796:9781448689330
3792:
3788:
3787:
3779:
3776:
3769:
3761:
3753:
3750:
3743:
3739:
3736:
3734:
3731:
3729:
3728:Mental abacus
3726:
3724:
3720:
3719:Doomsday rule
3717:
3716:
3712:
3710:
3708:
3704:
3700:
3692:
3690:
3688:
3682:
3674:
3672:
3665:
3663:
3661:
3657:
3653:
3649:
3645:
3640:
3636:
3632:
3628:
3627:Physiological
3624:
3620:
3613:
3611:
3607:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3572:
3571:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3546:
3545:
3539:
3537:
3533:
3527:
3524:
3521:
3518:
3515:
3512:
3511:
3510:
3507:
3504:
3500:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3466:
3465:
3458:
3454:
3451:
3447:
3444:
3440:
3437:
3433:
3430:
3426:
3423:
3419:
3416:
3412:
3409:
3405:
3402:
3398:
3395:
3391:
3389:
3385:
3384:
3383:
3375:
3372:
3369:
3366:
3363:
3362:
3358:
3355:
3352:
3349:
3346:
3345:
3342:
3339:
3332:
3330:
3328:
3323:
3317:
3309:
3288:
3285:
3280:
3277:
3272:
3268:
3261:
3240:
3239:
3238:
3216:
3213:
3208:
3205:
3200:
3196:
3189:
3186:
3183:
3162:
3161:
3160:
3138:
3135:
3130:
3127:
3122:
3118:
3111:
3108:
3101:
3100:
3099:
3082:
3078:
3074:
3049:
3045:
3041:
3014:
3011:
3006:
3002:
2998:
2995:
2992:
2989:
2986:
2981:
2977:
2969:
2968:
2967:
2948:
2945:
2940:
2932:
2929:
2926:
2916:
2915:
2914:
2912:
2908:
2887:
2884:
2881:
2878:
2866:
2865:
2864:
2845:
2842:
2837:
2823:
2822:
2821:
2819:
2811:
2780:
2761:
2749:
2748:
2747:
2744:
2730:
2727:
2721:
2718:
2712:
2692:
2686:
2682:
2679:
2672:
2648:
2645:
2641:
2635:
2627:
2623:
2619:
2615:
2611:
2608:Note that if
2606:
2604:
2578:
2575:
2573:
2565:
2562:
2559:
2556:
2554:
2543:
2540:
2537:
2532:
2529:
2526:
2520:
2517:
2514:
2512:
2494:
2493:
2492:
2490:
2462:
2459:
2449:
2438:
2430:
2418:
2417:
2416:
2414:
2406:
2402:Finding roots
2401:
2397:
2390:
2386:
2382:
2377:
2371:
2367:
2363:
2356:
2351:
2345:
2340:
2334:
2331:
2328:
2325:
2324:
2323:
2317:
2314:
2311:
2308:
2307:
2306:
2300:
2296:
2292:
2291:
2290:
2288:
2285:(i.e., 100 ±
2284:
2279:
2277:
2273:
2269:
2261:
2256:
2254:= 144 + 3,700
2253:
2250:
2249:
2248:
2242:
2238:
2234:
2233:
2232:
2230:
2226:
2221:
2215:
2213:
2211:
2207:
2203:
2199:
2194:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2163:
2161:
2153:
2146:
2142:
2138:
2136:
2135:
2131:
2127:
2123:
2121:
2120:
2116:
2112:
2108:
2106:
2105:
2101:
2097:
2093:
2090:
2086:
2085:
2077:
2072:
2069:
2066:
2065:
2064:Example: 125
2063:
2058:
2055:
2052:
2051:
2049:
2048:
2046:
2042:
2039:
2035:
2031:
2028:
2024:
2020:
2016:
2015:
2011:
2009:
2008:So 23 = 529.
2003:
2000:
1997:
1994:
1991:
1990:
1989:
1986:
1984:
1980:
1976:
1972:
1968:
1961:
1957:
1953:
1949:
1945:
1944:
1943:
1941:
1937:
1932:
1926:
1908:
1904:
1900:
1894:
1891:
1888:
1879:
1876:
1873:
1867:
1862:
1858:
1850:
1834:
1830:
1826:
1818:
1814:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1787:
1783:
1776:
1771:
1767:
1759:
1743:
1739:
1735:
1727:
1723:
1719:
1714:
1710:
1703:
1698:
1694:
1686:
1670:
1666:
1662:
1657:
1653:
1645:
1644:
1643:
1638:492 = 242,064
1637:
1636:
1635:
1629:
1628:
1627:
1621:
1620:
1619:
1613:
1611:
1609:
1605:
1601:
1597:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1566:
1559:
1552:
1549:
1546:
1543:
1540:
1537:
1536:
1532:
1529:
1526:
1523:
1520:
1517:
1516:
1512:
1509:
1506:
1503:
1500:
1497:
1496:
1492:
1489:
1486:
1483:
1480:
1477:
1476:
1472:
1469:
1466:
1463:
1460:
1457:
1456:
1453:
1450:
1448:
1439:
1436:
1433:
1429:
1426:
1423:
1419:
1413:
1410:
1406:
1403:
1400:
1394:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1368:
1365:
1364:For example:
1362:
1356:
1353:
1348:
1347:
1345:
1342:
1341:
1340:
1332:
1331:
1329:
1324:
1323:
1321:
1316:
1315:
1313:
1308:
1307:
1305:
1304:
1303:
1300:
1297:
1293:
1291:
1286:
1280:
1278:
1275:
1272:
1266:
1264:
1250:
1247:
1244:
1236:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1190:
1176:
1173:
1170:
1162:
1145:
1142:
1139:
1136:
1133:
1130:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1100:
1097:
1094:
1091:
1088:
1085:
1078:
1077:
1076:
1073:
1071:
1067:
1063:
1059:
1040:
1037:
1034:
1031:
1028:
1021:
1020:
1019:
1002:
999:
996:
989:
988:
987:
985:
980:
974:
973:
972:
955:
952:
949:
946:
943:
940:
931:
917:
914:
911:
908:
899:
885:
882:
876:
873:
870:
867:
864:
861:
858:
848:
847:
846:
825:
822:
819:
816:
810:
807:
804:
798:
795:
789:
786:
783:
777:
774:
768:
765:
762:
756:
753:
750:
747:
744:
737:
736:
735:
734:For example,
718:
715:
712:
709:
703:
700:
697:
691:
688:
682:
679:
676:
670:
667:
661:
658:
655:
649:
646:
639:
622:
619:
616:
613:
607:
601:
598:
595:
592:
582:
581:
580:
573:
557:
554:
551:
548:
542:
539:
536:
530:
527:
524:
521:
518:
515:
508:
507:
506:
481:
478:
470:
466:
462:
457:
453:
446:
443:
437:
434:
431:
421:
420:
419:
402:
399:
396:
393:
390:
387:
384:
377:
376:
361:
358:
355:
352:
349:
346:
343:
336:
335:
334:
317:
314:
311:
308:
293:
292:
291:
285:
283:
280:
278:
271:
267:
263:
261:
254:
251:
245:
243:
237:
235:
233:
229:
221:
217:
213:
209:
207:
200:
198:
192:
189:
186:
183:
180:
177:
176:
172:
171:
170:
164:
161:
158:
155:
152:
149:
148:
147:
145:
139:
131:
126:
124:
122:
117:
115:
111:
107:
103:
99:
95:
92:
88:
80:
70:
67:February 2017
60:
56:
52:
48:
44:
38:
37:
32:This article
30:
21:
20:
4126:
4122:
4088:(1): 103–7.
4085:
4081:
4046:
4037:
4028:
4014:
3981:
3977:
3971:
3938:
3934:
3928:
3895:
3891:
3885:
3852:
3848:
3842:
3830:. Retrieved
3826:
3816:
3805:
3785:
3778:
3752:
3696:
3684:
3669:
3617:
3608:
3604:
3592:log(7) ~ .85
3580:log(3) ~ .48
3577:log(2) ~ .30
3569:
3566:log(7) ~ .85
3563:log(3) ~ .48
3560:log(2) ~ .30
3543:
3534:
3531:
3508:
3505:
3501:
3498:
3462:
3456:
3449:
3442:
3435:
3428:
3421:
3414:
3407:
3400:
3393:
3387:
3381:
3340:
3336:
3319:
3236:
3158:
3031:
2965:
2910:
2906:
2904:
2862:
2817:
2815:
2777:known square
2745:
2625:
2621:
2617:
2613:
2609:
2607:
2599:
2486:
2446:known square
2410:
2395:
2388:
2384:
2380:
2375:
2369:
2361:
2354:
2349:
2338:
2321:
2315:= 8,600 + 49
2303:
2298:
2294:
2286:
2282:
2280:
2275:
2271:
2267:
2265:
2246:
2240:
2236:
2235:(50 −
2228:
2224:
2222:
2219:
2209:
2205:
2201:
2197:
2195:
2190:
2186:
2182:
2178:
2177:) = 50 − 100
2174:
2170:
2166:
2164:
2159:
2157:
2144:
2140:
2129:
2125:
2114:
2110:
2099:
2095:
2088:
2044:
2037:
2033:
2026:
2022:
2018:
2007:
1987:
1982:
1978:
1974:
1970:
1966:
1964:
1959:
1955:
1951:
1947:
1939:
1935:
1933:
1930:
1641:
1633:
1625:
1617:
1607:
1603:
1599:
1598:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1567:
1563:
1451:
1446:
1443:
1434:
1431:
1427:
1424:
1421:
1417:
1411:
1407:
1404:
1401:
1398:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1366:
1363:
1360:
1338:
1301:
1298:
1294:
1287:
1284:
1276:
1273:
1270:
1237:
1191:
1163:
1160:
1074:
1069:
1065:
1061:
1057:
1055:
1017:
981:
978:
970:
843:
733:
577:
496:
417:
332:
289:
281:
274:
269:
265:
259:
252:
249:
241:
231:
227:
225:
215:
211:
204:
196:
168:
141:
118:
113:
94:calculations
91:arithmetical
89:consists of
86:
85:
64:
41:Please help
33:
3623:mental task
2413:square root
2293:100(100 ± 2
2243:− 25)
2208:), add 100×
2047:to the end
1325:result: 349
1056:where 7 is
98:human brain
51:Wikiversity
4156:Categories
3978:Ergonomics
3770:References
3574:log(1) = 0
3557:log(1) = 0
3376:10 = 1000
3314:See also:
2812:Derivation
2758:root
2427:root
2147:+ 1) + 25
2029:are digits
1946:10 ×
1317:result: 49
102:calculator
59:Wikivoyage
3877:206324098
3855:: 12–24.
3652:cognitive
3338:= 74088.
3316:13th root
3262:≃
3206:−
3190:−
3184:≃
3128:−
3112:≃
2987:−
2930:−
2885:−
2762:≃
2680:−
2576:≃
2563:−
2557:≃
2541:×
2530:−
2521:−
2515:≃
2463:×
2450:−
2439:−
2431:≃
2335:93 = 8649
2162:near 50.
2001:= 520 + 9
1877:−
1811:−
1793:−
1720:−
1553:30 = 900
1533:29 = 841
1513:28 = 784
1493:27 = 729
1473:26 = 676
1412:Example:
1309:result: 9
1248:⋅
1217:⋅
1205:⋅
1174:⋅
1143:⋅
1131:⋅
1122:⋅
1110:⋅
1095:⋅
1089:⋅
1075:Consider
1068:and 3 is
1035:⋅
1000:⋅
953:⋅
947:⋅
915:⋅
883:⋅
874:⋅
862:⋅
823:⋅
808:⋅
787:⋅
766:⋅
748:⋅
716:⋅
701:⋅
680:⋅
659:⋅
608:⋅
540:−
522:×
463:−
435:×
347:−
315:×
55:Wikibooks
34:contains
4145:15716474
4110:23829063
4102:11135652
4006:38648671
3998:22849301
3963:25313835
3955:16435692
3920:30781628
3869:20381468
3713:See also
3699:Istanbul
3455:10 = 100
3359:5 = 125
2251:(−12) +
2239:) + 100(
2021:, where
1550:25 = 625
1547:20 = 400
1544:15 = 225
1541:10 = 100
1530:24 = 576
1527:19 = 361
1524:14 = 196
1510:23 = 529
1507:18 = 324
1504:13 = 169
1490:22 = 484
1487:17 = 289
1484:12 = 144
1470:21 = 441
1467:16 = 256
1464:11 = 121
1367:17 × 11
214:−
169:Example
144:operands
3912:9399313
3832:19 July
3738:Soroban
3703:Antalya
3662:lasts.
3373:9 = 729
3370:8 = 512
3367:7 = 343
3364:6 = 216
3327:coprime
2318:= 8,649
2257:= 3,844
2132:) + 25
2098:+ 5)(10
1290:integer
1064:, 2 is
1060:, 5 is
497:Where t
201:Factors
121:decimal
4143:
4108:
4100:
4004:
3996:
3961:
3953:
3918:
3910:
3875:
3867:
3793:
3452:down 3
3448:9 = 72
3445:down 1
3441:8 = 51
3438:down 3
3434:7 = 34
3427:6 = 21
3420:5 = 12
3417:down 3
3410:down 1
3403:down 3
3356:4 = 64
3353:3 = 27
2905:where
2489:square
2270:where
2124:= 100(
2023:a, b,
1538:5 = 25
1521:9 = 81
1518:4 = 16
1501:8 = 64
1481:7 = 49
1461:6 = 36
49:it to
4106:S2CID
4002:S2CID
3959:S2CID
3916:S2CID
3873:S2CID
3759:mod 9
3744:Notes
3413:4 = 6
3406:3 = 2
3350:2 = 8
3347:1 = 1
2731:6.416
2579:3.875
2566:0.125
2289:) is
2139:= 100
2117:+ 25
2113:+ 100
2109:= 100
2102:+ 5)
2094:= (10
2004:= 529
1498:3 = 9
1478:2 = 4
1458:1 = 1
57:, or
4141:PMID
4098:PMID
3994:PMID
3951:PMID
3908:PMID
3865:PMID
3853:1341
3834:2015
3791:ISBN
3721:for
3459:up 1
3431:up 1
3424:up 1
3399:2 =
3396:up 1
3392:1 =
3386:0 =
2769:mean
2616:and
2507:root
2297:) +
2091:+ 5)
2040:+ 1)
2025:and
2019:abc5
1958:) +
1447:i.e.
1018:and
984:FOIL
558:2625
47:move
4131:doi
4090:doi
3986:doi
3943:doi
3900:doi
3857:doi
3631:EEG
2370:m ×
2087:(10
2038:abc
2034:abc
1973:to
1622:492
1602:= (
1098:100
956:100
757:100
650:100
116:s.
112:or
4158::
4139:.
4127:15
4125:.
4121:.
4104:.
4096:.
4084:.
4036:.
4000:.
3992:.
3982:55
3980:.
3957:.
3949:.
3939:47
3937:.
3914:.
3906:.
3896:85
3894:.
3871:.
3863:.
3851:.
3825:.
2722:12
2624:-
2620:=
2533:15
2527:16
2193:.
2181:+
2128:+
2067:12
2045:25
1992:23
1985:.
1975:mn
1967:mn
1954:+
1952:mn
1940:mn
1938:,
1936:mn
1588:=
1576:=
1134:10
1072:.
1003:23
997:75
886:10
799:10
778:10
751:47
745:23
692:10
671:10
614:10
593:10
552:25
531:50
525:80
519:30
482:25
447:50
397:80
385:75
356:30
344:35
318:75
312:35
279:.
268:×
53:,
4147:.
4133::
4112:.
4092::
4086:4
4054:.
4040:.
4022:.
4008:.
3988::
3965:.
3945::
3922:.
3902::
3879:.
3859::
3836:.
3799:.
3762:.
3457:0
3450:9
3443:2
3436:3
3429:6
3422:5
3415:4
3408:7
3401:8
3394:1
3388:0
3289:a
3286:2
3281:x
3278:+
3273:2
3269:a
3258:t
3255:o
3252:o
3249:r
3217:a
3214:2
3209:x
3201:2
3197:a
3187:a
3180:t
3177:o
3174:o
3171:r
3139:a
3136:2
3131:x
3123:2
3119:a
3109:b
3083:2
3079:b
3075:+
3050:2
3046:b
3042:+
3015:x
3012:=
3007:2
3003:b
2999:+
2996:b
2993:a
2990:2
2982:2
2978:a
2949:x
2946:=
2941:2
2937:)
2933:b
2927:a
2924:(
2911:b
2907:a
2888:b
2882:a
2879:=
2875:r
2846:x
2843:=
2838:2
2833:r
2818:r
2789:)
2781:,
2773:(
2728:=
2719:5
2713:6
2693:.
2687:8
2683:1
2673:4
2649:n
2646:2
2642:d
2636:n
2626:n
2622:x
2618:d
2614:x
2610:n
2560:4
2544:4
2538:2
2518:4
2460:2
2396:n
2389:m
2385:m
2381:n
2362:n
2355:n
2299:a
2295:a
2287:a
2283:n
2276:n
2272:a
2268:a
2241:n
2237:n
2229:n
2225:n
2210:a
2206:a
2202:n
2198:n
2191:a
2187:a
2183:a
2179:a
2175:a
2171:a
2167:n
2160:n
2145:x
2143:(
2141:x
2130:x
2126:x
2115:x
2111:x
2100:x
2096:x
2089:x
2053:8
2036:(
2027:c
1983:n
1979:m
1971:n
1960:n
1956:n
1950:(
1948:m
1909:2
1905:a
1901:+
1898:)
1895:a
1892:+
1889:n
1886:(
1883:)
1880:a
1874:n
1871:(
1868:=
1863:2
1859:n
1835:2
1831:a
1827:+
1824:)
1819:2
1815:a
1808:n
1805:a
1802:+
1799:n
1796:a
1788:2
1784:n
1780:(
1777:=
1772:2
1768:n
1744:2
1740:a
1736:+
1733:)
1728:2
1724:a
1715:2
1711:n
1707:(
1704:=
1699:2
1695:n
1671:2
1667:n
1663:=
1658:2
1654:n
1608:x
1604:x
1600:x
1594:x
1590:x
1586:x
1582:x
1578:x
1574:x
1570:x
1251:d
1245:b
1223:)
1220:c
1214:b
1211:+
1208:d
1202:a
1199:(
1177:c
1171:a
1146:d
1140:b
1137:+
1128:)
1125:c
1119:b
1116:+
1113:d
1107:a
1104:(
1101:+
1092:c
1086:a
1070:d
1066:c
1062:b
1058:a
1041:d
1038:c
1032:b
1029:a
950:c
944:a
941:+
918:d
912:b
909:+
880:)
877:c
871:b
868:+
865:d
859:a
856:(
826:7
820:3
817:+
814:)
811:7
805:2
802:(
796:+
793:)
790:4
784:3
781:(
775:+
772:)
769:4
763:2
760:(
754:=
719:d
713:b
710:+
707:)
704:d
698:a
695:(
689:+
686:)
683:c
677:b
674:(
668:+
665:)
662:c
656:a
653:(
647:=
626:)
623:d
620:+
617:c
611:(
605:)
602:b
599:+
596:a
590:(
555:=
549:+
546:)
543:3
537:7
534:(
528:+
516:=
503:2
499:1
479:+
476:)
471:2
467:t
458:1
454:t
450:(
444:+
441:)
438:Y
432:X
429:(
403:Y
400:=
394:=
391:5
388:+
362:X
359:=
353:=
350:5
309::
305:x
302:E
270:b
266:a
232:a
228:b
216:b
212:a
69:)
65:(
61:.
39:.
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