3617:
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.
90:
268:
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
36:
3700:) 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.
3348:
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
3681:
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,
3616:
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,
3513:
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).
2493:
216:
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,
1306:
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
3652:
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
267:
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
855:
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:
2606:
3335:
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
2817:
3636:, 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.
1575:
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.
589:
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):
4090:
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".
3620:
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.
2315:
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:
217:
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).
3474:
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.
1455:
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),
2611:
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
1419:
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.
2432:
3720:(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.
1426:
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)
3648:. 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
847:
2508:
3546:
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.
1167:
3243:
740:
1307:
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.
3315:
2763:
3514:
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.
1282:
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.
261:
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.
1172:
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.
176:
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.
3767:
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
2513:
1645:
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:
115:. 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
3165:
1858:
1285:
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.
503:
167:
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.
1431:
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)
579:
907:
1767:
339:
119:. 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
2502:
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).
3716:, 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
2752:
2258:
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:
2869:
1932:
245:, 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.
3987:
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".
3038:
2911:
977:
647:
1244:
2714:
2674:
2972:
1441:
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)
1694:
939:
3107:
3074:
1062:
1024:
3555:
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:
424:
383:
1272:
1198:
2757:
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:
1416:
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.
1288:
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).
1296:
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.
2488:{\displaystyle {\text{root }}\simeq {\text{ known square root}}-{\frac {{\text{known square}}-{\text{unknown square}}}{2\times {\text{known square root}}}}\,}
2613:
2353:
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
1399:
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.
3901:
Brisswalter, J.; Arcelin, R.; Audiffren, M.; Delignieres, D. (1997). "Influence of
Physical Exercise on Simple Reaction Time: Effect of Physical Fitness".
301:
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:
1460:
15 − 2. Knowing that 15 is 225 and 2 is 4, simple subtraction shows that 225 − 4 = 221, which is the desired product.
170:
Apply the originally specified operation to the two condensed operands, and then apply the summing-of-digits procedure to the result of the operation.
3858:
Lambourne, Kate; Tomporowski, Phillip (2010). "The effect of exercise-induced arousal on cognitive task performance: A meta-regression analysis".
2601:{\displaystyle {\begin{aligned}{\text{root}}&\simeq 4-{\frac {16-15}{2\times 4}}\\&\simeq 4-0.125\\&\simeq 3.875\\\end{aligned}}\,\!}
751:
1413:
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.
3510:
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.
1092:
46:
164:
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.
3176:
2812:{\displaystyle {\text{root }}\simeq {\frac {{\text{mean}}({\text{known square}},{\text{unknown square}})}{\text{known square root}}}\,}
1410:
This technique allows easy multiplication of numbers close and below 100.(90-99) The variables will be the two numbers one multiplies.
1310:
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.
653:
3254:
3805:
3649:
1402:
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.
1393:
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.
4130:"Developmental Changes in Mental Arithmetic: Evidence for Increased Functional Specialization in the Left Inferior Parietal Cortex"
2289:
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.
1341:
Add the carried 1 to the highest valued digit in the multiplier, 7 + 1 = 8, and copy to the result to finish.
3733:
4177:
344:
As a preliminary step simply round the smaller number down and the larger up to the nearest multiple of ten. In this case:
2207:
For example, say one wants to square 48, which is 50 − 2. One subtracts 200 from 2500 and add 4, and get
3697:
3691:
3115:
4172:
1773:
435:
1629:
Take a given number, and add and subtract a certain value to it that will make it easier to multiply. For example:
1637:
492 is close to 500, which is easy to multiply by. Add and subtract 8 (the difference between 500 and 492) to get
522:
157:
and getting a result, the following procedure can be used to improve confidence in the correctness of the result:
3393:
Observe that there is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero:
862:
3654:
1700:
307:
3658:
4030:
2719:
2837:
2361:
This method is a straightforward extension of the explanation given above for squaring an integer near 100.
1864:
2983:
3641:
2880:
2639:
is their difference, it is more convenient to express this approximation in the form of mixed fraction as
945:
596:
89:
3352:
Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10:
1205:
2679:
1333:
Similarly add 7 + 5 = 12, then add the carried 1 to get 13. Place 3 to the result and carry the 1.
287:
3795:
3543:
This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11.
3340:
with 10 are somewhat easier since the digits are scrambled in consistent ways, as in the next section.
2642:
208:
The same procedure can be used with multiple operations, repeating steps 1 and 2 for each operation.
2930:
198:
Perform the original operation on the condensed operands, and sum digits: 2 × 7 = 14; 1 + 4 = 5
1372:
Another method is to simply multiply the number by 10, and add the original number to the result.
204:
5 = 5, so there is a good chance that the prediction that 6,338 × 79 equals 500,702 is right.
161:
Sum the digits of the first operand; any 9s (or sets of digits that add to 9) can be counted as 0.
4116:
4012:
3969:
3926:
3883:
3820:
3717:
1659:
913:
3079:
3046:
2387:
This method is a straightforward extension of the explanation given above for integers near 10.
1035:
1003:
253:
When the above situation does not apply, there is another method known as indirect calculation.
3944:
Murata, Atsuo (2005). "An
Attempt to Evaluate Mental Workload Using Wavelet Transform of EEG".
1303:
can be found by a series of additions to each of its digits from right to left, two at a time.
391:
350:
4151:
4108:
4004:
3961:
3918:
3875:
3801:
3743:
3670:
3666:
2394:
trailing digits) 407 + 7 = 414 414 × 4 = 1656 (leading digits; note this multiplication by
1251:
1177:
148:
120:
116:
1396:
For example: 24 x 11 = 264 because 2 + 4 = 6 and the 6 is placed in between the 2 and the 4.
4141:
4100:
3996:
3953:
3910:
3867:
2499:
3633:
173:
Sum the digits of the result that were originally obtained for the original calculation.
3645:
3332:
2409:
trailing digits) 79991 - 9 79982 × 8 = 639856 (leading digits) 79991 = 6398560081
57:
4166:
3887:
3738:
3729:
997:. F meaning first, O meaning outer, I meaning inner and L meaning last. For example:
4120:
4062:
4016:
3973:
3930:
4079:
4044:
3871:
184:
Assume the calculation 6,338 × 79, manually done, yielded a result of 500,702:
1942:
This method requires memorization of the squares of the one-digit numbers 1 to 9.
4000:
111:, with no help from any supplies (such as pencil and paper) or devices such as a
17:
2423:
994:
108:
104:
3957:
3914:
3661:
is a widely known task using mental arithmetic. This procedure, mostly used in
3043:
If 'a' is close to the target, 'b' will be a small enough number to render the
4063:"Mental Calculation World Cup - the World Championship for Mental Calculators"
4045:"Mental Calculation World Cup - the World Championship for Mental Calculators"
3833:
3637:
1325:
Add 5 + 9 = 14 so 4 is placed on the left side of the result and carry the 1.
112:
101:
1246:
is the addition of the product of the outer digits and the inner digits; OI.
4146:
4129:
3662:
3326:
201:
Sum the digits of 500702: 5 + 0 + 0 + (7 + 0 + 2 = 9, which counts as 0) = 5
4155:
4112:
4008:
3965:
3879:
842:{\displaystyle 23\cdot 47=100(2\cdot 4)+10(3\cdot 4)+10(2\cdot 7)+3\cdot 7}
3922:
3665:
experiments, suggests mental subtraction is useful in testing the effects
3709:
3629:
3704:
Memoriad – World Memory, Mental
Calculation & Speed Reading Olympics
1317:
The ones digit of the multiplier, 9, is copied to the temporary result.
3748:
3713:
3337:
2375:
trailing digits) 9997 - 3 = 9994 (leading digits) 9997 = 99940009
1300:
154:
131:
93:
Mental calculation has long been a component of mathematical education.
3946:
Human
Factors: The Journal of the Human Factors and Ergonomics Society
2405:
79991 = ? (79991 is -9 from 80000) (-9) = 0081 (
1162:{\displaystyle a\cdot c\cdot 100+(a\cdot d+b\cdot c)\cdot 10+b\cdot d}
3768:
3581:
From this information, one can find the logarithm of any number 1-9.
2368:
trailing digits) 1012 + 12 = 1024 (leading digits) 1012 = 1024144
2357:
Squaring any integer near 10 (e.g., 976 to 1024, 9976 to 10024, etc.)
189:
Sum the digits of 6,338: (6 + 3 = 9, so count that as 0) + 3 + 8 = 11
4031:"Mental Calculations World Championship – World Mental Calculation"
2924:
is the difference between the known root and the answer one seeks.
2371:
9997 = ? (9997 is -3 from 10000) (-3) = 0009 (
1463:
This method requires knowing by heart a certain number of squares:
4104:
88:
3506:
If the perfect cube ends in 9, the cube root of it must end in 9.
3503:
If the perfect cube ends in 8, the cube root of it must end in 2.
3500:
If the perfect cube ends in 7, the cube root of it must end in 3.
3497:
If the perfect cube ends in 6, the cube root of it must end in 6.
3494:
If the perfect cube ends in 5, the cube root of it must end in 5.
3491:
If the perfect cube ends in 4, the cube root of it must end in 4.
3488:
If the perfect cube ends in 3, the cube root of it must end in 7.
3485:
If the perfect cube ends in 2, the cube root of it must end in 8.
3482:
If the perfect cube ends in 1, the cube root of it must end in 1.
3479:
If the perfect cube ends in 0, the cube root of it must end in 0.
3238:{\displaystyle \mathrm {root} \simeq a-{\frac {a^{2}-x}{2a}}\,\!}
2364:
1012 = ? (1012 is +12 from 1000) (+12) = 144 (
1200:
is the product of the first digit of each of the two numbers; F.
735:{\displaystyle =100(a\cdot c)+10(b\cdot c)+10(a\cdot d)+b\cdot d}
3632:
of the proper level can lead to an increase in performance of a
1274:
is the product of the last digit of each of the two numbers; L.
3310:{\displaystyle \mathrm {root} \simeq {\frac {a^{2}+x}{2a}}\,\!}
2231:
This method requires the memorization of squares from 1 to 24.
3800:. United States: CreateSpace Independent Publishing Platform.
1365:−3478 × 11 = 3 3+4+1(8) 4+7+1(2) 7+8(5) 8 = −38258
29:
4084:
2616:. Thus, one should try rounding the estimated answer down.
2277:
This method requires the memorization of squares from 1 to
69:
2676:. Thus, in the previous example, the square root of 15 is
2390:
407 = ? (407 is +7 from 400) (+7) = 49 (
130:
Many of these techniques take advantage of or rely on the
61:
2398:
was not needed for integers from 76 to 124 because their
1368:
62473 × 11 = 6 6+2(8) 2+4+1(7) 4+7+1(2) 7+3(0) 3 = 687203
990:
to which is it is easy to add 21: 281 and then 800: 1081
512:
is the tens unit of the original larger number (75) and t
3606:
log(8) = log(2 × 2 × 2) = log(2) + log(2) + log(2) ~ .90
3530:
After the last three digits are taken away, 456 remains.
65:
3076:
element of the equation negligible. Thus, one can drop
1992:, adding 0 to the end and finally adding the square of
53:
4128:
Rivera, S.M.; Reiss, AL; Eckert, MA; Menon, V (2005).
3834:"How to Do Cube Roots of 9 Digit Numbers in Your Head"
3653:
mental tasks were completed concurrently, with a more
2727:
2687:
2650:
516:
is the tens unit of the original smaller number (35).
3712:, Turkey, in 2008. The second Memoriad took place in
3257:
3179:
3118:
3082:
3049:
2986:
2933:
2883:
2840:
2766:
2722:
2682:
2645:
2511:
2435:
1867:
1776:
1703:
1662:
1360:
Note the handling of 9+1 as the highest valued digit.
1254:
1208:
1180:
1095:
1038:
1006:
948:
916:
865:
754:
656:
599:
525:
438:
394:
353:
310:
2623:
is the closest perfect square to the desired square
2340:
93 − 7 = 86 (this gives the first two digits)
27:
Arithmetical calculations using only the human brain
2383:
10 (e.g., 276 to 324, 4976 to 5024, 79976 to 80024)
2343:(−7) = 49 (these are the second two digits)
2273:
Squaring an integer near 100 (e.g., from 76 to 124)
3309:
3237:
3159:
3101:
3068:
3032:
2966:
2905:
2863:
2811:
2746:
2708:
2668:
2600:
2487:
1926:
1852:
1761:
1688:
1266:
1238:
1192:
1161:
1056:
1018:
971:
933:
901:
841:
734:
641:
573:
497:
418:
377:
333:
3306:
3234:
3160:{\displaystyle b\simeq {\frac {a^{2}-x}{2a}}\,\!}
3156:
3029:
2963:
2902:
2860:
2597:
2196:. One knows that 50 is 2500. So one subtracts 100
241:are all smaller than the corresponding digits of
3533:456 is greater than all the cubes up to 7 cubed.
1953:being a two-digit integer, can be calculated as
1443:Now, the difference will be the first two digits
1433:Now, the difference will be the first two digits
3797:On a New Method of Multiplication and Shortcuts
2920:is a known root (4 from the above example) and
1853:{\displaystyle n^{2}=(n^{2}-an+an-a^{2})+a^{2}}
993:An easy mnemonic to remember for this would be
264:One place at a time is handled, left to right.
3657:at the higher level of physical workload. The
2426:of a number is to use the following equation:
2333:Another way to look at it would be like this:
1406:Multiplying two numbers close to and below 100
498:{\displaystyle (X\times Y)+50(t_{1}-t_{2})+25}
195:Sum the digits of 79: 7 + (9 counted as 0) = 7
153:After applying an arithmetic operation to two
3708:The first international Memoriad was held in
3644:, have been shown to be useful in indicating
3597:log(5) = log(10 / 2) = log(10) − log(2) ~ .70
3551:Approximating common logarithms (log base 10)
1357:999 × 11 = 9+1(10) 9+9+1(9) 9+9(8) 9 = 10989
8:
2614:inequality of arithmetic and geometric means
574:{\displaystyle =30\times 80+50(7-3)+25=2625}
3609:log(9) = log(3 × 3) = log(3) + log(3) ~ .96
3600:log(6) = log(2 × 3) = log(2) + log(3) ~ .78
3594:log(4) = log(2 × 2) = log(2) + log(2) ~ .60
902:{\displaystyle (a\cdot d+b\cdot c)\cdot 10}
3625:Mental arithmetic as a psychological skill
286:Many of these methods work because of the
4145:
3517:Therefore, the cube root of 29791 is 31.
3305:
3282:
3275:
3258:
3256:
3233:
3210:
3203:
3180:
3178:
3155:
3132:
3125:
3117:
3098:
3092:
3083:
3081:
3065:
3059:
3050:
3048:
3028:
3016:
2991:
2985:
2962:
2950:
2932:
2901:
2884:
2882:
2859:
2847:
2842:
2839:
2808:
2794:
2786:
2778:
2775:
2767:
2765:
2726:
2721:
2716:As another example, square root of 41 is
2686:
2681:
2649:
2644:
2596:
2534:
2516:
2512:
2510:
2484:
2476:
2463:
2455:
2452:
2444:
2436:
2434:
2043:Multiply this number by itself plus one:
2028:Take the digit(s) that precede the five:
1918:
1872:
1866:
1844:
1828:
1797:
1781:
1775:
1762:{\displaystyle n^{2}=(n^{2}-a^{2})+a^{2}}
1753:
1737:
1724:
1708:
1702:
1680:
1667:
1661:
1253:
1207:
1179:
1094:
1037:
1005:
949:
947:
917:
915:
864:
753:
655:
598:
524:
480:
467:
437:
393:
352:
334:{\displaystyle \mathrm {Ex} :35\times 75}
311:
309:
3821:multiplying two numbers close, below 100
3696:The first Mental Calculation World Cup (
3354:
1465:
1354:−54 × −11 = 5 5+4(9) 4 = 594
3786:
3760:
2747:{\displaystyle 6{\tfrac {5}{12}}=6.416}
2337:93 = ? (is −7 from 100)
47:instructions, advice, or how-to content
3794:Cheprasov, Artem (September 3, 2009).
3677:Mental Calculations World Championship
3536:The first digit of the cube root is 7.
2864:{\displaystyle \mathrm {r} ^{2}=x\,\!}
1927:{\displaystyle n^{2}=(n-a)(n+a)+a^{2}}
3033:{\displaystyle a^{2}-2ab+b^{2}=x\,\!}
2169:Suppose one needs to square a number
7:
2906:{\displaystyle \mathrm {r} =a-b\,\!}
2754:while the actual value is 6.4031...
2211:= 2304. For numbers larger than 50 (
1313:A step-by-step example of 759 × 11:
1299:The product for any larger non-zero
972:{\displaystyle {}+a\cdot c\cdot 100}
642:{\displaystyle (10a+b)\cdot (10c+d)}
2285:is the absolute difference between
2242:is between 26 and 74 inclusive) is
1239:{\displaystyle (a\cdot d+b\cdot c)}
986:(12 + 14) = 26, 26 × 10 = 260,
56:so that it is more encyclopedic or
3321:Extracting roots of perfect powers
3268:
3265:
3262:
3259:
3190:
3187:
3184:
3181:
3109:out and rearrange the equation to
2885:
2843:
2709:{\displaystyle 4{\tfrac {-1}{8}}.}
2180: = 50 −
1583: + 1) −
852:800 +120 +140 + 21 ----- 1081
315:
312:
25:
3650:Michigan Technological University
3559:log(a × b) = log(a) + log(b)
2669:{\displaystyle n{\tfrac {d}{2n}}}
2227:Squaring an integer from 26 to 74
2165:Squaring numbers very close to 50
1595: + 1 −
4085:Memoriad - World Mental Olympics
3682:fractions, and calendar dates.
429:The algorithm reads as follows:
34:
3734:calculating the day of the week
2967:{\displaystyle (a-b)^{2}=x\,\!}
2422:An easy way to approximate the
2176:The number may be expressed as
1344:Final product of 759 × 11: 8349
585:Multiplying any 2-digit numbers
3872:10.1016/j.brainres.2010.03.091
3539:The cube root of 456533 is 77.
2947:
2934:
2831:is the square root of x, then
2799:
2783:
1908:
1896:
1893:
1881:
1834:
1790:
1743:
1717:
1439:Alternate for first two digits
1233:
1209:
1138:
1114:
890:
866:
824:
812:
803:
791:
782:
770:
717:
705:
696:
684:
675:
663:
636:
621:
615:
600:
556:
544:
486:
460:
451:
439:
1:
3524:Find the cube root of 456533.
2238:(most easily calculated when
2054:Take above result and attach
2023:Squaring a number ending in 5
4080:Mental Calculation World Cup
4001:10.1080/00140139.2012.703695
3698:Mental Calculation World Cup
3692:Mental Calculation World Cup
3686:Mental Calculation World Cup
3562:log(a / b) = log(a) - log(b)
2874:One then redefines the root
2215: = 50 +
1938:Squaring any 2-digit integer
192:Iterate as needed: 1 + 1 = 2
3903:Perceptual and Motor Skills
2223:instead of subtracting it.
1689:{\displaystyle n^{2}=n^{2}}
934:{\displaystyle {}+b\cdot d}
271:Hence, the result is 2231.
4194:
3958:10.1518/001872005774860096
3915:10.2466/pms.1997.85.3.1019
3689:
3324:
3102:{\displaystyle {}+b^{2}\,}
3069:{\displaystyle {}+b^{2}\,}
2418:Approximating square roots
2379:Squaring any integer near
1617: − 1) + (2
1057:{\displaystyle ab\cdot cd}
1019:{\displaystyle 75\cdot 23}
297:The "Ends of Five" Formula
146:
2320:100(100 − 2(7)) + 7
2006:= 10 × 2(23 + 3) + 3
419:{\displaystyle 75+5=80=Y}
378:{\displaystyle 35-5=30=X}
221:Calculating differences:
3659:Brown–Peterson procedure
3612:log(10) = 1 + log(1) = 1
3527:The cube root ends in 7.
2200:from 2500, and then add
2089:Mathematical explanation
1267:{\displaystyle b\cdot d}
1193:{\displaystyle a\cdot c}
257:Look-ahead borrow method
3640:measures, specifically
3248:that can be reduced to
2446: known square root
1980:can be found by adding
3311:
3239:
3161:
3103:
3070:
3034:
2968:
2907:
2865:
2813:
2748:
2710:
2670:
2602:
2489:
2061:Example: 85 × 85
1976:Meaning the square of
1928:
1854:
1763:
1690:
1621: − 1)
1268:
1240:
1194:
1163:
1058:
1020:
982:i.e., in this example
973:
935:
903:
843:
736:
643:
575:
499:
420:
379:
335:
275:Calculating products:
138:Methods and techniques
94:
4178:Games of mental skill
4147:10.1093/cercor/bhi055
4049:www.recordholders.org
3669:can have on how long
3667:maintenance rehearsal
3565:log(0) does not exist
3344:Extracting cube roots
3312:
3240:
3162:
3104:
3071:
3035:
2969:
2908:
2866:
2814:
2749:
2711:
2671:
2603:
2498:The closer the known
2490:
2184:so its square is (50−
2009:= 10 × 2(26) + 9
1929:
1855:
1764:
1691:
1269:
1241:
1195:
1164:
1059:
1021:
974:
936:
904:
844:
737:
644:
576:
500:
421:
380:
336:
288:distributive property
117:a competitive context
92:
3331:Extracting roots of
3255:
3177:
3116:
3080:
3047:
2984:
2931:
2881:
2838:
2764:
2720:
2680:
2643:
2509:
2433:
1865:
1774:
1701:
1660:
1451:Using square numbers
1252:
1206:
1178:
1093:
1036:
1004:
946:
914:
863:
752:
654:
597:
523:
436:
392:
351:
308:
249:Indirect calculation
125:lightning calculator
4093:Nature Neuroscience
2402:= 1) 407 = 165649
1653:The proof follows:
1625:Squaring any number
1446:Final Product- 9215
1436:Final Product- 9215
1429:Final Product- yx15
1390:One last easy way:
237:When the digits of
54:rewrite the content
4173:Mental calculation
3718:Mental Calculation
3655:significant effect
3307:
3235:
3157:
3099:
3066:
3030:
2964:
2903:
2861:
2827:By definition, if
2809:
2744:
2736:
2706:
2701:
2666:
2664:
2598:
2594:
2485:
2081:12 × 13 = 156
1924:
1850:
1759:
1686:
1641:492 -> 484, 500
1350:Further examples:
1264:
1236:
1190:
1159:
1054:
1016:
969:
931:
899:
839:
732:
639:
571:
495:
416:
375:
331:
233:Direct calculation
121:mental calculators
98:Mental calculation
95:
3832:Dorrell, Philip.
3744:Mental calculator
3671:short-term memory
3630:Physical exertion
3520:Another example:
3391:
3390:
3303:
3231:
3153:
2977:Expanding yields
2806:
2805:
2804:known square root
2797:
2789:
2781:
2770:
2735:
2700:
2663:
2558:
2519:
2482:
2479:
2478:known square root
2466:
2458:
2447:
2439:
2350:Another example:
2162:
2161:
1999:For example, 23:
1579:This is because (
1568:
1567:
1292:Multiplying by 11
149:Casting out nines
143:Casting out nines
134:numeral system.
87:
86:
18:Mental arithmetic
16:(Redirected from
4185:
4159:
4149:
4124:
4067:
4066:
4059:
4053:
4052:
4041:
4035:
4034:
4027:
4021:
4020:
3984:
3978:
3977:
3941:
3935:
3934:
3898:
3892:
3891:
3855:
3849:
3848:
3846:
3844:
3829:
3823:
3818:
3812:
3811:
3791:
3774:
3771:
3765:
3355:
3316:
3314:
3313:
3308:
3304:
3302:
3294:
3287:
3286:
3276:
3271:
3244:
3242:
3241:
3236:
3232:
3230:
3222:
3215:
3214:
3204:
3193:
3166:
3164:
3163:
3158:
3154:
3152:
3144:
3137:
3136:
3126:
3108:
3106:
3105:
3100:
3097:
3096:
3084:
3075:
3073:
3072:
3067:
3064:
3063:
3051:
3039:
3037:
3036:
3031:
3021:
3020:
2996:
2995:
2973:
2971:
2970:
2965:
2955:
2954:
2912:
2910:
2909:
2904:
2888:
2870:
2868:
2867:
2862:
2852:
2851:
2846:
2818:
2816:
2815:
2810:
2807:
2803:
2802:
2798:
2795:
2790:
2787:
2782:
2779:
2776:
2771:
2768:
2753:
2751:
2750:
2745:
2737:
2728:
2715:
2713:
2712:
2707:
2702:
2696:
2688:
2675:
2673:
2672:
2667:
2665:
2662:
2651:
2607:
2605:
2604:
2599:
2595:
2582:
2563:
2559:
2557:
2546:
2535:
2520:
2517:
2494:
2492:
2491:
2486:
2483:
2481:
2480:
2477:
2468:
2467:
2464:
2459:
2456:
2453:
2448:
2445:
2440:
2437:
2095:
2094:
2084:So, 125 = 15,625
1988:, multiplied by
1933:
1931:
1930:
1925:
1923:
1922:
1877:
1876:
1859:
1857:
1856:
1851:
1849:
1848:
1833:
1832:
1802:
1801:
1786:
1785:
1768:
1766:
1765:
1760:
1758:
1757:
1742:
1741:
1729:
1728:
1713:
1712:
1695:
1693:
1692:
1687:
1685:
1684:
1672:
1671:
1607: + 1)
1571:Squaring numbers
1466:
1278:Multiplying by 9
1273:
1271:
1270:
1265:
1245:
1243:
1242:
1237:
1199:
1197:
1196:
1191:
1168:
1166:
1165:
1160:
1063:
1061:
1060:
1055:
1025:
1023:
1022:
1017:
978:
976:
975:
970:
950:
940:
938:
937:
932:
918:
908:
906:
905:
900:
848:
846:
845:
840:
741:
739:
738:
733:
648:
646:
645:
640:
580:
578:
577:
572:
504:
502:
501:
496:
485:
484:
472:
471:
425:
423:
422:
417:
384:
382:
381:
376:
340:
338:
337:
332:
318:
82:
79:
73:
38:
37:
30:
21:
4193:
4192:
4188:
4187:
4186:
4184:
4183:
4182:
4163:
4162:
4140:(11): 1779–90.
4134:Cerebral Cortex
4127:
4089:
4076:
4071:
4070:
4061:
4060:
4056:
4043:
4042:
4038:
4029:
4028:
4024:
3995:(10): 1166–79.
3986:
3985:
3981:
3943:
3942:
3938:
3900:
3899:
3895:
3857:
3856:
3852:
3842:
3840:
3831:
3830:
3826:
3819:
3815:
3808:
3793:
3792:
3788:
3783:
3778:
3777:
3769:
3766:
3762:
3757:
3726:
3706:
3694:
3688:
3679:
3646:mental workload
3627:
3553:
3346:
3329:
3323:
3295:
3278:
3277:
3253:
3252:
3223:
3206:
3205:
3175:
3174:
3145:
3128:
3127:
3114:
3113:
3088:
3078:
3077:
3055:
3045:
3044:
3012:
2987:
2982:
2981:
2946:
2929:
2928:
2879:
2878:
2841:
2836:
2835:
2825:
2777:
2762:
2761:
2718:
2717:
2689:
2678:
2677:
2655:
2641:
2640:
2593:
2592:
2580:
2579:
2561:
2560:
2547:
2536:
2521:
2507:
2506:
2469:
2454:
2431:
2430:
2420:
2415:
2410:
2403:
2385:
2376:
2369:
2359:
2354:
2323:= 100 × 86 + 49
2275:
2229:
2167:
2067:8 × 9 = 72
2025:
1940:
1914:
1868:
1863:
1862:
1840:
1824:
1793:
1777:
1772:
1771:
1749:
1733:
1720:
1704:
1699:
1698:
1676:
1663:
1658:
1657:
1627:
1573:
1453:
1448:
1408:
1384:170 + 17 = 187
1382:
1294:
1280:
1250:
1249:
1204:
1203:
1176:
1175:
1091:
1090:
1034:
1033:
1002:
1001:
944:
943:
912:
911:
861:
860:
853:
750:
749:
652:
651:
595:
594:
587:
521:
520:
515:
511:
476:
463:
434:
433:
390:
389:
349:
348:
306:
305:
299:
293:
284:
269:
259:
251:
235:
230:
214:
151:
145:
140:
107:using only the
83:
77:
74:
51:
39:
35:
28:
23:
22:
15:
12:
11:
5:
4191:
4189:
4181:
4180:
4175:
4165:
4164:
4161:
4160:
4125:
4087:
4082:
4075:
4074:External links
4072:
4069:
4068:
4054:
4036:
4022:
3979:
3952:(3): 498–508.
3936:
3909:(3): 1019–27.
3893:
3860:Brain Research
3850:
3824:
3813:
3806:
3785:
3784:
3782:
3779:
3776:
3775:
3759:
3758:
3756:
3753:
3752:
3751:
3746:
3741:
3736:
3725:
3722:
3705:
3702:
3690:Main article:
3687:
3684:
3678:
3675:
3626:
3623:
3614:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3579:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3552:
3549:
3541:
3540:
3537:
3534:
3531:
3528:
3525:
3508:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3472:
3471:
3464:
3457:
3450:
3443:
3436:
3429:
3422:
3415:
3408:
3401:
3389:
3388:
3385:
3382:
3379:
3376:
3372:
3371:
3368:
3365:
3362:
3359:
3345:
3342:
3333:perfect powers
3322:
3319:
3318:
3317:
3301:
3298:
3293:
3290:
3285:
3281:
3274:
3270:
3267:
3264:
3261:
3246:
3245:
3229:
3226:
3221:
3218:
3213:
3209:
3202:
3199:
3196:
3192:
3189:
3186:
3183:
3170:and therefore
3168:
3167:
3151:
3148:
3143:
3140:
3135:
3131:
3124:
3121:
3095:
3091:
3087:
3062:
3058:
3054:
3041:
3040:
3027:
3024:
3019:
3015:
3011:
3008:
3005:
3002:
2999:
2994:
2990:
2975:
2974:
2961:
2958:
2953:
2949:
2945:
2942:
2939:
2936:
2914:
2913:
2900:
2897:
2894:
2891:
2887:
2872:
2871:
2858:
2855:
2850:
2845:
2824:
2821:
2820:
2819:
2801:
2796:unknown square
2793:
2785:
2774:
2743:
2740:
2734:
2731:
2725:
2705:
2699:
2695:
2692:
2685:
2661:
2658:
2654:
2648:
2609:
2608:
2591:
2588:
2585:
2583:
2581:
2578:
2575:
2572:
2569:
2566:
2564:
2562:
2556:
2553:
2550:
2545:
2542:
2539:
2533:
2530:
2527:
2524:
2522:
2515:
2514:
2496:
2495:
2475:
2472:
2465:unknown square
2462:
2451:
2443:
2419:
2416:
2414:
2411:
2404:
2389:
2384:
2377:
2370:
2363:
2358:
2355:
2352:
2348:
2347:
2344:
2341:
2338:
2331:
2330:
2327:
2324:
2321:
2313:
2312:
2292:The square of
2274:
2271:
2270:
2269:
2266:
2263:
2256:
2255:
2234:The square of
2228:
2225:
2166:
2163:
2160:
2159:
2148:
2145:
2144:
2133:
2130:
2129:
2118:
2115:
2114:
2103:
2093:
2092:
2091:
2090:
2087:
2086:
2085:
2082:
2079:
2073:
2072:
2071:
2070:So, 85 = 7,225
2068:
2065:
2052:
2041:
2024:
2021:
2017:
2016:
2013:
2010:
2007:
2004:
1974:
1973:
1945:The square of
1939:
1936:
1935:
1934:
1921:
1917:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1875:
1871:
1860:
1847:
1843:
1839:
1836:
1831:
1827:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1800:
1796:
1792:
1789:
1784:
1780:
1769:
1756:
1752:
1748:
1745:
1740:
1736:
1732:
1727:
1723:
1719:
1716:
1711:
1707:
1696:
1683:
1679:
1675:
1670:
1666:
1651:
1650:
1643:
1642:
1635:
1634:
1626:
1623:
1603: + (
1591: + 2
1572:
1569:
1566:
1565:
1562:
1559:
1556:
1553:
1550:
1546:
1545:
1542:
1539:
1536:
1533:
1530:
1526:
1525:
1522:
1519:
1516:
1513:
1510:
1506:
1505:
1502:
1499:
1496:
1493:
1490:
1486:
1485:
1482:
1479:
1476:
1473:
1470:
1452:
1449:
1425:
1407:
1404:
1387:17 × 11 = 187
1381:17 × 10 = 170
1380:
1370:
1369:
1366:
1363:
1362:
1361:
1355:
1348:
1347:
1346:
1345:
1339:
1338:
1337:
1331:
1330:
1329:
1323:
1322:
1321:
1293:
1290:
1279:
1276:
1263:
1260:
1257:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1189:
1186:
1183:
1170:
1169:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1065:
1064:
1053:
1050:
1047:
1044:
1041:
1027:
1026:
1015:
1012:
1009:
988:
987:
980:
979:
968:
965:
962:
959:
956:
953:
941:
930:
927:
924:
921:
909:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
851:
850:
849:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
743:
742:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
649:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
586:
583:
582:
581:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
513:
509:
506:
505:
494:
491:
488:
483:
479:
475:
470:
466:
462:
459:
456:
453:
450:
447:
444:
441:
427:
426:
415:
412:
409:
406:
403:
400:
397:
386:
385:
374:
371:
368:
365:
362:
359:
356:
342:
341:
330:
327:
324:
321:
317:
314:
298:
295:
283:
273:
266:
258:
255:
250:
247:
234:
231:
229:
219:
213:
210:
206:
205:
202:
199:
196:
193:
190:
186:
185:
178:
177:
174:
171:
168:
165:
162:
147:Main article:
144:
141:
139:
136:
85:
84:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4190:
4179:
4176:
4174:
4171:
4170:
4168:
4157:
4153:
4148:
4143:
4139:
4135:
4131:
4126:
4122:
4118:
4114:
4110:
4106:
4105:10.1038/82831
4102:
4098:
4094:
4088:
4086:
4083:
4081:
4078:
4077:
4073:
4064:
4058:
4055:
4050:
4046:
4040:
4037:
4032:
4026:
4023:
4018:
4014:
4010:
4006:
4002:
3998:
3994:
3990:
3983:
3980:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3940:
3937:
3932:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3897:
3894:
3889:
3885:
3881:
3877:
3873:
3869:
3865:
3861:
3854:
3851:
3839:
3838:Thinking Hard
3835:
3828:
3825:
3822:
3817:
3814:
3809:
3807:9781448689330
3803:
3799:
3798:
3790:
3787:
3780:
3772:
3764:
3761:
3754:
3750:
3747:
3745:
3742:
3740:
3739:Mental abacus
3737:
3735:
3731:
3730:Doomsday rule
3728:
3727:
3723:
3721:
3719:
3715:
3711:
3703:
3701:
3699:
3693:
3685:
3683:
3676:
3674:
3672:
3668:
3664:
3660:
3656:
3651:
3647:
3643:
3639:
3638:Physiological
3635:
3631:
3624:
3622:
3618:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3583:
3582:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3557:
3556:
3550:
3548:
3544:
3538:
3535:
3532:
3529:
3526:
3523:
3522:
3521:
3518:
3515:
3511:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3477:
3476:
3469:
3465:
3462:
3458:
3455:
3451:
3448:
3444:
3441:
3437:
3434:
3430:
3427:
3423:
3420:
3416:
3413:
3409:
3406:
3402:
3400:
3396:
3395:
3394:
3386:
3383:
3380:
3377:
3374:
3373:
3369:
3366:
3363:
3360:
3357:
3356:
3353:
3350:
3343:
3341:
3339:
3334:
3328:
3320:
3299:
3296:
3291:
3288:
3283:
3279:
3272:
3251:
3250:
3249:
3227:
3224:
3219:
3216:
3211:
3207:
3200:
3197:
3194:
3173:
3172:
3171:
3149:
3146:
3141:
3138:
3133:
3129:
3122:
3119:
3112:
3111:
3110:
3093:
3089:
3085:
3060:
3056:
3052:
3025:
3022:
3017:
3013:
3009:
3006:
3003:
3000:
2997:
2992:
2988:
2980:
2979:
2978:
2959:
2956:
2951:
2943:
2940:
2937:
2927:
2926:
2925:
2923:
2919:
2898:
2895:
2892:
2889:
2877:
2876:
2875:
2856:
2853:
2848:
2834:
2833:
2832:
2830:
2822:
2791:
2772:
2760:
2759:
2758:
2755:
2741:
2738:
2732:
2729:
2723:
2703:
2697:
2693:
2690:
2683:
2659:
2656:
2652:
2646:
2638:
2634:
2630:
2626:
2622:
2619:Note that if
2617:
2615:
2589:
2586:
2584:
2576:
2573:
2570:
2567:
2565:
2554:
2551:
2548:
2543:
2540:
2537:
2531:
2528:
2525:
2523:
2505:
2504:
2503:
2501:
2473:
2470:
2460:
2449:
2441:
2429:
2428:
2427:
2425:
2417:
2413:Finding roots
2412:
2408:
2401:
2397:
2393:
2388:
2382:
2378:
2374:
2367:
2362:
2356:
2351:
2345:
2342:
2339:
2336:
2335:
2334:
2328:
2325:
2322:
2319:
2318:
2317:
2311:
2307:
2303:
2302:
2301:
2299:
2296:(i.e., 100 ±
2295:
2290:
2288:
2284:
2280:
2272:
2267:
2265:= 144 + 3,700
2264:
2261:
2260:
2259:
2253:
2249:
2245:
2244:
2243:
2241:
2237:
2232:
2226:
2224:
2222:
2218:
2214:
2210:
2205:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2174:
2172:
2164:
2157:
2153:
2149:
2147:
2146:
2142:
2138:
2134:
2132:
2131:
2127:
2123:
2119:
2117:
2116:
2112:
2108:
2104:
2101:
2097:
2096:
2088:
2083:
2080:
2077:
2076:
2075:Example: 125
2074:
2069:
2066:
2063:
2062:
2060:
2059:
2057:
2053:
2050:
2046:
2042:
2039:
2035:
2031:
2027:
2026:
2022:
2020:
2019:So 23 = 529.
2014:
2011:
2008:
2005:
2002:
2001:
2000:
1997:
1995:
1991:
1987:
1983:
1979:
1972:
1968:
1964:
1960:
1956:
1955:
1954:
1952:
1948:
1943:
1937:
1919:
1915:
1911:
1905:
1902:
1899:
1890:
1887:
1884:
1878:
1873:
1869:
1861:
1845:
1841:
1837:
1829:
1825:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1798:
1794:
1787:
1782:
1778:
1770:
1754:
1750:
1746:
1738:
1734:
1730:
1725:
1721:
1714:
1709:
1705:
1697:
1681:
1677:
1673:
1668:
1664:
1656:
1655:
1654:
1649:492 = 242,064
1648:
1647:
1646:
1640:
1639:
1638:
1632:
1631:
1630:
1624:
1622:
1620:
1616:
1612:
1608:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1577:
1570:
1563:
1560:
1557:
1554:
1551:
1548:
1547:
1543:
1540:
1537:
1534:
1531:
1528:
1527:
1523:
1520:
1517:
1514:
1511:
1508:
1507:
1503:
1500:
1497:
1494:
1491:
1488:
1487:
1483:
1480:
1477:
1474:
1471:
1468:
1467:
1464:
1461:
1459:
1450:
1447:
1444:
1440:
1437:
1434:
1430:
1424:
1421:
1417:
1414:
1411:
1405:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1379:
1376:
1375:For example:
1373:
1367:
1364:
1359:
1358:
1356:
1353:
1352:
1351:
1343:
1342:
1340:
1335:
1334:
1332:
1327:
1326:
1324:
1319:
1318:
1316:
1315:
1314:
1311:
1308:
1304:
1302:
1297:
1291:
1289:
1286:
1283:
1277:
1275:
1261:
1258:
1255:
1247:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1201:
1187:
1184:
1181:
1173:
1156:
1153:
1150:
1147:
1144:
1141:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1111:
1108:
1105:
1102:
1099:
1096:
1089:
1088:
1087:
1084:
1082:
1078:
1074:
1070:
1051:
1048:
1045:
1042:
1039:
1032:
1031:
1030:
1013:
1010:
1007:
1000:
999:
998:
996:
991:
985:
984:
983:
966:
963:
960:
957:
954:
951:
942:
928:
925:
922:
919:
910:
896:
893:
887:
884:
881:
878:
875:
872:
869:
859:
858:
857:
836:
833:
830:
827:
821:
818:
815:
809:
806:
800:
797:
794:
788:
785:
779:
776:
773:
767:
764:
761:
758:
755:
748:
747:
746:
745:For example,
729:
726:
723:
720:
714:
711:
708:
702:
699:
693:
690:
687:
681:
678:
672:
669:
666:
660:
657:
650:
633:
630:
627:
624:
618:
612:
609:
606:
603:
593:
592:
591:
584:
568:
565:
562:
559:
553:
550:
547:
541:
538:
535:
532:
529:
526:
519:
518:
517:
492:
489:
481:
477:
473:
468:
464:
457:
454:
448:
445:
442:
432:
431:
430:
413:
410:
407:
404:
401:
398:
395:
388:
387:
372:
369:
366:
363:
360:
357:
354:
347:
346:
345:
328:
325:
322:
319:
304:
303:
302:
296:
294:
291:
289:
282:
278:
274:
272:
265:
262:
256:
254:
248:
246:
244:
240:
232:
228:
224:
220:
218:
211:
209:
203:
200:
197:
194:
191:
188:
187:
183:
182:
181:
175:
172:
169:
166:
163:
160:
159:
158:
156:
150:
142:
137:
135:
133:
128:
126:
122:
118:
114:
110:
106:
103:
99:
91:
81:
78:February 2017
71:
67:
63:
59:
55:
49:
48:
43:This article
41:
32:
31:
19:
4137:
4133:
4099:(1): 103–7.
4096:
4092:
4057:
4048:
4039:
4025:
3992:
3988:
3982:
3949:
3945:
3939:
3906:
3902:
3896:
3863:
3859:
3853:
3841:. Retrieved
3837:
3827:
3816:
3796:
3789:
3763:
3707:
3695:
3680:
3628:
3619:
3615:
3603:log(7) ~ .85
3591:log(3) ~ .48
3588:log(2) ~ .30
3580:
3577:log(7) ~ .85
3574:log(3) ~ .48
3571:log(2) ~ .30
3554:
3545:
3542:
3519:
3516:
3512:
3509:
3473:
3467:
3460:
3453:
3446:
3439:
3432:
3425:
3418:
3411:
3404:
3398:
3392:
3351:
3347:
3330:
3247:
3169:
3042:
2976:
2921:
2917:
2915:
2873:
2828:
2826:
2788:known square
2756:
2636:
2632:
2628:
2624:
2620:
2618:
2610:
2497:
2457:known square
2421:
2406:
2399:
2395:
2391:
2386:
2380:
2372:
2365:
2360:
2349:
2332:
2326:= 8,600 + 49
2314:
2309:
2305:
2297:
2293:
2291:
2286:
2282:
2278:
2276:
2257:
2251:
2247:
2246:(50 −
2239:
2235:
2233:
2230:
2220:
2216:
2212:
2208:
2206:
2201:
2197:
2193:
2189:
2188:) = 50 − 100
2185:
2181:
2177:
2175:
2170:
2168:
2155:
2151:
2140:
2136:
2125:
2121:
2110:
2106:
2099:
2055:
2048:
2044:
2037:
2033:
2029:
2018:
1998:
1993:
1989:
1985:
1981:
1977:
1975:
1970:
1966:
1962:
1958:
1950:
1946:
1944:
1941:
1652:
1644:
1636:
1628:
1618:
1614:
1610:
1609:
1604:
1600:
1596:
1592:
1588:
1584:
1580:
1578:
1574:
1462:
1457:
1454:
1445:
1442:
1438:
1435:
1432:
1428:
1422:
1418:
1415:
1412:
1409:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1377:
1374:
1371:
1349:
1312:
1309:
1305:
1298:
1295:
1287:
1284:
1281:
1248:
1202:
1174:
1171:
1085:
1080:
1076:
1072:
1068:
1066:
1028:
992:
989:
981:
854:
744:
588:
507:
428:
343:
300:
292:
285:
280:
276:
270:
263:
260:
252:
242:
238:
236:
226:
222:
215:
207:
179:
152:
129:
124:
105:calculations
102:arithmetical
100:consists of
97:
96:
75:
52:Please help
44:
3634:mental task
2424:square root
2304:100(100 ± 2
2254:− 25)
2219:), add 100×
2058:to the end
1336:result: 349
1067:where 7 is
109:human brain
62:Wikiversity
4167:Categories
3989:Ergonomics
3781:References
3585:log(1) = 0
3568:log(1) = 0
3387:10 = 1000
3325:See also:
2823:Derivation
2769:root
2438:root
2158:+ 1) + 25
2040:are digits
1957:10 ×
1328:result: 49
113:calculator
70:Wikivoyage
3888:206324098
3866:: 12–24.
3663:cognitive
3349:= 74088.
3327:13th root
3273:≃
3217:−
3201:−
3195:≃
3139:−
3123:≃
2998:−
2941:−
2896:−
2773:≃
2691:−
2587:≃
2574:−
2568:≃
2552:×
2541:−
2532:−
2526:≃
2474:×
2461:−
2450:−
2442:≃
2346:93 = 8649
2173:near 50.
2012:= 520 + 9
1888:−
1822:−
1804:−
1731:−
1564:30 = 900
1544:29 = 841
1524:28 = 784
1504:27 = 729
1484:26 = 676
1423:Example:
1320:result: 9
1259:⋅
1228:⋅
1216:⋅
1185:⋅
1154:⋅
1142:⋅
1133:⋅
1121:⋅
1106:⋅
1100:⋅
1086:Consider
1079:and 3 is
1046:⋅
1011:⋅
964:⋅
958:⋅
926:⋅
894:⋅
885:⋅
873:⋅
834:⋅
819:⋅
798:⋅
777:⋅
759:⋅
727:⋅
712:⋅
691:⋅
670:⋅
619:⋅
551:−
533:×
474:−
446:×
358:−
326:×
66:Wikibooks
45:contains
4156:15716474
4121:23829063
4113:11135652
4017:38648671
4009:22849301
3974:25313835
3966:16435692
3931:30781628
3880:20381468
3724:See also
3710:Istanbul
3466:10 = 100
3370:5 = 125
2262:(−12) +
2250:) + 100(
2032:, where
1561:25 = 625
1558:20 = 400
1555:15 = 225
1552:10 = 100
1541:24 = 576
1538:19 = 361
1535:14 = 196
1521:23 = 529
1518:18 = 324
1515:13 = 169
1501:22 = 484
1498:17 = 289
1495:12 = 144
1481:21 = 441
1478:16 = 256
1475:11 = 121
1378:17 × 11
225:−
180:Example
155:operands
3923:9399313
3843:19 July
3749:Soroban
3714:Antalya
3673:lasts.
3384:9 = 729
3381:8 = 512
3378:7 = 343
3375:6 = 216
3338:coprime
2329:= 8,649
2268:= 3,844
2143:) + 25
2109:+ 5)(10
1301:integer
1075:, 2 is
1071:, 5 is
508:Where t
212:Factors
132:decimal
4154:
4119:
4111:
4015:
4007:
3972:
3964:
3929:
3921:
3886:
3878:
3804:
3463:down 3
3459:9 = 72
3456:down 1
3452:8 = 51
3449:down 3
3445:7 = 34
3438:6 = 21
3431:5 = 12
3428:down 3
3421:down 1
3414:down 3
3367:4 = 64
3364:3 = 27
2916:where
2500:square
2281:where
2135:= 100(
2034:a, b,
1549:5 = 25
1532:9 = 81
1529:4 = 16
1512:8 = 64
1492:7 = 49
1472:6 = 36
60:it to
4117:S2CID
4013:S2CID
3970:S2CID
3927:S2CID
3884:S2CID
3770:mod 9
3755:Notes
3424:4 = 6
3417:3 = 2
3361:2 = 8
3358:1 = 1
2742:6.416
2590:3.875
2577:0.125
2300:) is
2150:= 100
2128:+ 25
2124:+ 100
2120:= 100
2113:+ 5)
2105:= (10
2015:= 529
1509:3 = 9
1489:2 = 4
1469:1 = 1
68:, or
4152:PMID
4109:PMID
4005:PMID
3962:PMID
3919:PMID
3876:PMID
3864:1341
3845:2015
3802:ISBN
3732:for
3470:up 1
3442:up 1
3435:up 1
3410:2 =
3407:up 1
3403:1 =
3397:0 =
2780:mean
2627:and
2518:root
2308:) +
2102:+ 5)
2051:+ 1)
2036:and
2030:abc5
1969:) +
1458:i.e.
1029:and
995:FOIL
569:2625
58:move
4142:doi
4101:doi
3997:doi
3954:doi
3911:doi
3868:doi
3642:EEG
2381:m ×
2098:(10
2049:abc
2045:abc
1984:to
1633:492
1613:= (
1109:100
967:100
768:100
661:100
127:s.
123:or
4169::
4150:.
4138:15
4136:.
4132:.
4115:.
4107:.
4095:.
4047:.
4011:.
4003:.
3993:55
3991:.
3968:.
3960:.
3950:47
3948:.
3925:.
3917:.
3907:85
3905:.
3882:.
3874:.
3862:.
3836:.
2733:12
2635:-
2631:=
2544:15
2538:16
2204:.
2192:+
2139:+
2078:12
2056:25
2003:23
1996:.
1986:mn
1978:mn
1965:+
1963:mn
1951:mn
1949:,
1947:mn
1599:=
1587:=
1145:10
1083:.
1014:23
1008:75
897:10
810:10
789:10
762:47
756:23
703:10
682:10
625:10
604:10
563:25
542:50
536:80
530:30
493:25
458:50
408:80
396:75
367:30
355:35
329:75
323:35
290:.
279:×
64:,
4158:.
4144::
4123:.
4103::
4097:4
4065:.
4051:.
4033:.
4019:.
3999::
3976:.
3956::
3933:.
3913::
3890:.
3870::
3847:.
3810:.
3773:.
3468:0
3461:9
3454:2
3447:3
3440:6
3433:5
3426:4
3419:7
3412:8
3405:1
3399:0
3300:a
3297:2
3292:x
3289:+
3284:2
3280:a
3269:t
3266:o
3263:o
3260:r
3228:a
3225:2
3220:x
3212:2
3208:a
3198:a
3191:t
3188:o
3185:o
3182:r
3150:a
3147:2
3142:x
3134:2
3130:a
3120:b
3094:2
3090:b
3086:+
3061:2
3057:b
3053:+
3026:x
3023:=
3018:2
3014:b
3010:+
3007:b
3004:a
3001:2
2993:2
2989:a
2960:x
2957:=
2952:2
2948:)
2944:b
2938:a
2935:(
2922:b
2918:a
2899:b
2893:a
2890:=
2886:r
2857:x
2854:=
2849:2
2844:r
2829:r
2800:)
2792:,
2784:(
2739:=
2730:5
2724:6
2704:.
2698:8
2694:1
2684:4
2660:n
2657:2
2653:d
2647:n
2637:n
2633:x
2629:d
2625:x
2621:n
2571:4
2555:4
2549:2
2529:4
2471:2
2407:n
2400:m
2396:m
2392:n
2373:n
2366:n
2310:a
2306:a
2298:a
2294:n
2287:n
2283:a
2279:a
2252:n
2248:n
2240:n
2236:n
2221:a
2217:a
2213:n
2209:n
2202:a
2198:a
2194:a
2190:a
2186:a
2182:a
2178:n
2171:n
2156:x
2154:(
2152:x
2141:x
2137:x
2126:x
2122:x
2111:x
2107:x
2100:x
2064:8
2047:(
2038:c
1994:n
1990:m
1982:n
1971:n
1967:n
1961:(
1959:m
1920:2
1916:a
1912:+
1909:)
1906:a
1903:+
1900:n
1897:(
1894:)
1891:a
1885:n
1882:(
1879:=
1874:2
1870:n
1846:2
1842:a
1838:+
1835:)
1830:2
1826:a
1819:n
1816:a
1813:+
1810:n
1807:a
1799:2
1795:n
1791:(
1788:=
1783:2
1779:n
1755:2
1751:a
1747:+
1744:)
1739:2
1735:a
1726:2
1722:n
1718:(
1715:=
1710:2
1706:n
1682:2
1678:n
1674:=
1669:2
1665:n
1619:x
1615:x
1611:x
1605:x
1601:x
1597:x
1593:x
1589:x
1585:x
1581:x
1262:d
1256:b
1234:)
1231:c
1225:b
1222:+
1219:d
1213:a
1210:(
1188:c
1182:a
1157:d
1151:b
1148:+
1139:)
1136:c
1130:b
1127:+
1124:d
1118:a
1115:(
1112:+
1103:c
1097:a
1081:d
1077:c
1073:b
1069:a
1052:d
1049:c
1043:b
1040:a
961:c
955:a
952:+
929:d
923:b
920:+
891:)
888:c
882:b
879:+
876:d
870:a
867:(
837:7
831:3
828:+
825:)
822:7
816:2
813:(
807:+
804:)
801:4
795:3
792:(
786:+
783:)
780:4
774:2
771:(
765:=
730:d
724:b
721:+
718:)
715:d
709:a
706:(
700:+
697:)
694:c
688:b
685:(
679:+
676:)
673:c
667:a
664:(
658:=
637:)
634:d
631:+
628:c
622:(
616:)
613:b
610:+
607:a
601:(
566:=
560:+
557:)
554:3
548:7
545:(
539:+
527:=
514:2
510:1
490:+
487:)
482:2
478:t
469:1
465:t
461:(
455:+
452:)
449:Y
443:X
440:(
414:Y
411:=
405:=
402:5
399:+
373:X
370:=
364:=
361:5
320::
316:x
313:E
281:b
277:a
243:a
239:b
227:b
223:a
80:)
76:(
72:.
50:.
20:)
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