Master List of Mental Calculation Techniques

I have always been interested in learning Mental Calculation, but I think many beginners would agree with me when I say that I am unsure of where to look to learn techniques. There are a lot of books discussing different techniques such as Arthur Benjamin’s, the Trachtenberg System, and using an abacus. To aid with this problem and thus help spread the popularity of mental calculation, I wanted to ask seasoned mental calculators on the forum: What are the techniques you keep readily at hand to answer a question?

With your answers, we could compile a master list which would look like this

Multiplication:
Anchor Method
Trachtenberg System
etc.

Square Roots:
Duplex method
etc.

I think this would greatly help me and other beginners who would like to know what techniques are most helpful and what we should start learning if we’d like to excel at mental calculation.

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Do you mean for competition or for general use?

Either would be fine, I wouldn’t expect much discrepancy (Is there a discrepancy?). I think general use may be better because those best for competition could be handpicked by the person reading the list.

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google “VEDIC MATHS”

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It can be useful.

In the book The new art of memory, founded upon the principles taught by M. Gregor von Feinaigle: and applied to chronology, history, geography, languages, systematic tables, poetry, prose, and arithmetic there is a method to sum and multiply with points. I wasn’t looking for that it that moment, so didn’t pay much attention.

Wonder if there are more.

I have a copy of the book Vedic Mathematics, but I’m sure there are other (newer) techniques mental calculators use.

Not sure what you mean… is there a link/example you could point me to?

The book is quite old, it’s in archive.org, just google it.

In other words “mental calculators” Wikipedia, same techniques, no need to download. There is World Cup for mental calculators are going on.

I am currently reading your methods, after reading I will tell you more.

Here there is one easy case too -

34 × 38 (there is a difference of 4 numebr between them , so you have to find middle digit)

I can directly say middle digit is 36. But if you have trouble with that remember whatever the difference , add half of the difference.

Now the thing is square the middle digit (that probably you already know without calculating)
And subtract 4 from the square

36^2 = 1296

1292 (don’t you think it’s very very easy)

Other examples -

72 × 76 (there is a difference of 4 )

Find middle digit - 74
Square the digit and subtract - 5476 - 4 = 5472

Note : we subtracted 4 because there is a difference of 4 between them

subtract no = (Half of the distance)^2
= (4/2) ^2
= 2^2 = 4

Well, it’s only for understanding , you can remember that when the difference is 4 we have to subtract 4.

Now take a different example -

  1. 57 × 59 (here is difference of 2 )

Middle digit = 58
Square the number and subtract 1 = 3364 - 1 = 3363

  1. 52 × 58 (difference of 6 )

55^2 - 9

= 3025 - 9
= 3016 (3025 - 10 + 1)

That’s it.
Well, it’s really easy almost took me 1 to 2 sec to do this, only took me long time to explain.

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Addition and substraction

  • left to right
    578 + 125 = 678 + 25 = 698 + 5 = 703
  • complements (converting addiction to easier substraction and substraction to easier addition)
    760 - 487 = 760 - 500 + 13
  • abacus/soroban(The fastest method)

Small multiplication

  • by addition
    86 * 62 = 86 * 60 + 86 * 2
  • by substraction (When the last digit is high)
    86 * 69 = 86 * 70 - 86 * 1
  • factoring (It’s nice if the intermediate result has zero in the middle)
    67 * 24 = (67 * 3) * 8
  • base/Anchor (Works best when numbers are close together or the last digits add up to 10)
    34 * 38 = 30 * 42 + 4 * 8
  • difference of squares (Works best when numbers are close together or have the same last digit)
    67 * 73 = 70^2 - 3^2
  • grouping
    4546 * 7, calculate 45 * 7 = 315, you can use this result while calculating 46 * 7 = 315 + 7
  • end in 5
    28 * 45 = 14 * 90
  • near end in 5
    82 * 44 = 41 * 90 - 82
  • multiply by 25
    48 * 25 = 48 * 100 / 4
  • near 25
    84 * 26 = 84 * 100 / 4 + 84
  • multiply by 5
    428 * 5 = 428 * 10 / 2
  • multiply by 9
    234 * 9 = 2340 - 234
  • multiply by 11
    72 * 11 insert 7+2 between 7 and 2
  • multiply by 111
    72 * 111 insert twice 7+2 between 7 and 2
  • multiply by 37 or 74
    24 * 37 = 8 * 111
  • multiply by 143
    774 * 143 = 774 * 1001 / 7
  • factor and square
    39 * 78 = 39^2 * 2
  • factor and difference of squares
    62 * 87 = (31 * 2) * (3 * 29) = (30^2 - 1^2) * 6
  • factor and base
    27 * 69 = (27 * 23) * 3 = (20 * 30 + 7 * 3) * 3
  • near factor
    26 * 87 = 26 * (88 - 1) = (26 * 8 * 11) - 26
  • near base
    23 * 36 = (24 - 1) * 36 = (20 * 40 + 4 * 16) - 36
  • substraction and square
    23 * 77 = 23 * (100 - 23) = 23 * 100 - 23^2
  • two squares
    64 * 36 = 8^2 * 6^2 = (8 * 6)^2

Big multiplication

  • cross multiplication

Squares and cubes

  • squares formula (a + b)^2 = (a + 2b) * a + b^2
    72^2 = 74 * 70 + 2^2
  • squares end in 5
    75^2 first two digits are 7*(7+1), last two digits are 25
  • squares near 50
    52^2 first two digits are 25+2 last two digits are 2^2
  • squares near 500, 5000 etc. similar to the above
  • cubes formula (a + b)^3 =((a + 3b) * a + 3b^2) * a + b^3
    42^3 = (46 * 40 + 3 * 2^2) * 40 + 2^3

Small Division

  • long division (method learned in schools)
  • vedic division
  • division by 7 (after reaching fractional part, we can use memorized expansion of a fraction)
  • divisor end in 5
    5937 / 255 = 11874 / 510
  • factor divisor (It’s not necessarily faster, but it’s easier)
    3040 / 72 = (3040 / 8) / 9
  • ever shorter divisor (round off the divisor in each step)

Big division

  • cross(Fourier) division

Calendar

  • add up codes for century, year, month an day (Probably the fastest method)
  • Doomsday method

Square roots

  • Newton’s method (It seems fast for low accuracy)
  • duplex method (The best for high accuracy)

Cube roots

  • Newton method with Chebyshew’s correction (Explained in “Dead Reckoning: Calculating Without Instruments”. Probably the fastest method but requires good memory.)
  • Digit by digit (Explained here https://worldmentalcalculation.com/mental-cube-roots-algorithm/ . Slower method, but doesn’t require difficult division when calculating with huge precision. )
  • Logarithms method (Take log, divide by 3, inverse log. It can be fast if we know the logarithms up to 100)

Trygonometric functions and logarithms

  • It’s about memorizing values and some approximations. Explained in “Dead Reckoning: Calculating Without Instruments” book
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Your method is on my list as ‘difference of squares’. Your 76 * 79 example is wrong. This method works when one number is even and the other is odd.

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@Kamil

If you have to do this how you do that -

85 × 37

Like you did in your above example.

Edit -

Oh , sorry I didn’t notice that I did mistake,.I will remove that example.
And I know too, it works only for even number difference.

And I just read your comment, no it works when the difference between them is even or consider like this both are even or both are odd.

It is best if the second number is a multiple of 3.
Like 84 * 37 = 28 * 111 = 3108
Otherwise, this number can expressed as multiple of 3 plus minus 1.
Like 85 * 37 = (84 + 1) * 37 = 3108 + 37 = 3145
or 86 * 37 = (87 - 1) * 37 = 3219 - 37 = 3182

Alright then, I will use this only when other no. is multiple of 3.
Yeah, I know you are giving the example of 37 (note : except 86 isn’t a multiple of 3)

I prefer this -
85 × 37 (100-15 × 37)

3700 - 555 = 3145

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Right, I meant the even difference. Sorry, I’m new and I don’t know where to edit posts.

This is also a good way

Sorry , for again asking (i am just clearing my doubts)

I think you will saying again this , it will only work when the other number is even no. ??
Half the other number and double the no. that has 5 in the last.
And then multiply.

I am currently doing this type of things like this -
28 × 45 (multiple of 5 = 9 × 5)

28 × 9 = 252
252 × 5 = 1260

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Yes, the other number must be even.

How you did that??

44 is 45 - 1
We have 82 * 44 = 82 * 45 - 82 * 1
Now, we can use previous shortcut 82 * 45 = 41 * 90

But why,

82 × 44 (11 × 4)

You have to just multiply the no. by 11 and multiply by 4 (just 2 times double)

82 × 11 = 902
902 × 4 = 3608 (1804 = 3608)