Duplex method for squaring numbers very quickly

The speed the Indians calculate at the mental calculation world championships is ridiculous. I mean this in a positive way. Unbelievable!

Here is an example of the current world record holder Shashank Jain taking 10 square roots just a little over 1 minute:

Nodas explained in a separate post


that they do this using the duplex method and if you take some time to learn this, you will be very fast in squaring numbers and taking square roots.

I took some time to play around with this and it really speeds things up.

First, what is a duplex?
If a, b, c, d, e, f, etc. are digits, then
d(a) = a^2
d(ab) = 2ab
d(abc) = 2ac + b^2
d(abcd) = 2ad + 2bc
d(abcde) = 2ae + 2bd + c^2
etc.

In other words, from the outside working inwards, take the outermost digits, multiply them, double them, then take the next adjoining digits, multiply then, double them, then the next digits. etc.

If you are left with one single middle digit, square it.
Then add all these numbers.

If we want to square a 2 digit number: ab then the square is:
d(a) | d(ab) | d(b)

If we want to square a 3 digit number: abc then the square is:
d(a) | d(ab) | d(abc) | d(bc) | d©

If we want to square a 4 digit number: abcd then the square is:
d(a) | d(ab) | d(abc) | d(abcd) | d(bcd) | d(cd) | d(d)

Let’s start simple:
11^2 =
a=1
b=1
d(1) | d(11) | d(1) =
1 | 2 | 1 =
121

17^2 =
a=1
b=7
d(1) | d(17) | d(7) =
1 | 14 | 49 =
carry the 1 and the 4:
1+1 | 4+4 | 9 =
289

78^2 =
a=7
b=8
d(7) | d(78) | d(8)
49 | 2*56 | 64 =
49 | 112 | 64 =
carry 11 into the hundreds and 6 into the tens:
60 | 8 | 4 =
6084

alternatively:
78^2 = (80-2)^2
a=8
b=-2
d(8) | d(8, -2) | d(-2)
64 | 28-2 | 4 =
64 | -32 | 4 =
you want to end up with positive digits, so I turn 32 into -40+8 and carry the -40:
64 | -40 +8 | 4 =
carry the -40 into the thousands:
60 | 8 | 4 =
6084

Let’s square 345:
345^2 =
d(3) | d(34) | d(345) | d(45) | d(5) =
9 | 212 | 235 + 4^2 | 24*5 | 5^2 =
9 | 24 | 30 + 16 | 40 | 25 =
9 | 24 | 46 | 40 | 25 = (now handle the carries)
11 | 8 | 10 | 0 | 25 =
11 | 9 | 0 | 2 | 5 = (no carries left, we are done)
119025

A four digit number:
1234^2 =
d(1) | d(12) | d(123) | d(1234) | d(234)| d(34) | d(4) =
1 | 4 | 10 | 20 | 25 | 24 | 16 =
1522756

We can also do this two digits at a time:
12 34^2:
a=12
b=34
12^2 | 21234 | 34^2 =
144 | 816 | 1156 =
keep in mind that when working with 2 digits at a time, we work with hundreds between two ‘|’'s. So in 1156, we carry 11 and not 115:
144 + 8 | 16 + 11 | 56 =
152 | 27 | 56
1522756

Or with negative numbers:
788^2:
for 788 use 800 - 12, so:
a = 8 and b = -12:
d(8) | d(8 , -12) | d(-12)
64 | 2*-12*8 | 144 =
64 | -192 | 144 =
192 = 200-8, so -192 = -200+8.
64 | -200 + 8| 144 =
Do the carry negatively:
62 | 08 | 144 =
62 | 09 | 44 =
620,944

See how easy this is?

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Duplex = duplicated thread? (ignore me) (if you want)

Lol. Indeed, double post. I deleted one.

Good write-up Kinma, I think it provides food for thought but will take a little getting use to. Thank you for the write-up.

After reading Nodas’ post, of course the next thing in my mind was to research the duplex method. Thanks for the post, Kinma. You explain the process very well.

How do you use duplex method to do square roots?

I am working on a post for taking the square root using the duplex method.
Stay tuned.

Can you please provide some reference links where this method is named or called as “Duplex Method”. From what is described in this post it looks like a specific case of Vedic Multipication. Vedic Mathematics is a branch of Mathematics derived from Old Scriptures in India in Vedas.

There is a multiplication method in Vedic mathematics where if you put both the numbers same this method mentioned above is derived, and the Vedic method is even easier than this description to visualize using vertical and inclined lines through the numbers.

One link I found for the Vedic multiplication is at https://www.wikihow.com/Multiply-Using-Vedic-Math this place and many more can be found on Internet.

Just google it.
There are enough links about the duplex method for squaring numbers or taking roots of numbers.

Maybe I don’t understand your question though.

Let me elaborate the point a bit more. This method is mentioned as Duplex Method and a sequence of equations is provided as steps. I learned this method in Vedic Mathematics as Vedic Multiplication, which was taught as a sequence of visual vertical and inclined lines. Those steps help deriving the equations on the fly, that is you do not have to remember the equations by using those visual clues.

I want to add this information to this thread (and the general knowledge pool) so readers can watch and search the actual Vedic Multiplication system and then try applying it to its specific case of multiplying a number by itself (which makes it square)

If we can visualize the sequence of operations using the vertical and inclined lines I am mentioning one does not have to remember the equations or their sequence. This falls in place with our ability of being faster in spacial skills. Having visual location and order makes it faster to calculate.

Vedic Multiplication is not Duplex. However; I could see the duplex method as a special case of multiplying numbers with themselves.

Let’s keep this thread about the Duplex Method itself (when squaring numbers) en feel free to add to this thread if you think it helps people to calculate better.

I would love to hear your thoughts.

If you look at the original Book on Vedic Mathematics written by Jagadguru Shankaracharya Swami Bharatikrishna Tirtha you’ll find the term “duplex method” there. All other books/courses are derivatives of his original work and some use their own terminology, but it is originally called “duplex method” in Vedic maths.

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Thanks for this information in details. I will keep this in mind.

Keeping the topic going on, for me the equations to follow in order becomes slow process, hence I will try to use the visuals like for a two digit word ->[|X|] | - (product of last digits) , X - (product of cross digits added) , | - (product of first two digits.