
Babylonian Multiplication in the shower (2016) - plancien
https://www.iquilezles.org/blog/?p=4582
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plancien
I stumbled upon this old post from Iñigo Quilez, and this made my day.
Especially because I did not know about the Babylonian multiplication, let
alone its proof. The absence of explanation in the post made it into a nice
puzzle, I had to figure things out by myself. Maths teaching should more often
be like this kickstarted discovery...

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RajuVarghese
I spent some time looking at the geometric 'proof' in the photo. Was still
wondering why this method would have any advantage over our conventional way
of multiplying numbers because squaring would need multiplication too. Then I
realized that with a table of squares one could do the calculation quite
easily. To calculate the product of any 2 numbers from 1 to 100 one would need
the squares of the numbers from 1 to 200. Armed with this one can calculate
the product of any two numbers from 1 to 100 (there are 10'000 such
combinations) with the formula. Sweet!

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slongfield
And by the sum-of-odd-numbers theorem (sum of 2n-1 = n^2), you don't even need
to multiply to generate that table.

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RajuVarghese
True, that does remove the necessity of the table. However, I was checking how
the Babylonians did it and the article that I read seems to indicate that they
used base-60 tables. On a different note, does anyone know if this method has
been exploited on CPUs without multiplication circuitry in the ALU?

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ogogmad
According to Wikipedia ([1]), Charles Putney implemented the quarter-square
method for the 6502.

[1] -
[https://en.wikipedia.org/wiki/Multiplication_algorithm#Quart...](https://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplication)

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RajuVarghese
Thanks for that pointer! Fascinating to see the multiplication method invented
by the Babylonians being used on a microprocessor.

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082349872349872
Related:
[https://en.wikipedia.org/wiki/Proof_without_words](https://en.wikipedia.org/wiki/Proof_without_words)

Unfortunately the metamathematics of wordless proofs still needs words, maybe
someone can make a proof theory without words?

[https://www.maa.org/press/periodicals/convergence/proofs-
wit...](https://www.maa.org/press/periodicals/convergence/proofs-without-
words-and-beyond-pwws-and-mathematical-proof)

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pubby
I made a helper image:
[https://i.imgur.com/CHYYhn6.png](https://i.imgur.com/CHYYhn6.png)

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WantonQuantum
Thank you!

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krick
The idea would be more obvious if `a` woldn't be exactly `2b` in this example
(granted, author wasn't at liberty to choose values). Being previously unaware
of what is Babylonian multiplication, it took me quite some time to get what
exactly is being conveyed. It is too tempting to read it as "2 x 1" in this
case, which doesn't register as a very notable generic discovery. I suspect
many people who would find it otherwise interesting simply passed by before
understanding what he is trying to show us.

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woofie11
For those of us too lazy to think this through, do you care to post an
explanation?

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krick
Sure, here you go.

a — height of a white tile

b — width of a white tile

a x b — S₁, area of a white tile (this is what we are trying to "find")

(a - b)² — S₀, area of a black square

(a + b)² — S, area of a whole block (4 white tiles, 1 black tile)

Now, looking at the image it is obvious, that (S - S₀) = 4S₁.

