# Where does the “Visual Multiplication” technique originate from?

There is a geometric technique to perform multiplication of numbers.

But as the internet goes, it is hard to figure out who deserves the credit. What I've heard is

• A mayan technique
• From Vedic mathematics (possibly from the equally named book from Bharati Krishna Tirthaji)
• Used in Japanese schools to teach kids about multiplication.

I would love it if somebody could shed some light on the origins of this technique.

• I didn't want to imply that the way we learn it at school, say, in Europe is more complicated, but sometimes visualising a technique can help certain students grok something better. – flq Aug 3 '14 at 10:35
• True. But seing how many think this is much easier and straight-forward without realising that it's the exact same thing always reminds me of how little the average person understands something so simple as "What is multiplication, really?" I see it as an indicator of how mathematics education all over the world must be wrong somehow if most people miss something like this, and it makes me angry at the world in general. I wrote what I wrote to vent out general anger, and it wasn't actually directed at you in any way. – Arthur Aug 3 '14 at 10:42
• It's difficult to come up with a sicker way to teach kids multiplication. – Christian Blatter Aug 7 '14 at 9:52
• I think that the hypothesis that it's a Mayan technique can be ruled out. The MacTutor page on Mayan mathematics is pretty informative, and contains the statement, "We should also note that the Mayans almost certainly did not have methods of multiplication for their numbers and definitely did not use division of numbers." Some Mayan numerals do make use of sets of parallel horizontal lines, which may, at some point, have suggested to somebody a connection with visual multiplication, but I doubt there's anything to it. – Will Orrick Aug 12 '14 at 16:58
• This question has come up on MSE before: here and here. – Will Orrick Aug 15 '14 at 8:18

Let's make a table:

        300 +   20 +   1
------------------      ----> 10000 x        6  = 60000
200 | 60000 + 4000 + 200             1000 x    (15+4) = 19000
50 | 15000 + 1000 +  50              100 x (12+10+2) =  2400
4 |  1200 +   80 +   4               10 x    ( 8+5) =   130
1 x        4  =     4
-------------------------
81434


This is a very organized way to use the FOIL identity:

$$(100x + 10y + z)(100a + 10b + c) = 10^4a +10^3(ay+bx)+10^2(az+by+cx)+10(bz+cy)+cz$$

• This is a history of math question. It's not asking why the method works. – Will Orrick Aug 9 '14 at 3:23
• Considering that I did not ask for this but is a useful information nonetheless, would you be ok with me including it in the question (with proper credits) and y removing this answer? – flq Aug 13 '14 at 20:17
• what is "FOIL"? – Rolazaro Azeveires Nov 29 '17 at 13:24
• Also the Trachtenberg speed system puts an extra dot at intermediate places between each column to evaluate local square matrices.Believe it was from ancient Vedic maths; and there is also an unverified connection to the mayan civilization. – Narasimham Jan 31 '18 at 6:29
• @RolazaroAzeveires it's an acronym for "First, Outside, Inside, Last" which is a mnemonic for multiplying two linear terms of the form $(ax+b)(cx+d)$ to make a quadratic which is taught in North American schools. It's basically a specialized case of the distributive rule for exactly this form which tends to confuse students the moments they try to multiply polynomials with more terms. – CyclotomicField May 15 '18 at 14:19