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I saw this video on Facebook and I'm curious about this. I have very basic questions on this because I can use it but I can't understand the technique.

Link to the video explaining this method on Facebook: https://www.facebook.com/tbar9a/videos/824414374314985/

1) What is this method's name?
2) How does it work? What is the relation between the numbers and the intersections?
3) Is this working for any integer number times any integer number or are there limits?
4) Is there some problem with this problem like negative aspects? I don't know why it isn't standard way for people and students who have problems with math.
5) Why do we have to count the line's intersections from top to down?
6) How does someone invent this method? I have no clue how someone could think about this, not even by chance!

I'm sorry if this sounds very unintelligent or foolish. Also sorry if wrong network, I wasn't sure where to post it.

CodeShark
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2 Answers2

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I'll answer what I can:

2) The number of crosses in each area is the result of multiplying two digits.

3) It works, there are no theorical limits. You'll need tons of paper to multiply big numbers, though. Specially if the digits are high.

4) I think that this method is very slow. If you have to multiply, say, $7\times9$, you'll have to count $63$ crosses by hand. You'll take very long and it is very easy to miscount the points. Just my opinion.

5) You can also do it in any order, of course.

ajotatxe
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I'll just add the following to ajotatxe's answer:

1) It's a version of what the wikipedia page on multiplication algorithms calls "lattice multiplication" with multiplication of digits carried out by drawing a little grid and counting. The wikipedia page gives some historical information about lattice multiplication. I suspect that multiplying digits by drawing and counting is a 20th century pedagogical device, since it's very inefficient in practice, as ajotatxe points out.

2) In more detail, a decimal number is just a sum of multiples of powers of $10$: $4576 = 4\times10^3 + 5\times10^2 + 7 \times 10^1 + 6\times10^0$ and $341 = 3\times10^2 + 4\times10^1 + 1\times 10^0$. When you multiply these together you get a sum of products like $(5\times10^2) \times (4\times10^1) = 20 \times 10^{1+2} = 20\times 10^3$. The method is just using simple facts about rectangular grids to group together the products that contribute the same power of 10.

6) If you know how long multiplication works, this method becomes fairly obvious: it is just doing the addition of the carries and the additions of the columns in a different order. I believe that long multiplication and lattice multipliation are very similar to the process used to do multiplication with an abacus, so the basic ideas are very ancient.

Rob Arthan
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