Lattice Multiplication

[davidn]

Why on earth didn't anyone teach me about this when I was in school! I've been covering the whiteboard in these ever since I saw this going past on Facebook (I haven't yet found definitive proof that it's of Japanese origin).

After a couple of fascinated scribbles (please excuse the mouse handwriting), I realized that by counting the intersections, you're doing exactly the same thing as the normal long multiplication method of multiplying each position by each other position, just laying it out visually... but it's so satisfying to see it fall out of a tartan picnic rug like that. Besides, it's prettier than a column of numbers.

I will have to remember this for when I next have paper to hand but not a calculator, which in today's environment will be round about never.

Comments

[igorium.livejournal.com]

I never understood the exact algorithm behind it, nor why it works.
For 97x46 for example

[xaq.livejournal.com]

It basically breaks the numbers being multiplied into their 10s and 1s, makes 2 bunches of sticks in each color in those same numbers, then lays them out so they intersect.

Red sticks: 10s in the left bunch, 1s in the right bunch. In this case, we've got 9 sticks in the left and 7 in the right.
Blue sticks: 10s in the top bunch, 1s in the bottom. 4 on top, 6 on bottom.

Then, you simply multiply the number of sticks in each corner.

Top left corner: 10s x 10s, giving you your 100s result
Top right corner & bottom left corner: 10s x 1s, giving you 2 sets of 10s results
Bottom right corner: 1s x 1s, giving you your 1s results.

For the 97 x 46 example:
Top left: (9 x 4) 100s = 3600
Top right: (9 x 6) 10s = 540
Bottom left: (7 x 4) 10s = 280
Bottom right: (7 x 6) 1s = 42

Once you got that, you just add up your results.

3600 + 540 + 280 + 42 = 4462.


...Come to think of it, this basically works the same way as cross-multiplying an algebra equation.
(90+7)(40+6) = (90x40)+(90x6)+(7x40)+(7x6) = 3600 + 540 + 280 + 42 = 4462.

Heh. Definitely gonna have to show my mom this trick so she can pass it on to the math teachers up in Nome. Could prove quite useful to the kids up there...or at least keep their interest long enough to actually do their homework.

[ravenworks.livejournal.com]

http://en.wikipedia.org/wiki/Lattice_multiplication Looks like it arose all over, and the first recorded example of it is Arabic!

[xaq.livejournal.com]

Brain status:

[premchaia_pre4]

I feel like I should be trying to insinuate myself into the GOODS section of that image now.

[xaq.livejournal.com]

*shoves you into his backpack and latches it shut* Done and done!

[premchaia_pre4]

Unfortunately, your sudden motion of the creature sets off the Chirp Alarm. CHIRP!! CHIRP!! CHIRP!! After the third chirp, sixteen talons, a beak, teeth, two sharp-edged wings, and a tailspear all set to slicing up and skewering your backpack simultaneously in a whirling ball of doom. Within seconds, the backpack lies in ribbons on the floor along with its disgorged contents, and not long after that, so do you.

It's a sad thing that your adventure has ended here!!

[xaq.livejournal.com]

...You'd think I'd have seen that coming.


(Note to self: upgrade to orihalcon-plated backpack.)

[kibet.livejournal.com]

I have never seen that method before but would be good to use to explain multiplication to someone that can not do it in the linear way. I think I would suggest people to use it for larger number multiplication. The 97x46, is probably faster doing the 100*46-(3*46) but 121*124 is probably faster using the above method. The only problem I can see would be for the person to put the numbers together in the correct order as it goes from bottom right to top left. Japanese Kanji and Chinese characters typically go top left to bottom right in the writing order so could see children have difficulties with this method.

I would like to see one with division as I have yet to see a method that is straight forward to use and I am glad that I can deal with numbers in my head as I would have struggled otherwise.

[davidn]

Yes, it's easier on numbers with low values of every magnitude - I realized during the 97x46 example that what I was doing was exactly the same as the traditional way, especially as I had to multiply the top-right section and the bottom-left one just to count them efficiently.

I've never found a good long division either! The way that we were taught in school, to essentially just keep on trying and whittling it down to smaller and smaller numbers, always felt rather unsatisfying.

[premchaia_pre4]

It's much more elegant in binary where the choice is between writing a 1 or a 0 and the test is a basic comparison. You can partially emulate that in decimal: precompute (8B, 4B, 2B, B) for divisor B, then add bits to the result digit, which is at least an optimized “whittling-down” because not-adding a bit is very cheap. (5B, 2B, B) also works in a sort of mixed-base intermediary representation. Or, more commonly (I think?), you can round to two significant digits over one and get your initial estimate that way by inverted single-digit multiplication tables, then count up or down from there (there's probably a guaranteed error bound in there but I'm too lazy to compute it just now).

[davidn]

I do remember that I only truly understood long division the first time that I had to use the same technique for binary division. However, after that explanation I'm pretty sure I understand it less again.

[kibet.livejournal.com]

My mother once told me how she did it at school as she was shocked that I would just do the sum in the basic way but her way just looked like guessing!