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# Why lattice multiplication works

## Video transcript

the last video we did a couple of lattice multiplication problems and these Oh is pretty straightforward you have to do all your multiplication first and then do all of your addition but let's try to understand why exactly it worked it almost seemed like magic and to see why it worked I'm going to redo this problem up here and then also try to explain what we did in the longer problem so when we multiplied 27 so you write your 2 and your seven just like that times 48 I'm just doing exactly what we did in the previous video we drew a lattice we gave the two a column and the 7 a column just like that we gave the 4 row and we gave the 8 a row and then we drew our diagonal then the key here is the diagonals you can imagine otherwise we wouldn't be drawing them so you have your diagonals now the way to think about it is each of these diagonals are a number place so for example this diagonal right here that is the ones place that is the ones place the next diagonal I'll do it in this light green color the next diagonal right here in the light green color that is the tens place now the next diagonal to the left or above that depending how you want to view it I'll do in this little pink color right here you could guess that's going to be the hundreds place and then finally we have this little diagonal there and I'll do it in this light blue color that is the thousands place is the thousands place so whenever we multiply one digit times another digit we just make sure we put it in the right bucket or in the right place and you'll see what I mean in a second so we did 7 times 4 well we know it's 7 times 4 is 28 we just simply wrote a 2 and an 8 just like that but what did we really do and I guess the best way to think about it this 7 this is the 7 and 27 so it's just a regular 7 right it's just a regular 7 but this 4 it's the 4 and 48 so it's not just a regular 4 it's really a 40 right 48 can be re-written as 40 plus 8 this 4 right here actually represents a 40 so right here we're not really multiplying 7 times 4 we're actually multiplying 7 times 40 and 7 times 40 isn't just 28 its 280 and 280 how can we think about that we could say that's to hundreds to hundreds plus eight tens and that's exactly what we wrote right here notice this column or sorry this diagonal this diagonal right here I already told you it was the tens diagonal and we remote multiplied seven times for e we put the 8 right here in the tens diagonal so that means eight tens seven times forty is two hundreds we wrote a two and the hundreds diagonal and eight tens that's what the two eight years we actually wrote two hundred and eighty let's keep going when I multiply two times four you might say Oh 2 times 4 that's 8 what am I really doing this is the 2 and 27 this is really a 20 and this is really a 40 so 20 times 40 20 times 40 is equal to this 8 with two zeros is equal to 800 now what did we do we multiply 2 times 4 and we said Oh 2 times 4 is 8 we go to 0 and 1/8 just like that but notice where we wrote the 8 we wrote the 8 in the hundreds in the hundreds diagonal make this a different color we wrote it in the hunters diagonal so we literally wrote even though it looked like we multiply 2 times 4 and saying it's 8 the way we accounted for it we really did 20 times 40 is equal to 8 hundreds remember this is the hundreds diagonal this whole thing right there now we can keep going when you multiply 7 times 8 remember this is really 7 well this is the seven and twenty seven so it's just a regular 7 this is the 8 and 48 so it's just a regular 8 7 times 8 is 56 you write a six in the ones place 56 is just five tens and one six so it's five tens in the 10s diagonal and one 6 56 and then when you multiply 2 times 8 notice that's not really two times eight I mean we did right it's just 16 when we did the problem over here but we're actually multiplying 20 this is a 20 times 8 20 times 8 is equal to 160 or you could say it's 100 notice the one in the hundreds diagonal and six tens that's what 160 is so what we did by doing this lattice multiplication is we accounted all of the digits the right digits in the right places we put the six in the ones place we put the six the 5 and the 8 in the tens place we put 1 the 8 the 2 and the hundreds place and we put nothing right now in the thousands place then now that we're done with all the multiplication we can actually do our adding up and then you just keep adding and if there's something that goes over to the next place you just carry that number so six in the ones place well that's just a 6 then you go to the tens place eight plus five plus six is what 8 plus 5 is 13 plus 6 is 19 but notice we're in the tens place it's 1910s or we could say it's nine tens and 100 and we carry the one up here if you can see it into the hundreds place then we have now we had up all the hundreds is 100 plus 200 plus 800 plus 100 or what is this 1200 so you write 2 in the hundreds place 1200 is the same thing as two hundreds plus 1000 and now I only have one thousand and you're thousands diagonal and so you write that one right there that's exactly how we did in the same reasoning applies to the more complex problem we can label our places this was the ones place right there that was the ones and it made sense when we multiplied the 9 times the 7 those are just literally nines and seven it's 63 6 tens and three ones this right here is the tens diagonal then we got six tens and three ones when we multiplied nine times 80 remember 787 that's the same thing as seven hundreds plus eight tens plus seven just regular seven ones so this 9 times eight is really nine times 89 times 80 is seven hundred and twenty seven hundredths this is the hundreds place seven hundred and twenty-two tends to tens just right there and you can keep going this up here this is the thousands place this is the ten thousands I'll write it like that this is the hundred thousandths place under thousands and then this was the millions place this was the millions place so we did all of our multiplication at once accounted for things in their proper place based on what those numbers really are this caught this entry right here you looks like we just multiplied 4 times 8 and got 32 but we actually were multiplying 400 this is a 400 times 80 and 400 400 times 80 is equal to 3 2 & 3 zeroes is equal to 32,000 and the way you can we counted for it notice we put a 2 right there and what diagonal is that that is the thousands diagonal so we say it's 2003 ten thousands so three ten thousands and mm that's 32,000 so hopefully that gives you an understanding I mean it's nice to it's fun to maybe do some lattice multiplication and get practice but you know sometimes it looks like this bizarre magical thing but hopefully from this video understand that all it is is just a different way of keeping track of where the ones tens and hundreds place are with the advantage that it's kind of nice and compartmentalize it and take up a lot of space and it allows you to do all your multiplication ones and then switch your your brain into addition and carrying mode