Why lattice multiplication works

In the last video we did a couple of lattice multiplication problems and we saw it was pretty straightforward. You got to do all your multiplication first and then do all of your addition. Well, 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 then I'll also try to explain what we did in the longer problems. So when we multiplied 27-- so you write your 2 and your 7 just like that-- times 48. I'm just doing exactly what we did in the previous video. We drew a lattice, gave the 2 a column and the 7 a column. Just like that. We gave the 4 a row and we gave the 8 a row. And then we drew our diagonal. And the key here is the diagonal as 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 1's 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 10's place. Now the next diagonal to the left or above that, depending on 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 100's place. And then finally, we have this little diagonal there and I'll do it in this light blue collar. That is the 1,000's 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 that 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 in 27. So it's just a regular 7. But this 4, it's the 4 in 48. So it's not just a regular 4, it's really a 40. 48 can be rewritten 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, it's 280. And 280, how can we think about that? We could say that's two 100's plus eight 10's. And that's exactly what we wrote right here. Notice: this column or I'm sorry, this diagonal right here, I already told you, it was the 10's diagonal. And we multiplied 7 times 40. We put the 8 right here in the 10's diagonal. So that means eight 10's. 7 times 40 is two 100's. We wrote a 2 in the 100's diagonal. And eight 10's. That's what this 2 8 here is. We actually wrote 280. Let's keep going. When I multiply 2 times 4. You might say, oh, 2 times 4. That's 8. What am I really doing? This is the 2 in 27. This is really a 20 and this is really a 40. So 20 times 40 is equal to just 8 with two 0's. Is equal to 800. And what did we do? We multiplied 2 times 4 and we said, oh, 2 times 4 is 8. We wrote a 0 and en 8 just like that. But notice where we wrote the 8. We wrote the 8 in the 100's diagonal. Let me use a different color. We wrote it in the 100's diagonal. So we literally wrote-- even though it looked like we multiplied 2 times 4 and saying it's 8, the way we accounted for it, we really did 20 times 40 is equal to eight 100's. Remember, this is the 100's diagonal, this whole thing right there. And we can keep going. When you multiply 7 times 8. Remember, this is really 7-- well, this is the 7 in 27, so it's just a regular 7. This is the 8 in 48, so it's just a regular 8. 7 times 8 is 56. You write a 6 in the 1's place. 56 is just five 10's and one 6. So it's five 10's in the 10's diagonal and one 6, 56. Then when you multiply 2 times 8 notice, that's not really just 2 times 8. I mean we did write 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 1 in the 100's diagonal-- and six 10's. 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 6 in the 1's. We put the 6, the 5, and the 8 in the 10's place. We put the 1, the 8, and the 2 in the 100's place. And we put nothing right now in the 1000's 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 6 in the 1's place, well, that's just a 6. Then you go the 10's place. 8 plus 5 plus 6 is what? 8 plus 5 is 13. Plus 6 is 19. But notice, we're in the 10's place. It's nineteen 10's or we could say it's nine 10's and 100. We carry the 1 up here, if you can see it, it's in the 100's place. Now we add up all the 100's. 100 plus 200 plus 800 plus 100. Or, what is this? 1,200. Well you write 2 in the 100's place. 1,200 is the same thing as two 100's plus 1,000. And now you only have 1,000 in your 1,000's diagonal. And so you write that 1 right there. That's exactly how we did it. The same reasoning applied to the more complex problem. We can label our places. This was the 1's place right there. And it made sense. When we multiplied the 9 times the 7 those are just literally 9's and 7's and 63. Six 10's and three 1's. This right here is the 10's diagonal. Then we got six 10's and three 1's. When we multiplied 9 times 80-- remember, 787, that's the same thing as seven 100's plus eight 10's plus seven, just regular seven 1's. So this 9 times 8 really 9 times 80. 9 times 80 is 720. Seven 100's-- this is the 100's place. Seven 100's and 20-- two 10's just right there. And you can keep going. This up here, this is the 1000's place. This is the 10,000's. I'll write it like that. This is the 100,000's place. And then this was the 1,000,000's 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 entry right here, it 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 times 80 is equal to 3 2 and three 0's. Is equal to 32,000. And the way we counted for it-- notice, we put a 2 right there and what diagonal is that? That is the 1,000's diagonal. So we say it's 2,000 and three 10,000's. So three 10,000 and two 1,000's. That's 32,000. So hopefully that gives you an understanding. I mean it's fun to maybe do some lattice multiplication and get practice, but sometimes it looks like this bizarre magical thing. But hopefully from this video you understand that all it is is just a different way of keeping track of where the 1's, 10's, and 100's place are. With the advantage that it's kind of nice and compartmentalized, it doesn't take up a lot of space. And, it allows you to do all your multiplication at once and then, switch your brain into addition and carrying mode.