 Alright, so let's take a look at the multiplication of multiple digit numbers using base 10. So for example, we might take the product 3 times 156, and we'd like to show this both as a repeated addition, and then for the purposes of building up our standard algorithm, we want to see that as a digit-wise multiplication. So again, we don't specify a base, so because there's no base specified, we can assume we're working in base 10. We'll set down the number in our place value chart, so that's 100s, 510s, 6 1s, and I want to find 3 times that amount, so that means I'm going to take 3 of those. So I'll set down 3 of them, and I'll add them together. So, proceeding however I feel like, as long as I stay within each column, I have 100, 100, 100, that's 300s. I have 5, 10, 15, 10s, 6, 12, 18, 1s, and I can add as long as I stay within those columns. And finally, again, as a number, 300s, 15, 10s, 18, 1s makes perfect sense, although we might want to rewrite that in standard form, we need to then bundle and trade. And again, since this is base 10, we know that we can bundle 10 and trade for 1 in the next place over. So that 18 becomes an 8 and a 10, and I trade, and that makes 16. The 16 is 6 and 10, and I bundle the 10, and I trade, and combine, and there my final answer, 400s, 6, 10s, 8, 1s, 468. Now, if I want to show this as a digit-wise multiplication, again, the thing that I'm doing here is I'm taking 3 of each of these amounts. So I'm taking 3 1s, 3 5s, 3 6s, and I'm going to write them down in the appropriate places. And again, this time, I'll bundle and trade. This time, I'll actually save the combining for the end of the problem. So this 18 over here in the right-most column, that's, again, a 10 and an 8, and I'll move that 10 over. This 15 here is a 5 and 10, and again, I'll trade this 10 over for 1 more in the next place. And then the 300s that I had originally, well, I'll just keep those as my 3, and then I'll add them up. The thing to notice here for future references is this 18, when I did the bundling and trading, I rewrote as 1 here, 8 there. This 15, when I rewrote for bundling and trading, the 1 was in the next place, and the 5 was over here. And then the 3 is just a 3, so when I add, 468 is my product. Well, again, what this suggests is that I can perform the multi-digit multiplication by letting the 10s and higher digits of a product spill over into the next place, and then add the partial products. So for example, if I wanted to do 4 times 253, again, I'll set down my place value chart. I want 4 of each of these things. 4 2s, 4 5s, 4 3s. And I'll record those. 4 2s is 8, 4 5s is 20. So it'd be 20 here, but I'll write that as 2, 0. I'll let the higher digits spill over into the next place. 4 3s is 12. I should write that 12 here, but again, after I do the bundling and trading, the 1 ends up here. The 2 stays in the 1's place. And so there's my products, and I can now add them together. 8 and 2, that's 10, 1 and 2, 1,012 as my product. And this particular approach is sometimes referred to as the partial products multiplication. For example, let's take 7 times 193 using partial products. I'll set down my place value chart. I'll multiply. I have 7 1's, 7 9's, 7 3's. And I'll set those down 7 6321. So there's my partial products. And I'll add those together 1 351 as my product. And after you do this a couple of times, you don't really need that place value chart. The only thing, the only reason, the only thing that that place value chart actually adds is it tells you that I'm dealing with 1000s, 100s and 1s. But we knew that anyway, because we know how to read the number. And well, having this 7 times way over here is maybe a little inconvenient, so we're going to move that 7 over to a more convenient location. And so I might write my partial products as of this in this form. So again, here's my 7 times 193, 7 times 100, 7 times 90, 7 times 3. And there's my partial products. Add them together to get my complete product 1351.