 Okay, I've got two decimal numbers here, and I'm just going to walk through the process for forming each of these operations on the pair of numbers, just to remind us of how these operations work. So if I start with addition, I've got 52, 39, and I'm going to add 38, 56 to this. So, I start from the right, and I've got 9 plus 6. 9 plus 6 is 15, so I'll write down a 5, and I'm going to carry the 1. Now, I've got 1 plus 3 plus 5, which gives me 9. So, I'll write down the 9. There's nothing left to carry. Now, I have 2 plus 8, which gives me 10. So, I write down a 0, carry a 1. Now, I have 1 plus 5 plus 3, which again gives me 9. So, my result is 9,095. So, anytime I had a number that's larger than 10, I'm writing down the last digit, and I'm carrying a 1. Since I've only got two numbers to add together, I can only ever carry a 1. But otherwise, I'm looking at one pair of numbers at a time, and just moving from right to left doing the subtraction as normal. If I do subtraction, I'll start with the 52, 39, and I'm going to subtract 38, 56 from that. So, 9 minus 6 is simple. That gives me 3. But I can't subtract a 5 from a 3. 5 is larger than 3. So, I really need something larger than 5. I'm going to go borrow a 1 from this 2 over here. So, if I subtract 1 from my 2, it becomes a 1. But now, I have a 10 over here that I can add to my 3. So, I have 13. Now, I can subtract 5 from 13. That leaves me with 8. Now, I'll go over here. I've got 1 minus 8. I want a smaller than 8. So, I'm going to need to borrow something from my next position. 5 will become a 4, and the 1 will become an 11. Now, I can do 11 minus 8, leaves me with 3, and 4 minus 3 leaves me with 1. So, 52, 39 minus 38, 56 gives me 13, 83. Multiplication and division are going to be a little bit longer and harder, and we're going to do a lot of addition or subtraction in the process. So, if I start with 52, 39, and I want to multiply that by 38, 56, I'm going to start with the rightmost digit in my, the number on the bottom, and I'm going to multiply that by everything in the top number. So, 6 times 9 is 54. So, I write down the 4 and I'm carrying the 5. 6 times 3 is 18, and now I'll add in the 5. That will give me 23. 6 times 2 is 12, plus 2 is 14. 6 times 5 is 30, plus 1 is 31. Now, we'll do the same process again, but this time I'll be looking at the 5, multiplying it by everything on top. So, 5 times 9 is 45. So, I don't need this 5 anymore. I'm going to have a 4 that I carried. 5 times 3 is 15, plus 4 is 19. 5 times 2 is 10, plus 1 is 11. 5 times 5 is 25, plus 1 is 26. And again, with the 8. So, 8 times 9 is 72. Down the 2, carry a 7. 8 times 3 is 24, plus 7 gives me 31. 8 times 2 is 16, plus 3 is 19. And then, 8 times 5 is 40, plus 1 is 41. And finally, I'll repeat the process again for the 3. So, 3 times 9 is 27. Write down the 7, carry a 2. 3 times 3 is 9, plus 2 is 11. 1, carry a 1. 3 times 2 is 6, plus 1 is 7. 3 times 5 is 15. Now, I just have to add up all of these numbers that I've got. 4, 3 plus 5 is 8. 4 plus 9 is 13, plus 2 is 15. 5 carry a 1. 2, 3, 4, plus 7 is 11. So, 4, 10, 20. 2 plus 2 is 4, plus 1 is 5. Plus 7 gives me 12. Then, 1 plus 4 is 5, plus 5 is 10. And then, 1 plus 1 is 2. So, I get the number 20,201,584. If you're wondering why this algorithm works, think back to what these numbers meant when we were looking at base changes. Each of these positions actually has some meaning and carries a magnitude term with it. And when we're multiplying a couple of those magnitude terms together, each of them, it ends up shifting our number over. So, the shifting that we see here is actually an effect of the magnitude terms that we have attached to the larger numbers in the bottom number. So, if we go on and do this again for the division, I will start with 52, 39. And we're going to divide 38, 56 into this number. So, obviously this is not going to divide cleanly. We're going to have a long fraction attached to it. But we should be able to do this as well. So, 38, 56 is not going to go into 5. So, I can put a 0 there. It's also not going to divide into 52. Still not going to divide into 523. But it is less than 5239. So, it looks like it will go in about once. Twice would be 7,600 something. So, we should be able to subtract 38, 56 from this. And since we actually already did that arithmetic, we know that it's 1,383. So, if I wanted to do integer division, I would say that 52, 39 divided by 38, 56 goes in once with a remainder of 1,383. If I want to calculate a fraction though, I'm going to have to go on. So, first thing I'll do is I'm going to see, going to say that there's a whole bunch of zeros after this number, infinite number. So, I can go on calculating my fraction as long as I'd like. And I'm going to pull down one of those zeros. Now, my 38, 56 should go into 13,830 several times. And I'm going to guess that it goes in about four times. So, I'll start by figuring out what is 38, 56 times 4. So, I get 24, 22, 34, 12 plus 3 is 15. So, 15, 4, 24 was too large. I need to go back to this again with the multiple of 3. So, 3 times 6 is 18, 15 plus 1 is 16, 24 plus 1 is 25, and 9 plus 2 is 11. So, 11,568 is less than 13,000. So, I can write down a 3 as my next digit, and I will subtract 11,568. Now, when I do this subtraction, you know, I need to borrow something. So, this will be a 2. This will be a 10, giving me a remainder of 2, 2 minus 6. Now, I need to go borrow something. So, my 8 will turn into a 7, my 2 is now a 12. So, 12 minus 6 is 6, 7 minus 5 is 2, 3 minus 1 is 2, and 1 minus 1 is 0. So, now I have 22, 62, which is less than 38, 56, which is good. So, I'm, so I will pull down another 0. Now, I need to know how many times 38, 56 goes into 22,620, and should be at least 5 times. So, we'll try calculating for 5. So, 38, 56 times 5. 6 times 5 is 30, 5 times 5 is 25, plus 3 is 28, 5 times 8 is 40, plus 2 is 42, 5 times 3 is 15, plus 4 is 19. So, 19,280 is less than 22,000, and it looks like it should be enough less. If I added 3800 to this, I would get something around 23,000, which would be larger than this. So, I'm going to write down 5 as my next digit, and I'm going to subtract the 19,280. 0 minus 0 is 0, 2 minus 8, and I'll borrow something from my 6, turns into a 5, now I have a 12. 12 minus 8 gives me 4, 5 minus 2 is 3, and 22 minus 19 will also give me 3. So, 33, 40 is less than 38, 56, which is good, and I can pull down another 0. But since 33, 40 was pretty close to 38, 56, I'm going to guess that 38, 56 goes into this about 8 or 9 times. So, 10 times would be 38,560. If I subtract 4,000 off of that, that would leave me with about 34,000, so I'm going to guess that 8 times is about right. But I still have to do the arithmetic to find out just how large that actually is. So, 38, 56 times 8, 6 times 8 is 48, 8 times 5 is 40, plus 4 is 44, 8 times 8 is 64, plus 4 is 68, 3 times 8 is 24, plus 6 will give me 30. So, I would write down 30,848, and I'd put an 8 up here. So, this will become a 3, this turns into a 9, then I have a 1 subtract from. So, 10 minus 8 is 2, 9 minus 4 is 5, 3 minus 8, I need to go borrow some things again. 2, now I have a 13. 13 minus 8 will give me 5, 2 minus 0 is 2. So, I can continue doing this until either I have 0 left over, in which case any more division would be pointless, or until I get bored. This point I'm pretty much bored. We can say that 3,856 goes into 5,239, about 1.358 times, and some more change.