 So let's talk about the arithmetic of decimals, and the most important thing to remember about the difference between arithmetic with decimals and the arithmetic of whole numbers is that there is no actual difference between the two of them. The problem arises when you fail to understand what you're actually doing with whole number arithmetic, and it leads you to errors in performing decimal arithmetic. So, for example, let's take the problem add 173 plus 25, and if we just understand how to do addition by pushing digits around on paper, we know that we're supposed to write a line in the add-ins, and we add them together that's 8, 9, 1, and there's our sum. The problem is that when we try to do the same sum with decimals, well, I know I'm supposed to write a line in the add-ins, so I'll add them up, and I'll get the sum, and the decimal point goes someplace in here I don't know exactly where. And the problem here is two things. One, I have no idea where to put that decimal point, and the other one is that it's actually impossible to recover from this as a wrong answer. Now, the question to ask that is, well, why do we do the right alignment of the add-ins? And the reason we do that is we're really aligning the units. And this would be clear if we used a place-value chart, but otherwise, we note that in our addition 173 plus 25, the 3 and the 5 both indicate the number of ones, so for convenience, we want to actually place them in the same column. So there's our 173 and our 25, the 5 should go in the same column as the 3, and the 2 is the number of 10s, and so when I add them together, I'm going to do my addition correctly. If I add 1.73 and 2.5, in this case it is the 1 and the 2 that represent the number of ones, so I'm going to make sure that those two numbers end up in the same column, and now when I add, I am adding my 100s, my 10s, and my 1s, and I can perform this addition correctly and get my correct answer. Again, there is no difference whatsoever between decimal arithmetic and whole number arithmetic. The only thing that changes is we have this decimal point that tells us where the different units are going to be placed. So, for example, let's consider the problem 12.05 minus 4.87. Because there's no real difference between decimal arithmetic and whole number arithmetic, everything we did with whole numbers still works, so what can I do with this? Well, what I might do is I might use counting up to find the difference, and so here this difference is going to be, I'm going to start at 4.87 and go up to 12.05 and see how far I have to go. So what can I do? Well, 12.87, I might note that if I go up by 1300s, I get to 5. If I go up by 7, I get to 12, and if I go up by an additional 500s, I get to 12.05, and so all together I've had to go up 0.13 plus 7 plus 0.05 7.18. And this is actually a fairly standard cashier's trick because I have 487, 13 cents gets me to 5, $7 gets me to 12, 5 more cents gets me to 1205. And so here's a fairly standard cashier's trick for evaluating such a difference, 7.18. Well, I can do that an entirely different way, 12.05 minus 487. I can use this using counting past, and so I want to subtract 4.87, so I'll start by subtracting too much and returning the change. So here 12.05, I'll subtract 5. I'll subtract more than this amount, and that takes me down to 7.05, except this time I'll return that 1300s that was too much, and that takes me to 7.18, and so my difference, 12.05 minus 4.87, is going to be 7.18.