 What about multiplication and division of decimals? Well, again, everything that we do with whole number arithmetic translates into decimal arithmetic, provided we keep in mind that we are going to keep track of where the place values are. Again, all of arithmetic is essentially bookkeeping how many of each unit do we have. And so decimal multiplication really starts off, really relies on our commutative and associative properties of multiplication, together with the property of multiplication by the base. So, for example, let's take a look at 25.7 times 0.3. And the idea here is that I can include factors of 10 and 0.1 as I want to. If I include both of these 10 times 0.1, the product 10 and 0.1 is just going to be 1, so I can include as many pairs of 10 and 0.1 as I want to. Well, in this case, I might want to include one pair each. Commutativity and associativity allows me to rewrite this product this way. 25.7 times 10 is 257.3 times 10 is 3, and then the 0.1 stays long. Commutativity and associativity allows me to rewrite things this way. That gives me 771, 257 times 3, and then times 0.1 times 0.1 again. So that's going to give me my final product, 7.71. Now, all together, the thing that's worth noting is that this process is equivalent to ignoring the decimal point for a moment. That's 257 times 3, and then re-placing where that decimal is. And so to get this to a whole number, I had to multiply by 10. To make this a whole number, I had to multiply by 10. So I multiplied by 10 and by 10, so I have to divide by 10 and by 10. And so I do the product 257 times 3, and then I place the decimal point by restoring those factors of one tenth that I dropped down. Division is very similar, and again here we have to go back to the key idea for what a division is. A division is a product that has been reversed. So I have this product A times B equals to C. That gives me the division A is equal to C divided by B. Well, something I can do is if I multiply everything by the same number M, then again, associativity gives me the quotient A is C M divided by B M. So if A is C divided by B, A is also C times M divided by B times M, which we used as an earlier approach to dividing two numbers. So for example, let's say I want to divide 14.5 divided by 0.05. Well, by the preceding argument, I can multiply both numbers by the same amount without changing the actual quotient. So maybe I'll multiply both of these by 10. Well, multiplication by 10 is easy. That gives me 145 divided by 0.5. And I might be able to do this. On the other hand, maybe I don't want to. So I still have to work with decimals here. So let's change this a little bit more again. I can multiply both things by the same number and still get the same quotient. So again, if I multiply by 10 once more, I get 1450 divided by 5. And I can do that. That's 290. And there's my quotient.