 So let's talk about division of integers, and it's helpful before proceeding to introduce the idea of the absolute value of a number. And this is introduced as following, the absolute value of a, written this way, we place the a in between a set of vertical bars, is going to be defined as follows. If a is greater than or equal to zero, the absolute value is just the number itself. Otherwise, if a is less than zero, the absolute value is going to be the additive inverse of whatever the number is going to be. You can think about the absolute value as the number without any indication of a plus or minus. If we're talking integers, then you can think about the absolute value as being the whole number that corresponds to a given value. And if I introduce absolute values this way, I can find products much more easily in the following way. First off, if I want to find the product a times b, the first thing I'm going to do is ignore the signs of a and b completely and just multiply the absolute values. So I'm going to start off with c as being the product of the two absolute values. And then I'm going to assign c a value, whether it's going to depend on whether a and b are positive or negative. And I will make c positive if both a and b have the same sign. If they're both positive or both negative, then likewise, if one of them is positive and the other one is negative, then I'll make c have a negative value. And more generally, if c is a product of several numbers, c is going to be positive whenever there's an even number of negative factors because the negative factors pairs of them give us a positive factor. Again, pairs of positive factors, any positive factors doesn't chase a sign. And likewise, if there's an odd number of negative factors, then I'm going to have at least one positive and negative pair, and that's going to give me my negative. Now, for example, a is positive or negative in the product a times 5. And we note here that I'm multiplying two numbers and getting a negative value. And the only way that that can happen is if I have an odd number of negative factors. Well, the only factor that we have in sight is 5 is positive, which means that the other factor has to be negative. Now, how does this apply to division? So remember, there's two related definitions of division. One is that a divided by b is equal to c if and only if a is the product, c times b. And the other one is that a divided by b is some quotient with remainder if and only if a is b times quotient plus r where our remainder falls in this interval between 0 and b. For now, we'll focus on that first definition and let's see if we can find and prove product 24 divided by negative 3. So comparing our definition of division, we know that a divided by b equals c if and only if a is the product of c and b, which if I compare my definition of division to my statement, I see that a and 24 must be the same thing. b and negative 3 must be the same thing. And then c is whatever I end up with, well, I don't know what that is. But it's going to be the thing that's going to make this statement true. So, let's see, ignore the signs for a moment. I see that 24 is the product of 3 and something else. Well, that's something else has to be 8 and what I might try is just right in an 8 there. The problem is 24 equals 8 times negative 3. Well, that's not true. And the problem is 8 times negative 3 gives us the wrong sign that's negative 24. Because I want a positive product, I want to make sure that the two factors are going to both have the same sign. Well, I'm stuck with this negative 3 that emerged from the problem, which means that I have to change the sign of this 8. So, that tells me that 8 is going to be another factor which must also be negative. So, my c value has to be negative 8. And now I have the statement 24 divided by negative 3 is c, if and only if 24 equals negative 8 times 3. Well, this is true. Negative 8 times negative 3 is in fact 24. So, 24 divided by negative 3 is negative 8. And that allows us to fill in the details and complete our proof. So, again, I have to start with the thing I know to be true. 24 is negative 8 times negative 3. Then my definition of division allows me to invert that 24 divided by negative 3 gives me negative 8. And there's my simultaneous answer and proof. Now, remember, as with multiplication, we can find an algorithm for performing interdictory divisions. And so, again, to find the quotient A divided by B, essentially what we did is used the definition of multiplication as the inverse operation to addition. We found the quotient by ignoring the signs entirely. And then we assigned the quotient to sign depending on whether A and B are both positive or both negative.