 The importance of mathematics is its ability to quantify an abstract idea. And one of those notions is this idea of a difference. Now, if we permit signed numbers, we can subtract in any order that we want to. 8 minus 5 is 3, while 5 minus 8 is negative 3. A closely related concept is the difference between two numbers. And so we might say that the difference between 5 and 8 is 3. On the other hand, the difference between 8 and 5 is still 3. And the important thing to recognize is that the difference between two numbers is the larger number minus the smaller number. Now, that works if we know what the numbers are, so we can identify which one is larger and which one is smaller. But what if we want to know the difference between A and B? We can't write A minus B since we don't know which one is larger. The problem is we still need to be able to write down the difference between A and B. And so we can get around this problem of not knowing which one is the larger by using the notation. Now again, how you speak influences how you think. So you might look at this and think, the absolute value of A minus B. However, it's better to read this as the absolute difference between A and B. It's helpful to translate this into geometry. So if we take a number line, the difference between A and B is going to correspond to the distance between the two points. So let's say I want to find the absolute value of note weight. That's the absolute difference between 3 and negative 5. This absolute difference is the distance between 3 and negative 5 on the number line. So let's draw this out. I'll put down negative 5 and then 3 and we want to know the distance between these two points. One way we can determine this is that this point at 3 is 3 units to the right of 0. Meanwhile, this point at negative 5 is 5 units to the left. And that means the distance between the two points is 5 plus 3 or 8. Now we can come back to algebra. This concept of absolute difference and the distance between numbers leads to the algebraic concept of the absolute value of A written in our familiar fashion. And we'll note the connection. The absolute value of A, well A is the same thing as A minus 0. So the absolute value of A is the distance between A and 0 on the number line. And importantly, this relates the algebraic concept of the absolute value to the geometric concept of the distance between two points. So we might find the absolute value of 3 and the absolute value of negative 5. The absolute value of 3 is the distance between 3 and 0 on the number line. So we'll graph that. And that distance is going to be 3. The absolute value of negative 5 is the distance between negative 5 and 0 on the number line. So we can graph that. And that distance will be 5. Of course, part of the problem is that we might not want to mark up our screen because it's too difficult to erase after. So what if it's too difficult to draw a number line? We can go back to algebra and obtain a formula for the absolute value. And so if A is positive, then the absolute value of A is just A. Well if A is negative, then the absolute value of A is negative A.