Hi and welcome. My name is Anthony Varela, and today, I'd like to introduce absolute value. So we're going to define when absolute value is, take a look at how we can perform operations with absolute value, and then we'll talk about some special rules involving absolute value. So to get started, what is absolute value? Well, when we're talking about the absolute value of a number, this is the distance between that number and 0 on the number line.
So here, we have a number line. And let's place a number on this number line. How about 6? Well, what's the distance between 6 and 0 on this number line? We can see that the number 6 is a distance of 6 units away from 0. So we can write this then using absolute value bars as the absolute value of 6 equals 6.
All right. Well, now, let's place, then, a negative number on this number line. So let's take a look at negative 6. What's the distance between negative 6 and 0 on the number line? Well, we can see that the number negative 6 is also a distance of 6 units away from 0. So we could say that the absolute value of negative 6 equals 6.
So one thing that we notice about absolute value is that we can write this as what we call a piecewise function. You don't really need to know exactly what that means, but when we have the absolute value of a number, if that number is positive, it just remains the same. If that number inside the absolute value sign is negative, the absolute value of that is its opposite. So it goes from negative to positive. So the important thing, then, to remember about absolute value is that it's always non-negative.
All right. So now let's go ahead and add and subtract some numbers with absolute value. And the important thing here is to evaluate the absolute value first. So taking a look at this one, we have 3 plus the absolute value of negative 8 minus the absolute value of 5. And we need to evaluate the absolute values first. So I'm thinking, OK. The absolute value of negative 8 is 8, and the absolute value of 5 is 5. So let's make those replacements. And I have 3 plus 8 minus 5, which is 6.
Well, now let's take a look at this one. We have some things going on inside the absolute value bars. So sometimes, absolute value bars can act as grouping symbols or implied parentheses, something we should do first. So we need to evaluate 5 minus 2 and then 2 minus 8, and then take care of our absolute value.
So we have the absolute value of 3 plus the absolute value of negative 6. Well, I know that the absolute value of 3 is 3 and the absolute value of negative 6 is 6. So this reads now 3 plus 6, which I know equals 9.
All right. Now let's talk about some special rules involving products and quotients with absolute value. So let's talk about absolute value and multiplication. Here, I have the absolute value of 7 times negative 8. Well, I know that I have to multiply 7 and negative 8 first. So this is the absolute value of negative 56, which is a positive 56.
Now, thinking about a different way to express 56 using absolute value, I could say 56 equals the absolute value of 7 times the absolute value of negative 8, which leads me to our product rule involving absolute value. I can break down a number, ab, into two products, a and b. And if I'm taking the absolute value, then, of this entire product, I can break that down into the absolute value of a and multiply by the absolute value of b.
Now, let's look at this with division. And a similar thing exists here with division. If I'm taking the absolute value of 24 over 8, I know this is the absolute value of 3, which is, of course, just 3. And thinking about a different way to write the number 3, I can write this as the absolute value of 24, divided by the absolute value of 8. So I can create, then-- I can take a quotient, a over b, and if I'm taking the absolute value of that, I can rewrite that as the absolute value of the numerator divided by the absolute value of the denominator.
All right. So let's review our notes for today. We talked about absolute value as the distance between a number and 0 on the number line. It is always non-negative. And here is our piecewise definition of absolute value. If our number is a positive number or 0, the value doesn't change. If it's a negative number, we return its opposite when taking an absolute value, because it's always non-negative.
Absolute value buyers can act as grouping symbols. So make sure to do everything inside absolute value bars first. And then we talked about absolute value and products, where we can create two absolute value expressions here and multiply them together. And same thing with division. If we have the absolute value of a over b, we can break this into the absolute value of a over the absolute value of b.
Well, thanks for watching this tutorial on an introduction to absolute value. Hope to see you next time.