Meaning of absolute value

In this video, we're going to introduce ourselves to the idea of absolute value, which you can view as how far you are from zero.

So, for example, let's say that we have a bunch of people living on a street, and let's say that we say that the school is at zero. We could think about where other people live. So, let's say I live three miles to the right of the school—maybe that's east. So, this is where I live—so that's me right over here.

And that you live three miles to the left of the school. So this is where you would be: three miles to the left, or maybe we could say that's to the west of the school. Now, you could say that your position is at negative three miles relative to the school, and my position is at positive three miles relative to the school.

But what we might care about is just how far we are from the school. And so there you could say, well, we're each exactly three miles from the school. You're three miles from the school, and I am three miles from the school, and that's what absolute value is actually trying to get at.

So, for example, your position is negative three, but if we were to take the absolute value, which you denote by these two straight up and down bars around the number, then we say that's just going to be your distance from zero, which is just going to be three miles. Similarly, my position is at three miles—positive three miles. The absolute value of that is going to be, drum roll please, it is going to be three as well.

You might already see a pattern: if you're taking the absolute value of a negative number, you get the number without the negative, and if you take the absolute value of a positive number, it just equals itself again. But the reason why that is the case is because we're just saying how far that thing is from zero.

Let me give another example. Let's say that we have some type of a cruise ship, and that's the water right over there. This is the cruise ship—let me draw it. This big cruise ship, nice and big, one has water slides on it and whatever else, whatever you would expect from a huge cruise ship.

And let's say we think about the height of where the different floors are in the ship, and a natural place to think about is based on where the sea level is. So, we call that height zero. Now, there's this big waterfall pool-type thing up here. Let's just say it's a pool for simplicity on the roof, and let's say that that is at an altitude of 80 feet, so we would call that at positive 80 feet.

And let's say the engine room is right over here, and if you wanted to know its position, its height, well, it is going to be below zero—it's below sea level. Let's say it's at negative 20 feet, so that would give you its position. One is 80 feet above sea level, and then one is 20 feet below sea level.

But if you just wanted to know how far they are, well, you could take the absolute value. How far is that roof deck pool? When you take the absolute value of 80, you're going to get 80. How far is that engine room from zero? How far is it from sea level? Well, the absolute value of negative 20 is going to be 20.