Understanding Absolute Value

Pre-Algebra Tutorial

Understanding Absolute Value

Intro

When I first learned about absolute value, I thought it was a bit…weird. First of all, the notation for absolute value are these vertical bars |x| (literally called “vertical bars”) and the only thing it seemed to do was make a negative number positive. But absolute value means a lot more than just that.

Sample Problem

First, let’s review how absolute value works by answering these problems:

i. What is \lvert 5 \rvert?
ii. What is \lvert -7 \rvert?
iii. What is \lvert 0 \rvert?

i. -5, ii. 7, iii. 0


i. 5, ii. -7, iii. 0


i. -5, ii. -7, iii. 0


i. 5, ii. 7, iii. 0


Solution

Here’s one way of describing absolute value: “The absolute value of a number is its distance from 0”. This way of thinking will definitely get you to the right answer, but you might still question why its useful. Why do we need to know how far things are from 0? To make it a little more clear, let’s move away from abstract concepts like a number line and bring it in to the real world.

Say you’re at home and decide you want to get something to eat. Conveniently, there is a Wendy’s 2 blocks east and a McDonald’s 3 blocks west. Now you don’t want to spend a lot of time getting food, so you want to go to the restaurant that’s closer. So where do you eat? Well, you’re probably going to go to Wendy’s, right? Clearly 2 blocks is a smaller distance than 3 blocks, so, intuitively, the problem is really easy. But guess what? You just used the idea of absolute value to answer that question!

See, there are two pieces of information that describe the restaurants’ locations. One is distance, 2 and 3, and the other is direction, east and west. When you are talking about absolute value, you are only talking about magnitude, or a scalar. That is, you are talking about the number, not the sign (+ or -) or direction.

The reason why the “distance from 0” definition works as well is because, by definition, opposites (i.e. 2,-2 or 5,-5) have the same magnitude/size, they are just placed on opposite directions of the number line.

As a bonus, try to understand the intuition behind an expression like \lvert 5-7 \rvert and how it relates to the expression \lvert 7-5 \rvert. This should also give you some insight into subtraction and how it is related to absolute value.



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