Absolute value is an important concept in mathematics. The duality of absolute value makes this concept problematic and difficult for students to understand. However, this does not have to be this way. By looking at the absolute value of what it really is, that of the distance from a given point to 0 on a number line, we can put this abstraction in its proper perspective. Let’s explore this topic in more detail so you never have a problem again.
The absolute value of a number is simply its distance from 0 on a number line. The symbol used for the absolute value is the square brackets “| |” with a number or variable inside. So | 3 | = 3 because 3 is 3 units of 0 on the number line. The duality of absolute value comes into play because the absolute value of 3 and its additive inverse, or -3, are equal, that is, 3. Both 3 and -3 are 3 units of 0 on the number line.
The only thing to remember with absolute value is that if a number is positive, then the absolute value is equal to the given number; however, if the number is negative, then the absolute value is the negative or the opposite of the number. This seems too simple. So why does this concept present problems?
Well, put a variable in the absolute value expression and hell breaks out — literally. The reason is simple: a variable represents an unknown number. The keyword in the previous sentence is unknown. That is, we do not know if the variable represents a positive or negative number. Take the expression | x |. What does this amount to? Well, it all depends. Is x negative or positive?
If x is positive, then the expression | x | it is simply equal to x; however, if x is negative, then the expression | x | equals -x because the symbol “-” in front of x makes this quantity positive. Remember that two negatives turn into positives. Read the above again because this is where all the “stickiness” comes in. Most students will mistakenly say that | x | = x because they do not consider the duality of the absolute value. That is, when we do not know what is inside the absolute value symbol, we must consider both cases; that is, when what is inside is positive and when it is negative. If we do this, absolute value will never be a problem again. To clarify this, let x = 3. Then | x | = | 3 | = 3 = x; however, if x = -3, then | x | = | -3 | = – (- 3) = 3 = -x.
So don’t flinch when you see or hear absolute value. Just remember that all of this means the distance to 0 on a number line, and that both positive and negative cases need to be considered when dealing with a variable expression. If you do this, you will never flinch from such expressions. You can then add another pen to your math limit.
