Absolute value is an important concept in mathematics. The duality of absolute value makes this concept problematic and hard to grasp for students. Yet this need not be the case. When looking at absolute value for what it really is, that of the distance from a given point to 0 on a number line, we can put this abstraction into its proper perspective. Let us explore this topic in more detail so that it never presents a problem again.
The absolute value of a number is simply its distance to 0 on a number line. The symbol used for absolute value is the straight brackets “| |” with a number or variable placed inside. Thus |3| = 3 because 3 is 3 units from 0 on the number line. The duality of absolute value comes into play because the absolute value of both 3 and its additive inverse, or -3, are the same, namely 3. Both 3 and -3 are 3 units from 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 opposite of the number. This seems all too simple. So why does this concept present problems?
Well introduce a variable into the absolute value expression and all hell breaks out—literally. The reason is simple: a variable stands for some unknown number. The key word in the previous sentence is unknown. That is, we do not know whether the variable stands for a positive or negative number. Take the expression |x|. What does this equal? Well that all depends. Is x negative or positive?
If x is positive, then the expression |x| is simply equal to x; however, if x is negative, then the expression |x| is equal to -x because the “-” symbol in front of x makes this quantity positive. Remember two negatives become a positive. Read the preceding again because this is where all the “sticky-ness” comes into play. Most students will say erroneously that the |x| = x because they fail to consider the duality of absolute value. That is, when we do not know what is inside the absolute value symbol, we need to consider both cases; that is, when what is inside is positive, and when it is negative. If we do this, then absolute value will never be a problem again. To make this clear let x = 3. Then |x| = |3| = 3 = x; however, if x = -3, then |x| = |-3| = -(-3) = 3 = -x.
So do not cower when you see or hear absolute value. Just remember that all this means is the distance to 0 on a number line, and that one needs to consider both the positive and negative cases when dealing with a variable expression. If you do this, you will never shrink before such expressions. You can then add yet another feather to your mathematical cap.