Last week, in Number Sense 011, we watched a goat do a high wire act, and discovered opposite numbers. We also noticed it wasn't clear whether the goat moved or not, when the goat got back to the place he started.
One of the things we can do with opposite numbers is get rid of subtraction. Recall our original number line diagram for subtraction:
We can also use this property to subtract, by “covering up” part of the number we are subtracting from with the number we are subtracting, and seeing what's left
We could turn this into an addition problem, by turning both lengths into arrows, and pointing them in opposite directions
We defined addition as putting the arrows blunt end to pointy end, so adding 5 and -4 does exactly the same thing as subtracting 4 from 5. We no longer need a separate definition for subtraction.
A – B is the same as A + (-B)
Instead of subtracting B we add negative B.
Why is this a good thing? Well, when we add two numbers, the order doesn't matter. 4 + 5 is the same as 5 + 4. This property of addition, called the commutative property, doesn't work for subtraction. 5 – 4 is NOT the same as 4 – 5. When we change our subtraction to an addition, we can use the commutative property: 5 + (-4) is the same as (-4) + 5.
We are looking at doing more with numbers, solving more types of problems, than simple arithmetic. But when we attempt to solve those problems, we have to play by the rules. The commutative property is one of those rules. When we can use it, we have more options than when we can't. It's like playing chess, except, when you want to, you can say your knight is a bishop and move it like a bishop. You might not win all your games, but you'd have a big advantage over someone who didn't know how to do that.
Now, last week we posed the question: how far has the goat moved?
But, in between starting and ending, he went to the spot marked by the green arrow, turned around, and went back to his starting point.
Our observers got into an argument about this question, because the word “move” is ambiguous. Our close observer watched the goat “move” to the green arrow and back, while our lackadaisical observer simply noticed the start and end points, and concluded the goat didn't “move.”
Since they are both right (the goat did travel to the green arrow and back, and ended up where it started) we need some way to talk about, and to calculate, both observations.
We will define what our lackadaisical observer is looking at as “displacement.” Displacement is the distance from the starting point to the ending point. Since our goat ended up where it started, that distance is zero.
Had the goat not come all the way back, the displacement would be some positive number: an arrow pointing to the right.
If the goat continued past his starting point, the displacement would be some negative number: an arrow pointing to the left.
But what of our close observer, who watched the goat continuously. Can he say the goat moved?
No. To be fair, if our lackadaisical observer must give up the word “move” so must the close observer. Our close observer saw the distance traveled along a path.
But how do we calculate it? If we simply add those two arrows, we end up with the displacement. What we really want to do is to ignore the arrowheads and simply add up the two lengths. We need a new definition, something that gives us the unambiguous simplicity of counting numbers, that applies to this new and improved positive and negative integer number line.
Here it is:
The absolute value of a number is its value without regard to its sign. Or, the absolute value of a number is its distance from zero. (These two definitions are really the same, since the value of a number is the distance of that number from zero, and distance measurements are always positive.)
Absolute values are either zero or positive numbers. If we add the absolute values of those two numbers, we will get the distance traveled along the goat's path in getting from start to finish. If we simply add the numbers, we will get the goat's displacement.
Have fun in the comments.