Last week, in Number Sense 010, we added a whole bunch of numbers left of the zero mark on our number line, and called them negative numbers. We placed them on the number line the same way we placed counting numbers on the number line, with one important difference, instead starting at zero and repeatedly adding one to find our tick marks, we repeatedly subtracted one. This combined set of numbers, counting numbers, zero, and negative numbers, we call integers, and integers are closed under addition and subtraction (the counting numbers are only closed for addition and multiplication, since multiplication is repeated addition.) With our expanded number line, we can talk about opposites.
Suppose a goat goes for a walk. Further suppose that he starts his walk from the middle of a rope stretched taut between two mountains.
Also ignore the effects of gravity and the weight of the goat on even the strongest rope, so the rope is perfectly straight. Our sure footed goat is forced to walk in a straight line.
He walks for a certain distance, then he turns around (he's a very agile goat) and walks back to where he started.
Now we ask the question: how far has the goat moved?
We could get two different answers to that question. If someone observed the goat at the red arrow, before he started walking, then looked away, then looked back when the goat finished walking, that person would say, “The goat has moved zero distance, he's simply turned around.”
Someone else, who watched the goat the whole time, would argue, “No, no, first the goat moved all the way to that green arrow, then moved back, you just didn't see it happen, the goat moved twice the distance between the red and green arrows!”
The lackadaisical observer might respond, “Then the goat wasted his time and energy, because I can clearly see he hasn't really moved at all. He may have traveled out to that green arrow, but he's not there now, he is in the same spot as before.”
Are they both right? How can we clear this up?
First of all, we have to define our question's terms better than we have done. What, in fact, do we mean by “how far has the goat moved?”? Our close observer argues “moved” means “engaged in motion” while the lackadaisical observer argues that “how far” means the distance between start and end points, and gives an alternate word, “traveled,” for the idea of “engaged in motion.”
We can take a closer look at the goat's journey, and see that the two blue arrows, representing the trip out to the green arrow, and the trip back to the red arrow, are not pointing in the same direction. One is pointing left to right, the other is pointing right to left. They are pointing in opposite directions.
If we put these arrows next to a number line, to measure their length, we would find that they are exactly the same length.
But they are pointing in opposite directions. Originally, when we measured length of a bar on our number line, we started at zero and found the number at the other end of the bar. Now, we are measuring arrows and arrows start, by convention, at the blunt end, and end, by convention, at the pointy end.
In order to use the same start at zero idea, we need our new, improved number line.
Now both our arrows start at zero, and end at different places on the number line. The arrows go in opposite directions, so we define the numbers they end up at as opposite numbers.
Let's be a bit more precise. When a number is added to its opposite, the sum is zero.
We also need to be more precise as to “adding” since we are adding arrows now, instead of bars.
Adding two arrows means placing the blunt end of one arrow at zero, and placing the blunt end of the second arrow at the pointy end of the first arrow.
There are four ways this can play out:
Positive plus negative (blue arrows)
Positive plus positive (green arrows)
Negative plus negative (purple arrows)
Negative plus positive (red arrows)
(None of these are cases of opposite numbers, we are looking simply at adding arrows now.)
If we add opposite numbers, we will end up at zero. We can ignore adding positive to positive, and negative to negative, since both arrows are going in the same direction, we will never get back to zero.
So if we start with a positive number, say, 3, its opposite must have the same length but be in the opposite direction: -3
If we start with a negative number, say, -4, its opposite must be the same length but be in the opposite direction: 4.
Have fun in the comments.