In the last post, we had poor Carlos trying to catch the bus. To understand the third situation we need a bit of knowledge of square roots. You see, when we take square roots, we usually only allow positive numbers. For example:
But is not usually defined. Now, if you had to try to define , you would want it to be something like 3, since it looks kind of like it should be 3. However, you also want to differentiate it from the regular 3. So why not just add a symbol and call it:
Before you declare this approach crazy, consider this. What is 10-3 ?
10-3 = 7 of course.
What if we reverse the order. What is 3-10 ?
-7 of course.
Normally in subtraction you have a bigger number and you take away a smaller number. When the situation changes we simply introduce a new symbol and invent a new word, “negative”.
I hope that justifies my methods. If not, just humour me for now.
Returning to Carlos, when I use the formula for the first situation I get the values 2 and 5. As you can see from the graph, these are points of when the blue line (Carlos) red line (the bus) intersect.
When I use the formula for the second situation, I get the value 3.16. Again, this is the point where Carlos catches up with the bus (where the two lines intersect).
When I use the formula for the third situation I get an interesting answer. I get:
What does this solution mean? It can’t mean that Carlos catches the bus, because the lines do not intersect. However, Carlos does get close to the bus. The proper interpretation is as follows.
The first number, 2.5, means that after 2.5 seconds Carlos will be as close as possible to the bus. Before 2.5 seconds, he was gaining on the bus and after 2.5 seconds the bus gets further and further away. The second number, 1.9, means that at 2.5 seconds, Carlos is 1.9 meters away from the bus. This is the closest that Carlos will get to the bus. If Carlos had a friend on the bus with a really long pole, maybe he could grab on to the bus. But otherwise, he will miss it.
Even though Carlos does not make his bus, the math can still tell us interesting information about the situation. This is preferable to the usual “no solution” we are taught in school.
What I have used are the numbers more formally know as the complex numbers. While some people might say they are imaginary numbers, I say, that real or not, they are incredibly useful.