Prelude
With numbers, we started with counting numbers: {1,2,3,…}
After that we figured out we needed zero. After that we figured we needed negative numbers: {-1,-2,-3,…}
After that we figured we needed more numbers and we invented the “rational number” which is an integer divided by an integer. As an example, what we call “one and a half” – and on digital calculator it is 1.5 – is 3/2. Notice that both 3 and 2 are integers. That is as far as we need to go with math for this story.
The Chase
A tortoise wakes up and discovers, in horror, that he is next to a sleeping vampire rabbit.
“I got to get out of here!” he thinks in horror, and he desperately starts his trek to get away from the rabbit.
The speed of the tortoise is one womp per second. We won’t worry about what exactly the length of a womp might be. Tina Turner asked us to work with numbers that were nice and easy. Later on we can change over to nice and rough.
The tortoise travels a distance of a thousand womps and then the vampire rabbit wakes up, crazed and famished and he sees the tortoise.
“I’m gonna get you, yes I can!” the vampire rabbit shouts.
The speed of the vampire rabbit is 10 womps per second and he has the stamina to run several thousand romps before he stops running. Rabbits were made to run.
At this point the calculus in your pirate brain tells you that it is game over for the tortoise.
But the tortoise thinks he has a chance. Here is his train of thought:
In the time it takes the vampire rabbit to go 100 womps, I’ll go 10 womps.
When he gets to where I was, I will be further along.
In the time it takes the vampire rabbit to go 10 womps, I'll go 1 womp.
When he gets to where I was, I will be further along.
But the vampire rabbit catches the tortoise and defeats him.
The Math
Now, what happened with the math?
We can build a table of distances that the vampire rabbit had to travel:
100
10
1
0.1
0.01
And it goes on and on and on—to infinity. Yet, when we added all these distances together, they added up to the distance the vampire rabbit ran in order to catch the tortoise.
Now we turn to Algebra, and ask our friend Abu Musa Jabir Ibn Hayyan to do the calculation.
The distance the vampire rabbit travels is easier to put into a formula: r=10t
As an example, after five seconds the rabbit has traveled 50 womps. 10(5)=50.
The distance traveled by the tortoise is a little harder: t=t+1000
As an example, after five seconds the tortoise has walked 5 womps and he had a head start of 1000 so that adds together to make 1005 womps. 5+1000=1005.
The vampire rabbit catches the tortoise when his traveled distance equals the distance traveled by the tortoise. This happens when the following is true:
r=t
10t=t+1000
We now watch Hayyan do the algebra:
10t-t=1000
9t=1000
t = 1000/9
If you start doing the long division for 1000/9 you quickly notice a pattern…
111.111111111….
The value 1000/9 is a number and it contains an infinite number of 1’s—it has a 1 for every 1 that the tortoise thought would help him to escape.
The theme of this story isn’t about rational numbers, although if, in some way, this made them a little more fun—great!
The theme of this story is about something we do in our pirate brains.
Previously we threw a ball at the wall knowing it would hit the wall even though we had a “game” that might make someone believe it would take forever for the ball to reach the wall.
Now, we have a case where it might seem the math could keep a positive separation distance between the rabbit and tortoise, keeping the tortoise alive, and we sought to explore the scenario—having faith and belief that, when we played with the math, we would find something that both be interest and resolve the question.
In the future, this will happen again. These problems are like Krackens. We approach them cautiously, but with faith, and we defeat them.
