Saturday, October 20, 2012

In Math, Why is Nothing Something?

You may wonder in math why zero is a number? A number is a quantity, a unit to describe something in. So, why should nothing be considered a number? You can't use it to define distance or volume or weight, you can't assign it positive or negative, it even says in its name that it isn't a natural number. Why is it there?

Back when we found out that √2 isn't rational, we used something called proof by contradiction. We assumed that it was natural, and ran into trouble. Let's try that here. We will create our number line with the zero gone.

       -4   -3    -2   -1     1    2     3    4

First of all, try putting a dot where 1/4 would be. If you actually figured out proper guidelines to do it, plot -1/4 with those rules. If you successfully did this and 1/4 is bigger than -1/4, please explain it in the comments section. I would be curious to know.

The true proof is in the relationship in the numbers though. Find the difference between each number in the number line.

     4 - 3 = 1
     3 - 2 = 1
     2 - 1 = 1
 1 - (-1) = 2
-1 - (-2) = 1
-2 - (-3) = 1
-3 - (-4) = 1

Our intervals weren't correct, since there is a two left there. So, we need to figure out what number we forgot in there...

The zero has to be there! We cannot have a proper number line without it.

If you begin to think about other circumstances, you will realize that you need the zero there always. How do you have the origin of a Cartesian plane, or do the Fibonacci sequence backwards?

Using this method, you can also figure out why there isn't positive zero and negative zero. 0 - (-0) = 0, and there can't be an interval of zero in our number line either.

Bonus: In my game theory class this summer, we had a problem where the twelve of us were prisoners, and our teacher told us that he would free us if we could complete the following task:

He had two sets of fourteen cards, each one with a different number on it from one to fourteen. He also numbered each one of us from one to fourteen. He shuffled the cards, and put each one under a different card. So, card one doesn't have to have card one in it, and so on. He then had us discuss our strategy, and then come in one by one and choose an card. If you saw your number on it, you were finished, and if it didn't, you chose a second card. If you didn't get your number after seven cards, he gave all of the prisoners a strike and then reshuffled the cards. Each person went in one by one, and went through that process. We couldn't communicate at all while we were completing the task, we just waited until it was our turn, went in and chose the cards, and then walked out silently without telling anyone what we saw. After all fourteen of us went, we walked back in to hear that each one of us found our number in seven tries or less, and we were free.

There is no strategy that will guarantee you success, but we found a strategy that gets you very close. I want you to try to come up with a strategy that will give you the best odds possible, and comment it. After a month, I will post the best strategy I saw and the one that our class figured out.

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