How I view numbers

Let us consider the following game.
The Game
Two players. Player A and Player B. For the sake of simplicity, let us say I am player A, and you are player B.
There are 6 squares and we begin the game with you standing on Square 1. I ask you questions which can be answered in “Yes” or “No” and I instruct you to do certain tasks. There are two types of tasks. One which will require you to move from one square to another. And the other requires you to write a number on the Sheet of paper that you have.

Again, we shall assume that you have infinite supply of paper. And you can only use the symbols {0,1,2,3,4,5,6,7,8,9,/,i,.} while writing. So for the sake of convenience we use the standard notation for writing numbers.

The Rules:
Square 1 : I give you two numbers say a1 and b1. Now move to Square 2.

Square 2 : Can you say which of these numbers is smaller ?

If your answer is “Yes” then let us call the smaller
number as ‘a’ and the bigger number as ‘b’. Now go to Square 3.

If “No” we go back to Square 1.

Square 3: Are there any numbers between ‘a’ and ‘b’?
If your Answer is “Yes” then think of one such number which we shall call c. Now move to Square 4.
If your answer is “No”, then we move to Square 6.

Square 4 : Now write down on the paper the number ‘a’.
Move to Square 5 once you are done with writing the number.

Square 5: Are there any more numbers between ‘c’ and ‘b’ ?
If “Yes” then let us call ‘c’ as ‘a’ and ‘b’ as ‘b’. Now go to Square 3.
If “No”, go to square 6.

Square 6: The game is over.

The Objective
The objective of Player A, that’s me, is to ensure that I restrict the movement of the player B (you) and not allow you to go beyond a particular Square.

My ability to do this varies depending on the nature of the numbers that we are dealing with.

Nature of the numbers

When I gave you two numbers or asked you to think of a number, I didn’t specify what kinds of numbers we are talking about. So, what kinds of numbers are we talking about ? The ones that we all learnt in high school. Natural Numbers, Rational Numbers, Real Numbers and Complex Numbers. We shall restrict ourselves to these kinds of numbers for now.

How does it change anything for me or you?


1) Natural Numbers:

Suppose we are dealing with only natural numbers. Then, my strategy would be to give you two numbers which are as far away from each other as possible. I could start off with say something like a=1 and b=1267650600228229401496703205376. But here’s the deal, no matter how bad you decide to play, eventually you’ll end up picking a number ‘c’ between ‘a’ and ‘b’ such that ‘c’ = ‘b’-1. And after that ?
After that when I ask you are there any more numbers between ‘c’ and ‘b’ you have to say “No”.
And invariably you would reach this state. If you are smart, at the very beginning you would think of the number ‘c’ as ‘b-1’ so that the game gets over in just one round.

But the important point is that the game gets over and the number of numbers you have written on your piece of paper are only going to be finitely many. Further more, you will be able to cover all the squares. Thus I have absolutely no ability to control anything here.

2) Rational Numbers:

Now, if instead we play the game with rational numbers instead of natural numbers. This time, my strategy would be to pick any two rational numbers. And no matter what, you will be able to think of a number ‘c’ which lies between these two numbers. That is , suppose I think of a=4.3344 and b=4.3345, you can think of a c=4.33445. Thus this goes on and on. And as a result you will be only moving between squares 2 and 5. You will never get to reach the Square 6. No matter how small the difference between a and b is, it is still not zero. And hence you will be able to find a number between the two.

The reason why you lost the ability to cover all squares as compared to natural numbers is because unlike in the case of natural numbers, here we don’t have the ability to say what’s the previous or the next rational number. Because between every two rational numbers there is another rational number. We call this property as density.

Thus on your paper, you will have written “infinitely many” rational numbers. Well, if I were you I would be happy because I am getting to at least write so many numbers.

3) Real Numbers:

With real numbers, my ability to restrict you would have increased even further. With rational numbers if I was able to restrict you to just squares 1-5, this time around I can restrict you to only squares 1-4. How ?
Well I pick ‘a’ as square root of 2 or pi or “e” or one of these “irrational numbers” which are included in the class of the real numbers. What about ‘b’ ? Well, I don’t even need to worry about ‘b’. I can pick any number as ‘b’. Because you my friend, when you reach Step 4 and start writing out ‘a’, you’ll never finish. Nope. Say I give you a=sqrt of 2. And you say that a is 1.41421356. Well, not good enough, because when we multiply two of these we shall never get 2 but something less than 2.

Thus, here you wouldn’t even finish writing the number having started it. What is it that made you lose this ability ? Well, in case of the rationals, each number had some sort of a boundary which could not be extended further. Where as in case of irrational numbers, there is no such boundary. In fact you keep going and you’ll never hit the boundary. And whatever you have written so far always falls short of the mark.

Thus when you transitioned from Rationals to reals, you lost the ability to say where “a” number ends.

Of course, if you were an optimist you would be happy that you are at least getting to write something!

4)Complex numbers:

This is where I have all my powers! I won’t let you go beyond Square 2. Well, all I have to do is pick ‘a’ as square root of -1. And again, I don’t have to worry about what I pick as ‘b’. Because, no matter what you won’t be able to answer the question “Can you say which one is smaller?” [Note: Technically you still will be able to answer this, but not in the conventional sense. See here]

That’s because as we move from real numbers to complex numbers we don’t have a “number line” any more. And we would have left behind the natural ordering that exists in other varities of numbers.

And as a result, you do not get to move any further than Square 2. And as a result you don’t even get to use the infinite supply of paper that you have around!

This is how I have been looking at different classes of numbers!


About gautshen

A jack of many trades of which , Linux Kernel Programming puts food on the table. Also pursuing his PhD in the area Theoretical Computer Science at the Chennai Mathematical Institute. Is an avid reader interested in the Hindu traditions and philosophy. Loves Bicycling and Good Music. Name is Ranjal Gautham Shenoy.
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