Representing a Number

Date: 10/29/2001 at 18:03:29
From: M
Subject: Representing a number

How can you represent the number three without using any letters or 
variables or the digits 1 through 9? Only 0 and mathematical symbols 
are allowed. There are two possible answers.

Date: 10/30/2001 at 17:11:58
From: Doctor Ian
Subject: Re: Representing a number

Hi M,

Anything can represent anything else, so long as some group of people 
agrees on the representation. For example, you and I could agree that

  000

will represent the number three. If we do, then it represents the 
number 3, at least to us. (We could also agree that it represents the 
word 'elephant'... and that the word 'elephant' represents what we 
normally think of as a horse.) 

So it's simply incorrect to say that there are two possible answers.  
It would probably be correct to say that your teacher can only think 
of two possible answers, and wants you to guess what they are. It's 
certainly correct to say that there are _at least_ two possible 
answers.  

Having said that, some people - but not all - would agree that 0 
raised to the 0'th power is a way of expressing 1, in which case

  0^0 + 0^0 + 0^0 = 3

would be a way of expressing the number 3. Also, by convention, zero 
factorial is a way of expressing 1, so

  0! + 0! + 0! = 3

is a way of expressing the number 3. 

Note that we can combine these to get more ways of expressing 3:

  0^0 + 0^0 + 0! = 3

  0^0 + 0!  + 0! = 3

If we can use division and parentheses, 

  0! / (0! / (0! + 0! + 0!)) = 3

which is just a silly way of writing

      1
   ------- = 3
     1/3
  
which, of course, is just a silly way of writing 3.  And since

  3 = 1 / (1/3) 

    = 1 / (1 / (1 / (1/3)))

    = 1 / (1 / (1 / (1 / (1 / (1/3)))))

you can crank out as many representations of 3 as you want... or get a 
computer to do it for you. 

But the fun is just beginning:

              0!/0! + 0!/0! + 0!/0! = 3

             0! - 0! + 0! + 0! + 0! = 3

   0! - 0! - 0! + 0! + 0! + 0! + 0! = 3
  
And if you're familiar with set theory, you can use the notation 
#{...} to indicate the number of elements in a set.  For example, 

  #{a, b, c} = 3

which means that 

  #{0} = 1

So now we have another guest to bring to the party:

  0^0 + 0! + #{0} = 3

Note that when you get right down to it, this is really a question 
about how to represent 1, not 3.  

In general, it's hardly ever the case that there are only one or two 
ways to do something. The sooner you realize that - and begin to take 
advantage of it - the more interesting your life will become.

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/