Thursday, November 20, 2008

Geometrical representation of numbers

This post s a response to a comment I got today. I was asked if there are numbers that cannot be represented at all using geometry.
The answer to this question depends greatly on what you consider to be a number, and what you consider to be a geometrical representation.

What is a number?
One possible approach is to define that x is a number if and only if it belongs to R (the set of all real numbers). There are different ways to define this set, but it can be proven that all those definitions give the same set, so this definition of a number doesn't have any problems.
However, there are objects that don't belong to R but are considered numbers (at least by some people). The most obvious example is the complex numbers. It can be easily shown that the "number" sqrt(-1)=i doesn't belong to R, so if we want to consider it as a number we need to extend our definition to include all the complex numbers - in other words x is a number if and only if it belongs to C (the set of all complex numbers).
It turns out however that even this definition can be extended. In addition to i we have other imaginary "numbers" - the infinitesimals (a non negative number smaller than any positive number) and infinity. Neither of them belong to R or to C.
Surprisingly, even this is not the end. There are also cardinal numbers - a cardinal number is basically a size of a set, so for finite sets this is just a natural number. For infinite sets a cardinal is a "number" (and there are infinitely many different sizes of infinite sets, so there is an infinite os such cardinals) that doesn't belong to any of the sets mentioned above (except for the cardinal of the set of the natural numbers).
It is possible to bring more examples of objects that can be called numbers but, in my opinion the best definition is the first one - x is a number if and only if it belongs to R. But you are free to chose the one you like.
Geometrical representation
It is important to notice that there are two different definition of what a geometrical representation is. The definitions are:
1. A number has a geometrical representation if there is a point on the real line that corresponds to this number.
2. A number has a geometrical representation if a line segment of a corresponding length can be constructed geometrically (using compass and straightedge alone).

Since R can be viewed as the real line, it follows immediately that all the numbers in R has a geometrical representation according to the first definition. It turn out that if you extend this definition of the numbers to include all C, there is also a geometrical representation, because every complex number can represented by a point on a plane. Infinitesimals, infinity and cardinals don't have such a representation.

The second definition is much more strict. The ancient Creeks never asked if there are numbers that cannot be represented in such a way, but it turned out (at about the 16 century I think) that there are such numbers. To better understand this, lets first see some examples. Lets look on the following numbers - 2, 9, sqrt(2) , pi.
It is obvious that we can construct the first two. All we need to do is to decide what we call a line segment of length one, and we are done. We can now draw 2 such segment to get 2 and 9 to get 9. For sqrt(2) it is a bit more complex, we need to make a right triangle with sides 1 and then we will get that the third side is sqrt(2). Generally it has been proven that a number that is a root of a polynomial with rational coefficients can be somehow constructed under the restrictions we put on ourselves. It was also proven that a number that is not a root of any such polynomial cannot be constructed in such a way. Not so long ago (in the last century) it was shown by Cantor that most numbers (numbers according to the first definition) are not constructible. Such numbers are called transcendental.
It is important to note that numbers that belong to C also can be constructed (not all of them, but some of them) this happens because C and R are sets of the same cardinality so you can assign a number in R for any number in C.

No comments: