multijimbo

I recently gave a mathematics talk at the Science Cafe at the Southwestern Arms in Southampton, on the most beautiful formula in mathematics.  The formula is simple, involving only 7 symbols:

e^{i π} + 1 = 0

(I know that there appear to be 10 symbols in the above, but that’s an artefact of how we express a formula without the fancy tools that mathematicians use to make their formulas appear to the eye as works of art.  The symbols ^, { and } should be viewed as invisible, with ^ denoting exponentiation and { } merely delineating what terms form the exponent.)

So, why is this formula beautiful?  And what does beautiful even mean in the context of a formula?   This is one of those questions that if you ask half a dozen mathematicians, you might well get half a dozen reasonable answers.  (In fact, you might get upwards of a dozen.)  I have an answer with which I am comfortable, but this is just me.

I would like to start with the statement that e and π are complicated numbers, and I would like to explain this statement in a bit of detail.  Mathematicians like to categorise and characterise things, and in this case we characterise numbers.  There are lots of different ways of doing this, but this way is very well known.

Think of it as a sort of sifting process.  We begin with the positive numbers, 1 and 2, 3 and 1005, 8 and 10126341763912362, which we can view as the boulders that we sift out first.  Such numbers are often referred to as counting numbers, as they are the numbers of things.  For an explanation of this, I refer to the expert known as the Count.

Contained in the act of counting is one of the great abstractions of mathematics, which is the abstraction of number.   Specifically, how we move from 3 apples or 3 oranges to the concept of 3, divorced from 3 of any specific objects. Closely related to this is the concept of definition.  How do we define 3 as separate from 3 of something or somethings?  All I will say about this is that we mathematicians have ways, but they are not ways to be described here.

What can we do with positive integers?  One direction is to extend to negative integers, but that’s a conversation and a construction for another day and another time.  A different direction to go is introduce complex numbers, but that’s also a conversation for a completely other day.  A third direction is to use the positive integers break the set of all positive real numbers into different pieces.

The first step is to consider the rational number, the fractions, which are the quotients of integers.  As simple as the positive integers are, the rational numbers are almost as simple, in the sense that each rational number is completely specified by 2 integers, the numerator and denominator, to use terms to take us back to warm spring days with mathematics at the front of the room and sun shining outside the window, tempting us from our lessons.

Being able to specify a rational number by 2 integers raises the (not completely) obvious question, which asks, what numbers can we specify using finitely many integers, and how might this act of specifying work.  There are many ways, but one is to build a polynomial, which is an expression of the form

p(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0

where the integers in questions are n, a_n, a_{n-1}, …, a_1 and a_0.  The a_n, …, a_0 are the coefficients of the polynomial and n is its degree.  We can specify a number Z (almost completely) by asking the question, does there exist a polynomial p(x) of some degree n with some collection a_n, a_{n-1}, a_1, a_0 of coefficients, so that p(Z) = 0.  It is an old fact, not as old as Euclid and his geometry but old nonetheless, that for such a polynomial p(x), there are only finitely many numbers which have the property that they yield the value 0 when they take on the role of x in the polynomial p(x). (Yes, I am performing a bit of a fudge here, but I am trying to be a bit informal.)

These numbers Z, for which there exists some polynomial p(x) as described above, with integer coefficients, so that p(Z) = 0, are the algebraic numbers.  Rational numbers are always algebraic (a proof left to the reader, but one I’ll be happy to discuss at a later point should a request come in), and some irrational numbers are also algebraic.  The classical example of an irrational but nonetheless algebraic number, going back to a different ancient Greek mathematician known as Pythagoras, is the square root of 2.

Why go through all this sifting, from integers to rational numbers and from there to algebraic numbers, is that the algebraic numbers form the largest class of real numbers that can be specified by a finite collection of integers.  Beyond the algebraic number lies the vast wilderness of the transcendental numbers, the jungle where π and e both live.

Enough for one day.  I’ll finish this particular story soon and we’ll see where we go from there.

~ by Jim Anderson on 13 October 2014.

Posted in mathematics
Tags: mathematicians, positive integers, rational number