the numerology of unrelated constants 2

A long time ago, I wrote about the most beautiful equation in mathematics and the numbers involved, π and e.  I’ve started slowly working through my backblog and realized that I never finished that particular tale, as I promised.

Looking back on it, it strikes me that I’m not sure I was writing to an appropriate audience.  This is part of a larger internal ponderation I’m having, about writing mathematics in general and writing mathematics for an audience of non-mathematicians in particular.

I have come to believe that spending so much of time thinking about mathematics, and teaching mathematics to others, has shaped how I approach many things.  When I engage in discussions about rules and regulations, process and procedures, I like the ground to be firm beneath my feet.  I like to know the definitions of the terms we use in those discussions, and the rules of argument that we use in our deliberations.

But I think that it does sometimes get in the way when I’m trying to write.  I tend to write very slowly, and there are times I think that my slowness of writing stems from this need to have a clear picture, from beginning to end.  This is relatively straightforward in a piece of mathematics, but much less clear in a work of fiction.  Or a blog.

So back to the matter at hand.  In the previous section, I was talking about levels of complication of numbers.  We started with the rational numbers, quotients of integers.  I then defined the algebraic numbers.  These numbers have the technical definition that they are solutions to equations of the form p(x) = 0, where p(x) is a polynomial with integer coefficients.  Another, heuristic, explanation is that the algebraic numbers are those numbers which can (essentially) specified by a finite collection of integers, much in the same way that rational numbers are specified by pairs of integers.

These levels of complication are related to one another.  Every rational number n/m is an algebraic number, where the polynomial is p(x) = mx – n (since p(m/n) = n (m/n) – m = m – m = 0).  In fact, the rational numbers are the simplest algebraic numbers, if we take as our yardstick of simple to be the degree of the relevant polynomial.

But not every algebraic number is a rational number.  The classical example, classical in the sense of Pythagoras and the ancient Greeks and things we have known for a very very long time, is √2 with associated polynomial p(x) = x^2 – 2.

There are lots of directions we can go from here.  One direction is the question, are there any real numbers which are not algebraic.  Another direction is the question, what other flavours of numbers are there.  And you won’t be surprised to find out, there are many.  Mathematicians have been busy for centuries exploring numbers and their properties.

So how do we demonstrate the existence of real numbers that are not algebraic numbers?  There are hard to construct explicitly, such is their nature.  My favorite demonstration takes us on a detour into a different, and difficult, topic, which is the strange nature infinity.




~ by Jim Anderson on 21 May 2016.

5 Responses to “the numerology of unrelated constants 2”

  1. I loved your piece but failed dismally to understand the middle section on mathematics. I then became enthralled at the idea that numbers can have properties. I realise that my notion of a number is limited to it being a count of things. So if I see 7 crows, what other properties can that ‘7’ offer me ? Are the properties related to crow things or the number itself? I have for a while realised that I have no idea what a number is and what is real and not real about a number?

    • And it is in this speculation that things get interesting. Though this doesn’t directly address your point, and it doesn’t answer your question of what other properties that 7 can offer you, it is going from 7 crows to 7 on its own where the power of mathematics makes itself manifest. Mathematicians abstract, and part of that abstraction is to free numbers from operations such as counting. Once we don’t have to count with numbers, that is once we accept that numbers can exist of and for themselves, we can start playing with the very idea of number, to real numbers and the number line, to complex numbers, quaternions and beyond.

  2. It’s more than possible this blog is not meant for someone like me. So apologies to those who know all these things!

  3. […] As we begin the year 2017, I like many others have spent some time looking back on the year just come to a close, and perhaps years farther back, and I find pieces of unfinished business.  One big piece involves something that I’ve tried my hand at a bit, not as much as I want to, and this piece is explaining hard bits of mathematics, interesting bits of mathematics, to people who are not mathematicians.  If you wish to read the current state of my efforts here, you can find them in the numerology of unrelated constants 1 and the numerology of unrelated constants 2. […]

  4. […] touches on what I started to do, and never finished, in the numerology of unrelated constants 1 and the numerology of unrelated constants 2, the completion of which has now gone back on the LIST OF THINGS TO […]

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