I’ve been hit twice today by the majesty of numbers, by their capacity to throw up surprising results that fly in the face of all expectations.

The first time was when I was working through some simultaneous equations earlier. Nothing special there, until something happened that made me very nervous about pursuing any future career in economics, illustrating to me what a disaster could ensue by the very smallest slip-up in my modelling.

Take into consideration the following two systems:

55 x + 144.018 y = 144
89 x + 144.022 y = 233

What’s the only difference in these two systems? There is one, but it’s only slight.

The coefficient on the y changes from 144.018 to 144.020. That’s tiny.

To put it into context, it’s the same scale as a child growing from 4 foot 8.7 inches to 4 foot 8.7008 inches. In other words, it’s so insignificant as to be negligible, an unremarkable increase of 8 thousands of an inch.

Yet this tiny, barely noticeable change rips asunder any similarity between the two systems. When you solve the first equation, you find that the following values hold for the two parameters:

x = -159.2, y = 100

There’s nothing untoward there, nothing that would stand out. That’s until you solve the second equation, the one in which we’ve increased the coefficient on y by 0.00138%.

x = 18.8, y = -10

The variables have changed sign and have reduced in size by a magnitude between nine and ten!

That’s preposterous to me. One tiny little change can have such a major repercussion!

Imagine if a government advisor were to slip up in one of his equations. As we can see here, it only takes something so small to completely turn things on their head.

Chaos theory stems from the line about a butterfly flapping its wings causing a hurricane in another part of the world. This revelation from these two near-identical equations makes me think the same. It also makes me very dubious about the accuracy of my own maths into the future. Amazing.

I don’t need to make up something like that equation to come up with something mindblowing though. Nature seems to have handled that all itself.

If you were to ask people which numbers are the most important in the whole universe, it’s a pretty fair bet that the first four would be 0, 1, e, and π. I also imagine that people would consider i, the imaginary number that gives the square root of -1, to be considered “an important number”, even if they’ve never seen it in action.

Yet all these things can come together beautifully. Raise e to the power of iπ, add 1 to it, and you get 0!

That’s right, , a tidy formula if ever I saw one, genuinely works. Wow!

And all this talk about the wonders of maths brings me round to a book review …

50 Mathematical Ideas You Really Need To Know About by Tony Crilly

This was a surprise gift from Radio as an apology. She actually didn’t need to apologise at all, but she used some twisted form of guilt-ridden logic to blame herself for my neglecting to pay attention to a big sign, which resulted in The Moosemobile being separated from its master overnight.

Well, this is probably my favourite of the books that she’s bought me.

We’re both a fan of popular books about mathematics. I have a few, and have really enjoyed learning about the history of the subject over the last few millennia.

This book looks at 50 different things, presenting four pages on each of the headings.

What it means is that you can cover logarithms, game theory, chaos, fractals, curves, and so on, without having to invest a few days in reading a book dedicated to just one of the things.

I can’t really glamourise it, but it was fun, and one that I thoroughly enjoyed. Thanks Radio!

Tags: Mathematics, Number

This entry was posted on Friday, October 17th, 2008 at 11:02 pm and is filed under Books, Economics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.