And this is The wonderful world of mathematics, written , and concerning Musings
At the Wolves LUG meeting last night, we got into a discussion of mathematics (see, I told you these meetings were exciting). Jono took the Luddite position that maths was extremely boring and of no relevance to the real world, a position that I imagine he's not the only person to hold. So I tried convincing him that maths was interesting and had a point, but didn't get very far. The attempt to demonstrate its interestingness was by relating a couple of maths anecdotes. Firstly, Euler's formula, eiπ + 1 = 0, which relates the five most fundamental constants in mathematics and is the clearest evidence I know of of some kind of underlying order in the universe, since e and π are both transcendental and i is imaginary and yet they combine to make 1. Jono liked that one. Secondly, I talked about Hilbert's Hotel, since I think stuff to do with infinities is fascinating and the idea of a full hotel being able to accommodate one new guest, an infinite number of new guests, and an infinite number of coaches each with an infinite number of guests on board is just mind-blowing. That one didn't go down too well, so I abandoned my third attempt, which was to talk about the difference between aleph-null and the continuum (which is handy since I can't remember how to prove there is such a difference without Cantor's diagonal proof and you need pen and paper to demonstrate that). So, I throw the question open. Since I've had snarky maths comments every year when I play guess-the-age on my birthday (2003 2004 2005), I assume there are some mathematically capable people reading this. Tell me some examples of how mathematics is beautiful and simple and elegant that can be used to convince a non-maths person what you see in the tumbling world of numbers. Note that the last part is important; if you think that Andrew Wiles' proof of Fermat's last theorem is elegant and beautiful then I don't want to hear about it. I spoke a little about axioms, with the intention of then going on to Russell's Principia Mathematica and then knocking it all down with Gödel, but we never got that far.
The second thing to demonstrate is that maths is really relevant to the real world and has a point. I talked a little about how pure maths came up with i as a pointless theoretical concept and it then turned out to be useful in electrical engineering, but we never got very far into that. So, again, the question's open. Demonstrate to a non-maths person why maths is important to the real world. Answers involving the phrases "joy of discovery" or "sacred guild of scholars" or similar are not wanted here. These also have to be semi-constructive demonstrations: when Dan presented the argument that "maths describes quantum physics and that's where computers come from", there was no clear recognition by our Luddite audience that that actually meant anything. How does maths make quantum physics work? Speak on, maths readers.
Comments
Tony B
Rory Parle
Tony: I don't think anyone has ever looked at Euler's equation for the first time and though "oh yeah, it's so obvious now". It's not at all obvious, but it's true. It comparable to taking, say, your favourite five pieces of architecture in the world, say the Sydney Opera House, the Empire State Building, the Golden Gate Bridge, that spiky cathedral in Barcelona, and the Taj Mahal and stuck them all together and found out that it made a perfect sphere.
The wonder of Euler's equation is that it seems astonishingly unlikely.
Stuart: My usual recourse is to find a book written by someone more convincing than me and steal something from it. Ian Stewart is quite good at getting the point of maths across. Try 'Game, Set, and Math' or 'Math Hysteria'.
Rory Parle
I will proofread before posting in future. Grammar in my previous post is left as a exercise for the reader.
Cathy
I'd recommend Ian Stewart as well, though he does tend to assume quite a mathsy background sometimes.
The nice real-world example I remember from a book of his that I read is using chaos theory to predict and generate random white noise signals, which can then be used to rather effectively cancel out actual noise in the background of recordings.
Sina Motamedi
I'm a 3rd year math major at the University of Toronto. I always love the simple math properties that seem unlikely (at first). There are two things I always show to friends of mine who know nothing about mathematics.
1) Prove that 1.9999 (infinite 9's) equals 2.
2) Share the following pattern: The sum of odd integers add up to perfect squares.
1 = 1
1+3 = 4
1+3+5 = 9
1+3+5+7 =16
1+3+5+7+9 = 25
1+3+5+7+9+11=36
etc.
I just noticed that for the first time a few months ago, and though it's simple, I just found it quite remarkable.
sil
Rory: love the sphere comment. Love it. That's exactly my feeling on the matter :)
Sina: I can feel the edges of a proof as to *why* the sum of odd integers adds to a square. If I can hash that out I might use it as an example.
mrben
Back before he was famous, Dave Gorman did a bit of standup all based around a maths class, which was really funny (to 'mathies' and 'non-mathies' alike).
The Whizzkids handbooks (remember those!?!) had nice proofs showing algebraically how 1=0 and 2=1, which always made me laugh.
Alex
O... math... i think it is NOT WONDERFUL... remember i had a lot of problems with it at school =).
Tim
To prove that the sum of odd numbers is a square, you need to know sigma notation:
The sum of the odd numbers from one to (2n-1) = 1 + 3 + 5 + ... + (2n-1)
sigma (r=1 to n) 2r-1 = 2*sigma(r) - sigma(1)
= 2*(1/2*n*(n+1)) - n
= n(n+1)-n
=n^2 + n - n
=n^2
Therefore the sum of the odd numbers from 1 to 2n-1 = n^2. so the sum of a consecutive series of odd numbers, starting at 1, must be a square.
I'm currently doing Further Maths A-level. I knew it would come in handy some day.
Gary Fleming
I don't think anyone apart from Cathy has produced anything interesting that relates to the real world in a practical sense. Do you honestly think that things adding up to other things, regardless of how much of a coincidence it can't be, meets the criteria? I do not. Keep your sums of squares and maths constants.
Cathy was hinting in a direction that would convince most people: signal processing. Signals are a bunch of waves mixed together that do great things, like make your mobile and tv work. Does it get more relevant than that?
What if you're a musician? Got a tremolo or wah-wah pedal? Those, my friends, are just the signals from your guitar passing through a careful signal and getting just the right sort of interference.
So many practical applications.
Sina Motamedi
To keep things easy, you can 'prove' (or show) that the sum of odd integers add up to perfect squares by using pictures. Start with a 1x1 square. In order to create a larger square, you must add 3 to the corner (and you'll get a 2x2). To make the next square, you'll need to now add 5 to the corner (get a 3x3). Next you'll have to add 7 to get a larger square (4x4). Continue to add odd integers to create the next possible square. This is how the Pythagoreans proved this property.
* 1 = 1
* *
* * 1+3 = 4
* * *
* * *
* * * 1+3+5 = 9
* * * *
* * * *
* * * *
* * * * 1+3+5+7 = 16
Darren Griffith
Cool. Math geeks like myself, all congregating around your blog post.
You guys might enjoy a song I really got a kick out of. Jonathan Coulton wrote a song about the Mandelbrot Set, and it rocks. You can find it here, midway down the page:
http://jonathancoulton.com/songs
madhur sambhar
Hi there
Great friends, i am an Indian , i love mathematics, done my graduation in Pure Mathematics, but post grads in comp applications.... but couldn't resist my temptation to read your great blog.... Hilberts Hotel is nice.
madhur sambhar
Hi
As an application of Pure Mathematics you can use to fight terrorism.... i know bit absurd but see the following link
http://web.mit.edu/newsoffice/2005/math-terrorism-0406.html
We can use the lattice theory to fight terrorism and predict results. You can also visit
http://gillinc.blogspot.com/2004/10/mathematics-of-terrorism.html
which tells how ordered sets helps us tracking that..... Mathematics is simply melodious but you need to have knowledge of combining good permutation of notes.
-regards
Madhur sambhar
"Creators Concern is conquest of nature
Parasite's concern is conquest of men " -Howard Roark
andy (lofty)
Fibonnacci's sequence /spiral
Theseries:
1+2=3
2+3=5
3+5=7
5+7=13
7+13=20
etc. is fibonacci's sequence. Guess you knew that - but a good example of the beauty of maths for non-mathematicians is that this series comes up in things like the spiral patterns of seeds on a sunflower head, and the angles of plant leaves going up the stem.
abhishek
well a little bit interested in maths ,thanks sambhar for letting me know that maths could even be used for fighting terorism.
I'm sure I'd think maths is great, if i understood it. I don't. One look at an equation causes my brain to fog. I once tried to read "The Anthropic Cosmological Principle" by Barrow and Tipler; some of which was fascinating. But it's full of equations that I completely failed to understand. I also find stuff like cryptography fascinating - I just don't understand 70% of it.