Wednesday, July 21, 2010

That crazy formula

I rarely consider mathematical formulas on their own to be beautiful things, but there is something quite special about one formula in particular, here it is:
e^i*pi + 1 = 0
The formula is saying 'if you rotate 1 by 180 degrees then you get -1'.
It is a very satisfying formula, it contains a plus, a multiply and a power, and it contains possibly the four most useful numbers in mathematics, 0, 1, pi and e.

Another reason why it is a special formula came to me yesterday when I was investigating how the units of measurements combine in formulas.
If we square a velocity (in metres per second) we get a value in square metres per second squared. So if you write the units as a vector [metres, seconds] then a velocity is [1, -1] and a velocity squared has units [1, -1]^2 = [2, -2].
What if you multiply two measurements together? What units do you get if you multiply a velocity by an acceleration? You get [1, -1]*[1, -2] = [2, -3].
Can you see the pattern? Well here it is... if you take a measurement to the power x then you multiply the units by x. If you multiply two measurements then you add the units together. It seems that the units operate one 'level' down from the operation on the measurements themselves.
To clarify this idea of a 'level' we need to look at hyper operators. Here they are:
a (1) b means a + b
a (2) b means a * b
a (3) b means a ^ b
So what seems to be the case is that the units of measurements A (n) B are a (n-1) b (where lower case means units of).

So what happens if you add two measurements together? what happens to the units? Well what are the units of a velocity plus a time? or a position plus a velocity? A tough question... you could say that the resulting measurement is unitless, but that is actually wrong, a gradient (e.g. height/width) is unitless. The sum of two measurements seems to be 'less than' unitless, unlike a unitless measurement which has a vector [0,0,..,0], the sum isn't even that, it isn't even the empty set because it isn't a set, it is just empty, void.
However, you can still sum measurements if they have the same units... an acceleration plus an acceleration still has units [1, -2].

This is interesting because it means that we have defined a (0) operator, not only that but it is different from the normal 0 hyper operator (which is succession), we have:
a (0) b = void if a different to b
a (0) b = a if a same as b (or it can equal b, same thing)
I like this new version of the 0 hyper operator (maybe we should call them ultra operators from now on), because a (0) b is a binary operator and is symmetrical.
The best part about this operator is that it has an obvious meaning that we use in every equation ever written, it is the equals operator! It is actually an important change to how '=' is interpreted. Rather than thinking of a=b being a boolean expression, it is numerical expression that either returns the equal value, or simply doesn't return. A bit like this:
operator = (a, b)
if (a==b)
return a;

Anyway, where was I? Oh yes, this brings me around to the other reason why the formula at the top of the page is special. If we write it using ultra operators it looks like this:
e (3) i (2) pi (1) 1 (0) 0

It is quite a simple formula wouldn't you agree?

It makes me wonder whether there is some more that can be learnt from it.

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