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eip=-1
By Alexandra on November 2, 1998:

Can anyone explain what this MEANS? I see that it's amazing - but what is it actually saying?

By Simon Munday (sjm78) on November 2, 1998:

Hello Alexandra,

You're quite right - it is a pretty amazing equation. It's probably one of the most beautiful in mathematics because it relates the numbers e, i and p, which are some of the most important (and apparently unrelated) numbers you'll come across mathematically, in a delightfully simple way.

The equation is simply a special case of the following
eiz = cos z + i sin z (equation +),

where z is any complex number. This comes from the definition of ez, cos z and sin z as complex power series. Don't worry if you don't know what complex power series are - they are used to generalise the definitions of the functions mentioned to cases where, for example, sin x = opp/hyp (should be familiar) don't make sense.

I will rewrite equation (+):

reia = r cos a + i r sin a. (equation ++)

(where r is a non-negative real number and a is any real number). This shows that there are two ways of writing any complex number. The first way is z = x+iy, which is the RHS with x = r cos a , y = r sin a. Here, you can see that |z| = (x2 + y2)1/2 = (r2cos2 a + r2 sin2 a)1/2 = r(cos2 a + sin2 a)1/2 = r, and arg z = tan-1 (y/x) = tan-1 (sin a / cos a) = tan-1 (tan a) = a + 2np, where n is some integer, so r = |z| and a is one of the values of the multivalued function arg z. (Notice that z is unchanged if we add a multiple of 2p to its argument).

The other way is given by LHS of (++): z = reia where r = |z| and a = arg z + 2np. Now -1 is a complex number (it just so happens that its imaginary part is 0), so it can be written in both of the forms above. The first form works with r = 1, a = p (cos p = -1, sin p = 0), so the second form of -1 is ei×p.

The best way to look at complex numbers is the picture called the complex plane (or the Argand diagram). This is just a generalisation of the familiar number line: Draw the number line and draw another line perpendicular to it through 0. The first line is called the real axis, the second the imaginary axis. The picture is essentially the same as the x-y graphs you're used to. The complex number z = x+iy can be represented as a vector (or directed line segment) joining 0 and the point (x,y). This vector has length r and makes an angle a with the positive real axis. In this picture multiplying two complex numbers corresponds to multiplying the lengths of the vectors and adding the angles they make with the positive x axis. (Can you prove this?). Thus -1 = e{ip} simply means that -1 is the complex number in the Argand plane of length 1 making an angle p (180 degrees if you're not familiar with radians) with the positive real axis.

I hope you followed all that. Why don't you draw yourself a copy of the Complex Plane and draw some complex numbers on it. In particular look at a number and its square, cube etc. roots (start with real numbers first). What problems occur because of the multivalued property of the argument? (hint: you'll have to define the argument to lie in some interval of length 2p). Have fun - Complex Analysis (which is what this is the very start of) is one of the most beautiful and important areas of Pure Mathematics.

By Simon Munday (sjm78) on November 2, 1998:

Hi Alexandra,

I read your other question just after I'd finished answering this one, and I see that you already knew nearly all the stuff I explained. Sorry to be so patronising - I hope some of it helped anyway!

Simon

By Alexandra on November 8, 1998:

Thank you! I did know quite a lot of that but some did help.
A.