# Complex numbers and coordinates

1. Apr 26, 2005

### BobG

If you have a 2-D vector in polar coordinates (a magnitude R and an angle theta) you can convert it to Cartesian coordinates with the following equation:

$$x + yi = R e^{\theta i}$$

Or from Cartesian to polar by:

$$(R,\theta) =ln (x + yi)$$

Why does this work? I just can't quite envision this. cosh and sinh have a similar relationship in that you could almost treat them as trig functions even though they're based on e.

2. Apr 26, 2005

### mathman

cosh(ix)=cos(x)
sinh(ix)=isin(x)

I don't get (R,a)=ln(x+iy), where a=angle.

3. Apr 26, 2005

### BobG

Oops. My bad. The natural log gets the angle (the imaginary part of the result), but R is e^(real part).

The definition of cosh is:

$$cosh z = \frac{e^z + e^{-z}}{2}$$

and, in practice, it does have the relationship you described.

The real question I had is why the relationship between sin, cos, and e?

4. Apr 26, 2005

### master_coda

Because $\cos\theta+i\sin\theta=e^{i\theta}$. That's probably not the explanation you wanted to hear; but I'm afraid I don't know any deeper motivation, other than just providing a proof that this is true.

And your relationship between x+yi and re^(i theta) still doesn't seem correct.
$r=\sqrt{x^2+y^2}$ and $\theta=\arctan(y/x)$

Last edited: Apr 26, 2005
5. Apr 26, 2005

### Data

Yeah, a proof is really the best you can get. You can define $e^{\alpha x}$ for complex $\alpha$ as the unique solution to

$$\frac{d^2u}{dx^2}=\alpha^2 u, \ u(0)=1,\ u^\prime(0)=\alpha$$

so $e^{ix}$ is the unique solution to

$$\frac{d^2u}{dx^2}=(i)^2u = -u, \ u(0)=1, \ u^\prime(0)=i$$

which (from the definitions of $\sin$ and $\cos$ as solutions to other DEs) yields the solution $\cos{x} + i\sin{x}$, and the identity.

6. Apr 27, 2005

### BobG

Found it (or at least the source where I can figure it out).

Thinking about it, I remembered that, before John Napier, people used trig tables as multiplication tables (using the principles of the sum/difference identities for trig functions) and that that was the inspiration behind Napier's logarithms. He wanted to invent an easier way of multiplication and division - such high level mathematics that were beyond the capability of the average person.

Sure enough, his original logarithms were based on an analogy:

He later revised his logarithms to set "log 1 = 0" and converted his logs to base 10, a revision that made them much more practical for the original purpose of logarithms - multiplication and division.

Until the invention of the slide rule, common logs still didn't quite do the trick (he should have gone one step further and used the decibel scale - phsychologically, it would have made multiplication using common logs feel as simple as it actually is). The invention he was most noted for during his lifetime was "Napier's bones". Those were little rods, normally made of ivory, with numbers on them that could be manipulated to perform multiplication, division, squares, and roots. Some of the really smart people memorized all of the numbers inscribed on the bones and just kind of mentally carried a set of "Napier's Bones" around with them. That's a pretty impressive memory - you'd think no one could remember all of those numbers. Maybe that's why kids have to start memorizing them in second or third grade, except now we call them something dull like "multiplication tables".

Edit: That leaves out how we got our "natural logs". Looking at Napier's work, Euler realized the significance of Napier's original logarithms and revised them to today's format (setting ln 1=0, etc.)

Last edited: Apr 27, 2005