Complex numbers and coordinates

[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.

 

[mathman]

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

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

 

[BobG]

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

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

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?

 

[master_coda]

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

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 $\theta=\arctan(y/x)$

 

[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.

 

[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:

Unlike the logarithms used today, Napier's logarithms are not really to any base, although in our present terminology, it is not unreasonable (but perhaps a little misleading) to say that they are to base 1/e. Certainly they involve a constant 10^7 which arose from the construction in a way that we will now explain. Napier did not think of logarithms in an algebraic way, in fact algebra was not well enough developed in Napier's time to make this a realistic approach. Rather he thought by dynamical analogy. Consider two lines AB of fixed length and A'X of infinite length. Points C and C' begin moving simultaneously to the right, starting at A and A' respectively with the same initial velocity; C' moves with uniform velocity and C with a velocity which is equal to the distance CB. Napier defined A'C' (= y) as the logarithm of BC (= x), that is

y = Nap.log x.

Napier chose the length AB to be 10^7, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 10^2 sin X.

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.)