Perhaps the following will help with Feynman’s algebra lecture. When I first read this material I thought it was one of the most remarkable things that I had ever seen. After thinking about it I realized that much of what comes out is inevitable, but it takes someone with remarkable insight, such as Feynman, to put something like this together.
As is probably obvious, this is not any sort of proof. It’s more like a mathematical laboratory demonstration which, if understood, can yield considerable insight into the properties that it discusses.
The basic approach is to define a few basic operations, initially for positive integers. However, manipulations using these operations will yield expressions which have no meaning unless new types of numbers are introduced, and so Feynman introduces negative integers, fractions, irrational and transcendental numbers, then i, the square root of -1, and finally complex numbers. Along the way he asserts that each operation continues to be valid for each type of new number. He also manages to derive the values of two fundamental constants, i and π. I’d quibble with the description of where these came from as mathematical prestidigitation. I’d also maintain that the logic behind all of this is very carefully laid out, but it does take a little digging and working out of things to see how it all really works, and a few of the simpler steps are omitted. I’m an engineer, not a mathematician, so I’m perfectly happy with this demonstration. Perhaps some who are better mathematicians than I am may quibble with some of the things I say here.
Let’s look first at why ln10 appears the way it does.
For any positive real number p other than 1, the following turns out to be true:
\lim_{q\rightarrow 0}p^q = 1+kq where k is a constant.
What is k? Note that:
10^q=(e^{ln10})^q =e^{qln10}≈1+qln10 for small values of q
So for base 10, k=ln10 and in general, for any positive real base p other than 1,
p^q=(e^{lnp})^q=c^{(qlnp)}=1+qlnp for sufficiently small values of q.
This is why the ln10 shows up in very high roots of 10. One could, in priniciple, find the natural log of any positive real number other than 1 this way.
It’s easy to show that (log_ba)(log_ab)=1 so finding ln10 allows computation of log_{10}e and the value of e.
Note that for e, k=lne=1 so for very small values of q:
e^q≈1+q
Going further to Euler’s equation:
First Feynman discusses the equivalence of i and -i, so that dividing one complex number by another is equivalent to multiplying the numerator by the complex conjugate of the denominator.
Multiplying a complex number c+d^i=10^{gi} by its complex conjugate c-d^i=10^{-gi} yields:
(10^{gi})(10^{-gi})=10^{gi-gi}=10^0=1=(c+di)(c-di)=c^2+d^2
This sounds a lot like a specific case of the Pythagorean theorem:
sin^2θ+cos^2θ=1
so it seems reasonable that a positive real number (other than 1) raised to a power of i can be represented by two orthogonal components (real and imaginary) whose resultant is always 1.
The demonstration already showed that for large roots of 10, the value of the root can be approximated by:
1+(ln10)(1/1024)s
where s is a number showing how far beyond the 1024th root one wishes to go (so for the 2048th root, for instance, s = ½).
So, assuming that this relationship holds for complex numbers as well, then:
\sqrt[1024]{10^i}≈1+(ln10/1024)i
By squaring these successively one can get back to 10^i and from this table one can also get log_{10}i which is approximately 0.66826 (the only major typo that I’ve noticed in this chapter).
Deriving a set of complex numbers by taking successive powers of 10^{i/8} yields a set of numbers whose overall magnitude is always 1 but whose real and imaginary components fluctuate. As one thinks about this it may become less surprising that they oscillate and trace out the sine and cosine. It turns out that the period of these, when e is used as the base, corresponds to 2∏, so
e^{iθ}=cosθ+isinθ
and e^{i∏}+1=0