Is Feynman's Lecture on Algebra Still Relevant Today?

[Hetware]

Is anybody familiar with Feynman's lecture on algebra? Vol I Chapter 22? The first time I read it many years ago, I didn't grasp the development of Euler's formula. The result was astounding. And more so due to the way he developed it. In subsequent readings, I skipped over that part because I had learned other means of deriving the same formula. This time through, I was holding on by a fingernail, but still didn't get it.

Euler's formula:

$e^{\text{i\omega t}}= \cos \text{\omega t} + i \sin \text{\omega t}$

Is there another source that takes the approach Feynman took in his lecture? It's pretty clear that Feynman's development is "old school". It's also clear that it's worth understanding.

Whatever happened to the slide-rule?

 

[chiro]

Whatever happened to the slide-rule?

People will pull it out when the EMP knocks out all the computers when the world ends.

 

[SteveL27]

People will pull it out when the EMP knocks out all the computers when the world ends.

Yes, we'll be using slide rules to dig roots and tubers out of the ground to eat. It'll be that bad when the grid fails.

 

[chiro]

Yes, we'll be using slide rules to dig roots and tubers out of the ground to eat. It'll be that bad when the grid fails.

That'll happen right after the initial zombie apocalypse where all the armed civilians take them out.

 

[micromass]

Is anybody familiar with Feynman's lecture on algebra? Vol I Chapter 22? The first time I read it many years ago, I didn't grasp the development of Euler's formula. The result was astounding. And more so due to the way he developed it. In subsequent readings, I skipped over that part because I had learned other means of deriving the same formula. This time through, I was holding on by a fingernail, but still didn't get it.

Euler's formula:

$e^{\text{i\omega t}}= \cos \text{\omega t} + i \sin \text{\omega t}$

Is there another source that takes the approach Feynman took in his lecture? It's pretty clear that Feynman's development is "old school". It's also clear that it's worth understanding.

Whatever happened to the slide-rule?

Maybe you can tell us which part you did not understand of Feynman's approach, we might be able to fill in the gaps.

 

[Hetware]

Maybe you can tell us which part you did not understand of Feynman's approach, we might be able to fill in the gaps.

Are you familiar with the discussion? It's rather involved, and not well stated in some parts. I am referring to section 22-4, starting with Table 22-1 and following. His presentation is rather inexact, and clearly not rigorous. In addition, there are some typos which don't aid in understanding.

I'll need to review it a few times. I was just hoping for a more formal and lucid development.

 

[SteveL27]

Are you familiar with the discussion? It's rather involved, and not well stated in some parts. I am referring to section 22-4, starting with Table 22-1 and following. His presentation is rather inexact, and clearly not rigorous. In addition, there are some typos which don't aid in understanding.

I'll need to review it a few times. I was just hoping for a more formal and lucid development.

For those of us who are curious but haven't seen this discussion, can you summarize it? I'm familiar with the development of the trig functions from the power series definition of the exponential. I suppose that's the standard way and Feynman has some other interesting approach? I'd be curious to know what it is.

 

[Hetware]

For those of us who are curious but haven't seen this discussion, can you summarize it? I'm familiar with the development of the trig functions from the power series definition of the exponential. I suppose that's the standard way and Feynman has some other interesting approach? I'd be curious to know what it is.

He starts off showing how to develop a table of base 10 logarithms and, using some mathematical prestidigitation, he demonstrates that a residue bubbles up as you distill the numbers further and further. That residue is log_e10.

My quip about slide-rules was not random. I believe Feynman's audience would have understood him more readily than those of us who have only used a slide-rule once or twice in our lives.

 

[Hetware]

This is one of the things that I don't fully appreciate from his development:

Lim$_{i\rightarrow{∞}}$ $\left(10^{1\left/2^i\right.}-1\right)2^i$ = ln10

 

[engineerdave]

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 $$e^{i∏}+1=0$$

 

[lurflurf]

There is no Feynman's Lecture on Algebra that is from Feynman's Lecture on Physics. To define exp(z) for complex numbers we chose some properties from the real exponential to preserve and define the complex exponential from that. This is not a proof or derivation, but just a definition. For most reasonable choices we find that
exp(i x)=cos(x) + i sin(x)

For example if we require
exp(x+y)=exp(x)exp(y)
exp'(0)=1
we get the usual complex exponential

 

[engineerdave]

my post was an attempt to clarify hetware's confusion about certain points in the feynman lecture. if you're not familiar with that lecture, entitled "algebra" in the books, then you'll find it difficult to interpret the posts in context.

hetware gave the reference in his initial post. feynman originally gave this lecture, then titled "some interesting properties of numbers," at los alamos during the war.

 

[mathwonk]

it is indeed hard to improve on a lecture by feynman. I will not try.