
The Feynman Lectures on Physics: Algebra (1963) - signa11
http://www.feynmanlectures.caltech.edu/I_22.html
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jihadjihad
One thing I will always appreciate about Feynman as a teacher is that he never
shies away from telling you _why_ things work the way they do. I've spoken
with a lot of people who "hate math" and it's almost always because they never
had a teacher that went into the whys and wherefores behind equations, tables,
graphs, etc.

Here we have a plainspoken yet rigorous explanation of what algebraic
operations _actually mean_ (addition is iteratively adding 1 _b_ times;
multiplication is iteratively adding _a_ _b_ times; exponentiation is
iteratively multiplying by _a_ _b_ times, etc.).

I wish more students would have the chance to build up such intuition, for
such intuition is the key to keeping oneself interested in math and not
finding that one "hates" it.

~~~
acpetrov
I stopped enjoying math for about this reason. My university tried to teach
with proofs, which to me is about as far from an intuitive understanding as
possible

Looking for recommendations on lectures / course that goes over linear algebra
in this way

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Ntrails
> I stopped enjoying math for about this reason. My university tried to teach
> with proofs, which to me is about as far from an intuitive understanding as
> possible

Interesting, because until I had the proofs for eg differentiation, or heck
even the quadratic formula - they were meaningless rote learnings. The proof
_is_ the intuition imo

~~~
aschismatic
Exactly! Proofs are why I love mathematics so much. There's nothing quite like
the great "ah-ha!" moment when you find that one leap of logic that topples
the rest of the dominoes in a proof. Of course, I understand that some people
are better at teaching proofs by involving students in the discovery. Maybe
the parent commenter was taught proofs in that same rote manner. I've had
professors that have done that, and it's like someone sucked all the fun and
learning out of the subject.

From the preface of Computation: Finite and Infinite Machines by Marvin L
Minsky:

> The reader is therefore enjoined not to turn too easily to the solutions;
> not unless a needed idea has not come for a day or so. Every such concession
> has a price—loss of the experience obtained by solving a new kind of
> problem. Besides, even if reading the solutions were enough to acquire the
> ability to solve such problems (which it is not), one rarely finds a set of
> ideas which are at once so elegant and so accessible to workers who have not
> had to climb over a large set of mathematical prerequisites. Hence it is an
> unusually good field for practice in training oneself to formalize ideas and
> evaluate and compare different formalization techniques.

~~~
Fr0styMatt88
For proofs, could you recommend any good resources for a beginner? Is there a
'beginner proof' that's great to start with?

I figured out I actually like maths waaaay after I'd left uni. From that time
at uni I have a vague memory of proofs being something like a whiteboard full
of equations that I got lost somewhere in.

I have a vague feeling that what I'm thinking of is 'formal proofs', but I'm
not sure.

~~~
gumby
Euclid. He tried to prove theorems of basic plane geometry (hence "euclidean
geometry). Since we all have an intuitive understanding of (at least the
basics of) plane geometry you can look at the work and not have to also learn
the domain.

People recommending the classics can come off as pretentious so I will add
that I am serious: a modern book of Euclid's methods should be quite
accessible.

As a followon bonus: Minsky's and Papert's 1967 book "Perceptrons" (the one
that said you can't do XOR with a single-layer network, though you can with a
multilayer one) that lead to 25 years of lack of interest in neural networks
is entirely about using neural networks on Euclid. So you can go from one to
the other!

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mr_gibbins
And this is what gets me about Feynman, why I am a huge admirer of his work.

Take a look at the footnote. An offhand comment of how to calculate the square
root of any N using a short iterative loop that you can compute by hand.

I'm no mathematician (clearly) but I had no idea that this method even
existed. I've just had to try it out.

And it's a footnote!

I have the book of Feynman's computation lectures and they're dated, but most
of the theory is still relevant and his writing remains accessible to anyone.

~~~
grey--area
Here's the method that's being applied:
[https://en.wikipedia.org/wiki/Newton%27s_method](https://en.wikipedia.org/wiki/Newton%27s_method)

It's an iterative method for finding x such that f(x)=0 for a given function
f, in this case f(x) = x^2 - N, where N is the number you want the square root
of.

The special case has such a simple expression, though, that I'm tempted to
commit it to memory.

Edit: Another nice example, for when you have access to a calculator that can
compute powers but not logarithms: a' = a - 1 + x/c^x will converge the base-c
logarithm of x (though it will converge slowly if you start very far from the
solution).

~~~
fastbeef
At it's core, it's also what drives Carmack's Fast Inverse Square Root
([https://en.wikipedia.org/wiki/Fast_inverse_square_root](https://en.wikipedia.org/wiki/Fast_inverse_square_root)).

Yes, yes, I know Carmack didn't invent it.

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jameshart
There’s something delightfully physicsy about deriving Euler’s identity, which
famously deals with the irrational and the imaginary, by brute force
calculation using log tables and fudging the fifth decimal place. But the
approach of sneaking backwards into the algebraic continuation of e^x by
actually doing the arithmetic is a great way to viscerally _get_ that it’s all
just the same set of mathematical tools.

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michelpp
If you can find it an audio recording of this lecture is also available and it
is one of the best lectures in the series. Feynman really brings forward the
"soul" of algebra and the illustrates how powerful the art of abstraction is.
Audible has all the audio lectures for sale, which is how I got them on my
kindle, and I have also seen them floating around the net in other formats. I
wish Caltech would release the audio material like they did the books!

~~~
arunix
The article has a link to a page which has the audio:

[http://www.feynman.com/the-animated-feynman-
lectures/](http://www.feynman.com/the-animated-feynman-lectures/)

~~~
michelpp
The audio in the link appears to only be a 5 minute excerpt, not the full 50
minute lecture.

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enriquto
The Feynman lectures is one of my favourite books ever. Yet, this chapter
specifically is the least enjoyable for a mathematician. Somehow I find he's
trying to explain very simple concepts using way too many words, which is the
opposite style of the rest of the book.

~~~
jonjacky
It's deliberate. In his book _The Character of Physical Law_ , Feynman
discusses some differences in how physicists and mathematicians approach math.

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xref
I was reading the 1993 bio Hard Drive about Bill Gates recently (great to go
back and read old books centered on tech, really makes you remember why people
hated Bill/Microsoft...and he’d barely gotten started!) and it mentions
several times that Gates would watch these lectures in what he considered his
“down time”

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gerbilly
My high school math teacher taught us a class where he followed this general
outline, building up the number systems based on which equations we couldn't
solve.

Now I know where he must have gotten it from.

I think it must have been around grade 10 or 11.

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mrcactu5
He does a numerical computation of the complex powers of 10 (to five decimal
places) and they still are on the unit circle. And this was done in 1963.

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smcl
This specific Feynmann Lecture popped up very recently either here or on
lobste.rs. I have since bookmarked the "FLP" series as I really want to come
back to it, but I'm curious if there is any significance to why the Algebra
lecture in particular was submitted a couple of times. Is it just coincidence,
or was it a particularly good one?

~~~
tim333
It's one of my favourites. Aside from Feynman's clear and enthusiastic style,
I find it's remarkable you can start from counting 1,2,3 and by a bit of
reasoning come up with logs, complex numbers and:

>This, then, is the unification of algebra and geometry.

I can't help wondering if you could go further and come up with some physics.

~~~
crispyambulance
That material is covered in all STEM curriculums as part of the Calculus
sequence.

For most students, the sublimeness of getting to the Reals and beyond tends to
get lost in the grind of lectures, problem-sets and exams.

Feynman had a knack for getting straight to the essence of a subject, that's
why people love him so much.

As for algebra + geometry with physics, there are people who have cut a path
to that (see texts by Hestenes), unfortunately, it requires some mathematical
sophistication and most curriculums simply don't have the time to reach that
level for undergrads, given the way that they're structured.

