Fermat's Little Theorem (2013) [video]
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It contains proofs of very many results from elementary number theory, like Fermat's Little Theorem here. The reason this book is different from any other is because the proofs are all combinatorial, like counting beaded necklaces or otherwise. It is also a Dover book, meaning it costs very little. It is also one of the books I used as reference to create the elementary number theory course that I sometimes teach.
Letter, pipe, letter, superscript. Yeah, real nice. Clear as mud.
And don't start down the road of thinking everything can be made unambiguous. It is context dependent, and always will be. Look at computer code, look at something deep in the bowels, and tell me what the "+" sign means. Is it integer? Float? List? Or has it been over-ridden by some library you've loaded somewhere else? Even finding where it's been defined can be a challenge. You have to trust that the author has played fair, and that your intuitive understanding that "+" is doing something consistent with the concept of "+" in numbers is close enough.
You could write out all the math notation in longhand, but it won't aid your understanding. If you actually follow the video then you will follow the notation. But there's a reason why we say "read like math" - it takes time and effort to gain understanding. It's not a novel, it's not a comic.
The only way to learn mathematics
is to do mathematics.
-- Paul Halmos
"There is no Royal Road to geometry,"
-- Euclid, in reply to King Ptolemy's
request for an easier way of learning
mathematics.If the notation destroys your understanding, then you didn't understand it in the first place, you only had a vague sense of satisfaction with a pseudo-understanding. This is akin to thinking you understand an algorithm, but not being able to code it.
I have every sympathy with people who find the notation a barrier, because so did I. Over the years, indeed, the decades, I have come to appreciate notation as a way of concisely expressing deep and complex ideas. Try not using notation - your brain quickly turns to mush as you try to follow hundreds of words that could otherwise be expressed in a mere dozen symbols.
Archimedes in "Measurement of the circle" wrote:
The area of any circle is equal to a
right-angled triangle in which one of
the sides about the right angle is equal
to the radius, and the other to the
circumference, of the circle.
Area = (1/2)⋅r⋅(2πr)= π⋅r^2
Edited to change the example from Newton's Principia to Archimedes
If someone is truly notation-phobic then there is nothing to be done.
Having the notation means that those in the middle ground have a chance to see real maths in action, and may be inspired to do more. The genuinely notation-phobic will never advance in math. For those who know the theorem this is a cute visualisation. People will only learn when they are ready, and all pedagogy needs to be targeted at those who are ready for it.
I suspect we agree more than people reading this might think, but we might disagreed over the perceived purpose of the video. I see it as getting people engaged with the thought processes, and then showing that it's really math, and look, here are the formulas that come out of it. It's an opportunity to learn about the formulas. For the genuinely notation-phobic, there is no real hope, except for them to see that this is a cool thing their phobia is preventing them from understanding better, so maybe getting over the phobia would be worthwhile.
I could do a complete case analysis, but this isn't the place, and I don't have the time. Would you suggest not having the notation at all? What would then be the point of the video? Who would gain?
https://www.khanacademy.org/computing/computer-science/crypt...
The simplest proof I know of Fermat's Little Theorem is induction, assuming that we already know the Binomial Theorem. Suppose a^p == a. (== is my symbol for "is congruent to mod p" in this post.) Expand (a+1)^p, and note that every binomial coefficient except the first and last has p in the numerator but not the denominator. Hence (a+1)^p == a^p+1 == a+1.
Cleaning up the proof from there is trivial.
In programming we slowly gain a big grab-bag of patterns and approaches to certain problems and build an intuition of what to apply where. It doesn't nearly cover your full experience but helps break problems down.
Do you find there is an analog to this with theorems? If so what's the essentials from your 'grab bag' and, beyond just reading more, what practices help build your feeling of where to use them?
Tim Gowers writes well about how to build mathematical knowledge, techniques, and a library of tools.
1. If you're going to take a test for school, you can recognize patterns pretty easily. For example, Harvard has a 3-day Qualifying Exam as a PhD requirement, given twice a year. Once, after looking at the exam pages for the first two days, I said "Hmm, there hasn't been a Schwarz Reflection Principle question yet."
The next evening, I got a nice hug from Lisa Mantini. :)
2. If you're doing actual research math, it's of course much harder. For example, the theorem in my thesis was scooped by a few months by Neyman and Mertens. What's more, they did it as a special case of a fixed-point theorem, while I just proved it by brute force.
And now some simpler anecdotes from my undergraduate days.
3. In my E&M course, we were supposed to work out fields for a cylindrical wire with an off-center cylindrical hole. This was a LOT easier in curvilinear coordinates than Euclidean. When I showed that to the professor in his office, he literally stood up and applauded.
4. In an abstract algebra course, there was a typo in the book for one of the exercises as to whether a certain polynomial was reducible. I solved the hard form by transforming the problem into one about a polynomial with matrix variables.
I took that story with me when I went to graduate school. Ron Livne's eyes lit up, and he transformed it again into something about -- well, into something about transformations on the polynomial's complex roots.
My second sentence above is a bit of a handwave. If you want to get formal about it, here's one way: suppose d isn't a multiple of 5; then note that there's an integer a such that ad = 1 (mod 5). Then ad rotations have the same effect as 1 rotation; but ad rotations do nothing, so d=1.
Now the handwaving is concentrated in the "then note that ..."; if that isn't sufficiently obvious, consider ad for a=0,1,2,3,4 and note that no two can be equal mod 5 because if ad=bd mod 5 then (a-b)d is a multiple of 5, but neither factor is a multiple of 5 and 5 is prime. So: five numbers, all of them different mod 5, so we must have one each of each congruence class mod 5; in particular, one of them is 1 mod 5.
He also get formulas like: pq divides a^pq - a^p - a^q + a
Incidentally, assuming you mean p and q to be distinct primes, this latter formula is almost just another instance of Fermat's little theorem: we have that q divides (a^p - a)^q - (a^p - a) and p divides (a^q - a)^p - (a^q - a) (that's the little theorem), and then that q divides (a^(pq) - a^q) - (a^p - a)^q and p divides (a^(pq) - a^p) - (a^q - a)^p (that's essentially the binomial theorem); so both p and q divide a^(pq) - a^p - a^q + a.
(you can google it for a link to a supplier)
Well worth the read. Tells the whole story of the famous "Fermat's last Theorum" problem and is extremely well written.
Fermat's Little Theorem: For any integer a and prime p, a^p is congruent to a mod p.
Fermat's Last Theorem: For any integer n > 2, there are no integers a, b, and c such that a^n + b^n = c^n.
The Little Theorem has been known for hundreds of years and is easy to prove. The Last Theorem, on the other hand, was only finally proven in 1994 by Andrew Wiles after countless failed attempts by mathematicians before him.
There is also an excellent long interview with Ken Ribet, who proved a result that inspired Wiles to take on the proof: https://www.youtube.com/watch?v=nUN4NDVIfVI