Reddit reviews Journey through Genius: The Great Theorems of Mathematics
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William Dunham has a great book,Journey through Genius: The Great Theorems of Mathematics, about this.
Journey Through Genius is an excellent book that I read when I was an undergrad. If I remember correctly, it focuses on ten (or so) major results and goes through in detail the motivations behind them and the work leading up to them. I found it really interesting. As an undergrad, I took a course on Philosophy of Mathematics (which basically amounted to the history of Math). The prof had written his own textbook for the course and it's available here (scroll down the page until you get to "6. The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development. Kluwer, 1999." and then click the links below). You may also want to spend some time browsing the extensive MacTutor History of Mathematics website - not a book, but incredibly thorough.
/u/cm362084 already recommended The Millennium Problems by Keith Devlin, which is literally exactly what you're looking for. If you're interested in other great books about math, 2 I'd recommend are Journey through Genius by William Dunham and Fermat's Enigma by Simon Singh.
Journey through Genius is organized such that every other chapter is some important proof (detailed out step-by-step), and the remaining chapters provide the historical/biographical context for those proofs. There are some interesting stories included in the book such as how mathematicians in the middle ages would keep their techniques secret, since there was a chance that another mathematician would come to town and challenge them to a math duel.
Fermat's Enigma tells the story of how Andrew Wiles was able to prove Fermat's Last Theorem, which states that there are no integer solutions to the equation a^n + b^n = c^n for n>2. (This one was a century problem last century, but since it was solved, there was no need to list it as a Millennium Problem.) This is a bit more storytelling than actual math, though Singh doesn't shy away from going a little bit into detail about the underlying math.
The last book to consider is The Poincare Conjecture by Donal O'Shea. The Poincare Conjecture was one of the Millennium problems and was recently solved. I should point out that I can't recommend this book personally because too much of it went over my head. That says more about me than the book, though, so I don't want to leave it off the list just because I was too dumb to get it. :) I never took any classes in topology, so I may want to read up on that and give this book another shot.
Journey Through Genius, I couldn't put it down, it goes through some of the greatest/most well known proofs in math. It is a book that goes into detail and while one may need to reread a section a couple times to comprehend, it does a great job of explaining what is going in
http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_sp-atf_title_1_1?ie=UTF8&qid=undefined&sr=8-1&keywords=journey+through+genius
Journey Through Genius: Exploring the Great Theorems of Mathematics. - William Dunham
Journey through Genius: The Great Theorems of Mathematics https://www.amazon.com/dp/014014739X/ref=cm_sw_r_cp_api_i_uOd3CbD8DH8CN
A great read that does walkthroughs of proofs and breakthroughs. Highly recommended.
You seem like the kind of person who'd enjoy a book like Journey through Genius, by William Dunham.
http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_1_1?ie=UTF8&qid=1346777582&sr=8-1&keywords=william+dunham
If you want to teach probability or statistics, take a look at Gelman's Teaching Statistics: A Bag Of Tricks. I've used material from there to good effect.
Edit: Maybe also take a look at better explained.
Edit2: Also Dunham's Journey Through Genius. Very inspiring and fun.
I have not read many books explicitly devoted to the history of mathematics, such as those recommended in this math.stackechange post #31058, so I will refrain from recommending any of them. Instead, I'd like to mention a few books that do well discussing aspects of mathematical history, although this is not their main focus.
Hope this is helpful!
I am currently reading a fantastic book which might be interesting for you. It is called Journey through Genius. The book starts from the beginning of math and presents hand picked theorems in a very engaging way. Background information on the great mathematicians and what drove them to come up with these proofs in the first place makes the information stick long after reading. I also second PuTongHua who recommended Better Explained.
Journey through Genius: The Great Theorems of Mathematics is also good too, with many historical mathematicians and their contributions. The author William Dunham, actually gave a lecture on Newton and Euler at my university a few weeks ago too.
One of the best books I've read that places mathematical discoveries in their historical contexts: Journey Through Genius. Dunham tells the story of math through different great theorems - why they were historically important, why they are important today - and then walks you through the proof. My copy is at school, so I can't say anything more tonight, but give it a shot.
Good luck!
Not a book, but I can share a few videos that I've found inspirational during some rough times with mathematics...
Fermat's Last Theorem -- This is a documentary on Andrew Wiles' proof of Fermat's last theorem. It's also probably the most emotional video I've ever watched about math. Highly recommended.
Fractals -- This is a neat NOVA documentary on fractals. In particular, it provides some inspiring history regarding Mandelbrot's discovery and journey with this subject.
Everything is relative, Mr. Poincare -- Another exceptional and inspiring documentary.
The only book I can recommend is Journey Through Genius by William Dunham, which provides an excellent treatise on the history of mathematics. From the book description
> Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics.
It's a very good read, and not too gigantic. Good wishes your way, mate.
I read a book called Journey Through Genius for a math history class. I enjoyed it quite a bit and it matches what you are looking for.
I highly recommend Jouney Through Genius by William Dunham. It covers the great theorems in history from Euclid to Cantor, and the writing style is very engaging and accessible. It has a perfect five star record on amazon with over a hundred reviews which is pretty rare.
Journey through Genius - The Great Theorems of Mathematics - the beauty and elegance of the theorems he goes through is overwhelming. I read it in high school, and I remember having a strong, emotional reaction to some of them. It seems strange so many years later, but I think it's the only book that's ever made me cry. YMMV
Strangers to Ourselves is about how our unconscious shapes who we are despite what our conscious mind realizes/processes/understands.
Another great book that will hone your math and your vocabulary is Journey Through Genius.
https://smile.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X?sa-no-redirect=1
This book was like a revelation to me.
Introduced me to non-Euclidean geometry and all other kinds of crazy shit.
Working your way through a beginning discrete math class is kind of an overview of the history of math. But here are some stand-alone books on it. Writing quality varies.
The World of Mathematics
A History of Mathematical Notation. Warning: his style is painful.
Journey Through Genius
The Princeton Companion to Mathematics. A reference book, but useful.
Check out some pop math books.
John Derbyshire's Prime Obsession talks about today's most famous unsolved problem, both the history of and an un-rigorous not-in-depth discussion of the mathematical ideas.
There's also Keith Devlin's Mathematics: The New Golden Age, which, to quote redditor schnitzi, "provides an overview of most of the major discoveries in mathematics since 1960, across all subdisciplines, and isn't afraid to try to teach you the basics of them (unlike many similar books)."
Flatland by Edwin A. Abbott is an interesting novel about dimension and immersion. An absolute classic, first published in 1884.
You should also check out the books on math history.
Journey Through Genius covers some of the major mathematical breakthroughs from the time of the Greeks to modern day. I enjoyed this one.
Derbyshire wrote one too called Unknown Quantity: A Real and Imaginary History of Algebra which I've heard is good.
And finally, you should check out at least one book containing actual mathematics. For this I emphatically recommend Paul Halmos' Naive Set Theory. It is a small book, just 100 pages, absolutely bursting with mathematical insight and complexity. It is essentially a haiku on a subject that forms the theoretical foundation of all of today's mathematics (though it is slowly being usurped by category theory). After sufficient background material is introduced, the book covers the ever-important Axiom of Choice (remember the Banach-Tarski paradox?), along with its sisters, Zorn's Lemma and the Well Ordering Principle. After that it discusses cardinal numbers and the levels of infinity. The path he takes is absolutely beautiful and his experience and understanding virtually drips from the pages.
Oh yeah, there's an awesome reading list of books put out by the University of Cambridge that might be of interest too: PDF warning.
I think looking at the history of math is a great starting point. Where did all the ideas come from? How were they formed? Who were these people that came up with them? What inspired them?
A good read (I thought) on this subject was Journey through Genius:
http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X
For a pretty good introduction to a lot of different mathematics, try this book. Journey Through Genius is one of my favorite books. I learned a lot in high school about proof and the history of mathematics and mathematicians. It does a wonderful job of introducing the counter-intuitive concept of countability and sets of infinite numbers.
I've always liked Journey Through Genius. It's pretty small, ~280 pages of paperback novel size, but it covers a nice selection of mathematical history and thinking. It's not comprehensive, but it's a very good introduction to math history. It starts in 440 BCE (Hippocrates) and ends in 1891 CE (Cantor).
Paperback version is only $12: http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X
My big-picture recommendation would be to learn proofs. Can you read and understand proofs? If asked to justify a proposition, can you produce a coherent, rigorous proof that unambiguously communicates your understanding to others? Learning how to do proofs is, as I mentioned above, its own skillset, one that's a necessary condition for being able to do any kind of serious mathematics.
From my perspective, if you're interested in mathematics, then the specific content—i.e., analysis vs. abstract algebra, combinatorics vs. number theory—is secondary to two things:
After all, you don't want to burn yourself out with something you find boring, nor do you want to overwhelm yourself by trying to bite off more than you can chew (such as EGA).
Proof-writing is simply the lifeblood of doing mathematics, and the only effective way to learn this skill is by practice.
One book I often recommend to people in your situation is Journey Through Genius: The Great Theorems of Mathematics by William Dunham. It's not really a textbook, but it's still a really interesting introduction to mathematics. It's a bit like a sampler plate, too, since it covers examples from all sorts of topics: number theory, set theory, calculus, and others, as I recall.
One of the challenges of learning how to write proofs is that it can be difficult to do so as an autodidact; it really helps to get feedback from people who can help you sharpen both your thinking and your writing. This next recommendation may be more logistically challenging (or expensive) for you to pursue, but I'd nonetheless recommend that you look into summer math camps whose focus is teaching fluency with proofs. Three in particular include the following:
All three have good reputations, but you might have personal preferences that would lead you to prefer one over the other two. All three, as I understand it, use number theory as the entry point to teach you how to think about math abstractly, though Hampshire's program is a bit more eclectic than the number theory-specific focus of the Ross and PROMYS programs. Which program, if any, might be your best fit could turn as much on outside issues like when their sessions are held, how much you'd have to pay (US$3,800–4,000 for program costs alone, from what I could tell), and travel logistics (especially if you'd be an international student), separate from any narrowly mathematical considerations. Oh, and another advantage to attending one of these programs is that you're surrounded by fellow students like you who are really interested in mathematics. There's no way to replicate the value of that from any single textbook, no matter how inspired.
Anyway, that's a starting point. If you have local, regional, or national mathematics competitions—e.g., AMC, ARML, as well as other assorted city, state or provincial, or regional competitions—then that's another good entry point into interesting math. From my experience, the main advantage of math contests is that they expose you early to concepts you might not otherwise see for years, and, again, you get to spend time with fellow math students like you.
Competitions, whether individual or team-based, often have more of a proof-based focus than, say, typical the typical high school curriculum (with the exception of geometry), but "contest math" has the danger of students inferring a distorted picture of what it takes to become a mathematician. Namely: you do not have to be a prodigy in math competitions in order to become a good mathematician, let alone a mathematician. Separately, if there's a Math Circle near you, that might be another valuable resource.
As a high school student who will have already completed Calculus BC before your senior year, you might be able to take college-level classes next year, assuming there's a nearby college or university that has some kind of arrangement with your high school. (Some public school districts even cover your tuition, too.) If that's an option for you, free or not, then I'd recommend coordinating with your school's guidance counselor and a professor in the math department to discuss your options.
Oh, and as an obliquely-related topic: if you have time, now would be a good time to teach yourself how to use LaTeX (or one or more of its siblings) to typeset mathematics. (LaTeX may be useful to you if you pursue other scientific field, too, but it's especially useful in math.) If you're serious in pursuing math going forward, you'll inevitably be using LaTeX, and better to get a head start today on scaling its learning curve.
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I'm sure I will think of half a dozen more suggestions an hour from now, but I'll leave things here for now. I hope something above will help, and good luck!
> You can do it with one triangle, but it's ugly ugly algebra.
This is the example that came to my mind. From William Dunham's Journey Through Genius:
> So, Heron's formula provides us with another proof of the Pythagorean theorem. Of course, this proof is incredibly more complicated than is necessary-rather like traveling from Boston to New York by way of Spokane....
An ugly proof, and a great line.
I highly recommend "Journey Through Genius" by William Dunham for people with an interest in math, but maybe with not much background yet.
Each chapter talks about one of the great theorems in math, starting with the ancient Greeks and ending with Cantor. The chapter explains some history behind the problem, and provides motivation for why the question is interesting. Then it actually presents a proof. It's a great way of getting exposure to new ideas, proofs, and is a nice survey of a wide range of math. Plus, it's well-written!
Personally, I don't think learning something like, say, category theory makes sense unless you've had some more higher math that will provide examples of where category theory is useful. I love abstraction as much as the next mathematician, but I've learned that it's usually useless unless you have a set of examples that help you understand the abstraction.
Journey through Genius: The Great Theorems of Mathematics.
Livro bem maneiro, explica alguns pontos com profundidade, contudo de uma forma que você consegue seguir o raciocínio sem maiores problemas. Achei um excelente livro pra vc descobrir algo para se aprofundar.
Na minha experiência a física é o melhor caminho para você aprender matemática, pois dá um "contexto" interessante na aplicação do conhecimento (incidentalmente, sou advogado, contudo leio sobre física há uns 15 anos).
Mas tem alguns detalhes: i) certos pontos de matemática não estarão na física (quem sabe no futuro, como certos aspectos da simetria, que eram tópicos de matemática pura até descobrirem ser uma ferramenta na física quântica) ; ii) certos campos da matemática são muito profundos quando aplicados a uma física (sendo esta passível de adaptação para leigos); ou seja: certos livros de física que vc não terá dificuldade alguma em compreender omitem a maior parte da matemática por ser bem hardcore (muita dedicação para um autodidata, embora possível de ir atrás).
For books that will help you appreciate math, I recommend Journey Through Genius by William Dunham for a general historical approach, and Love and Math by Edward Frenkel and Prime Obsession by John Derbyshire for specific focuses in "modern" mathematics (in these cases, the Langlands program and the Riemann Hypothesis).
There's a lot of mathematical lore that you'll find really interesting the first time you read it, but then it becomes more and more grating each subsequent time you come across it. (The example that springs most readily to mind is how the Pythagorean theorem rocked the Greeks' socks about their belief in numbers and what the brotherhood supposedly did to the guy who proved that irrational numbers exist). For that reason, I recommend reading only one or two books that summarize the historical developments in math up to the present, and then finding books that focus on one mathematician or one theorem that is relatively modern. In addition to the books I mentioned above, there are also some good ones on the Poincare Conjecture and Fermat's Last Theorem, and given that you're a computer science guy, I'm sure you can find a good one about P = NP.
I highly recommend Journey Through Genius.
/u/tick_tock_clock's list includes Euclid's Elements, and I second that recommendation. I read the first couple books of that when I was taking Geometry in 9th grade and found it very interesting.
Thanks for the reply! I'm doing my AS-levels in Maths, Further, Physics, and computing. Also, very interested to hear your recommendation for Physics. For math im looking at these books:
Tell me what you think
Journey through Genius, by William Dunham
If you are interested in the history of mathematics I highly suggest reading Journey through Genius: The Great Theorems of Mathematics. It's a wonderful book that gives you insight about the persons behind the maths we take for granted today and what tools they had available at the time.
Your specific questions about ellipses and hyperbolas is not covered but parabolas and cones are mentioned.
I read this book https://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X and programming helped!
The Christian Bible, the Quran, and the Torah. Don't read them as though these are hostile texts that are trying to impose upon you, read them as though they're mythologies that can be enjoyed, and appreciate their content, as you will have a better understanding of culture and the people around you. You might be surprised just how much of the common and mundane is actually in reference to the bible in our lives. You need to understand people and this is how you do it.
Outside that, I recommend a few good books on math history, not necessarily math education books. Try out Flatland (and it's sequels by other authors) and The History of Pi. I particularly enjoyed Journey Through Genius.
Journey through Genius: The Great Theorems of Mathematics - Willian Dunham
Very accessable walk through of about a dozen famous proofs from Euclid to Cantor.
I got halfway through your post and immediately thought of Journey through Genius. It really is an excellently-written text which presents precisely what you're looking for. I'd definitely check it out.
Dunham, Gleick, Pagels, Bell
To answer your second question, KhanAcademy is always good for algebra/trig/basic calc stuff. Another good resource is Paul's online Math Notes, especially if you prefer reading to watching videos.
To answer your second question, here are some classic texts you could try (keep in mind that parts of them may not make all that much sense without knowing any calculus or abstract algebra):
Men of Mathematics by E.T. Bell
The History of Calculus by Carl Boyer
Some other well-received math history books:
An Intro to the History of Math by Howard Eves, Journey Through Genius by William Dunham, Morris Kline's monumental 3-part series (1, 2, 3) (best left until later), and another brilliant book by Dunham.
And the MacTutor History of Math site is a great resource.
Finally, some really great historical thrillers that deal with some really exciting stuff in number theory:
Fermat's Enigma by Simon Sigh
The Music of the Primes by Marcus DuSautoy
Also (I know this is a lot), this is a widely-renowned and cheap book for learning about modern/university-level math: Concepts of Modern Math by Ian Stewart.
I like Journey Through Genius. It is completely elementary, requiring nothing beyond perhaps a semester of basic algebra. It presents some amazing theorems and emphasizes both the creativity and the logical rigor required to achieve them. I can't remember every theorem, but I know it includes Pythagorus' Theorem, the irrationality of the square root of two, Euclids geometry, the infinitude of the primes, some number theory of Fermat, Isaac Newton on the Binomial Theorem, the quadratic equation and the solution of the third and fourth degree polynomials by radicals and why this requires complex numbers, an exploration of complex numbers, and some non-Euclidean geometry. All that whilst requiring, as I said, no mathematical maturity whatsoever, and being quite easy and enjoyable to read. I highly recommend it.