(Part 2) Best mathematics history books according to redditors
We found 520 Reddit comments discussing the best mathematics history books. We ranked the 169 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
![An Imaginary Tale: The Story of [the square root of minus one]](https://m.media-amazon.com/images/I/51plploOO1L.jpg)
Russel also wrote an introduction to mathematical logic just because of this...
http://www.amazon.com/Introduction-Mathematical-Philosophy-Bertrand-Russell/dp/1420938401
> It seems to my simple mind that we're over complicating a simple answer.
The purpose of the 162 page proof was effectively to create a axiom upon which we could base higher levels of mathematics, because we as mathematicians, logicians and philosophers want math to be founded on something other than an arbitrary rule set, we want a provable rule set than can be used to expand our knowledge.
This is a very loaded question, but I'll do my best to answer it.
Disclaimer: I'm a lowly undergrad.
Math is made up of many different fields, so it's hard to assess its development in a linear fashion. The main areas are: algebra, geometry, analysis, logic/set theory/foundations, number theory, and applied math.
Now, the calculus that you study was mostly invented by the end of the 17th century, but interestingly, ideas about limits didn't really get formalized until the 19th century. Cauchy was instrumental in formalizing the definition of the derivative, and Riemann in formalizing the equivalent definition of the definite integral.
If you were to continue to take math classes in an undergraduate curriculum, you might take any of the following: linear algebra, abstract algebra, more calculus (multivariable, etc.), differential equations, discrete math, real analysis, complex analysis, basic functional analysis, topology, set theory, non-euclidean geometry, graph theory, number theory, probability and statistics, etc.
But even if you took most courses available to you as a college student, you would still be about 100 years behind mathematics as it stands today. You could argue the exact number of years, as different fields have evolved at different rates, but the point is that math has exploded in variety over the last century.
The last person to truly grasp "all" areas of mathematics of his time is probably someone like Gauss or Euler. Today, you're lucky if you can master a subfield of your subfield.
I'll now try to give you a quick overview of how math has developed since the invention of calculus. Apologies if it sounds vague (that's because it is).
The invention of calculus itself was a gradual process. Leibniz and Newton are credited with the invention largely because they collected the known information at the time and managed to come up with the Fundamental Theorem of Calculus, along with a ton of notations. Newton also applied it to his physics, which began a long relationship between calculus and the natural sciences. Up until this time, most math was geometric, rather than algebraic. Complex geometric proofs were drawn up to solve problems that would later be solved simply with algebra. Analytic geometry had only been invented a few years prior by Descartes, and though it seems obvious to us today, it was revolutionary at the time.
Progress in calculus was made very quickly in the 17th century - before long, all of the calculus you learn (and then some) was invented, though it lacked the formal foundations that it has today. The Bernoullis helped to found the calculus of variations, which is a topic usually reserved for grad school. That gives you an idea of how much math there truly is. In the 18th century, most of the progress was made in further developing the calculus, while other fields of math were being born. Euler and Gauss helped to popularize the notion of complex numbers. Euler considered the Königsberg bridge problem which would later inspire graph theory and topology. Lagrange and others made discoveries in what would become the field of abstract algebra.
The 19th century brought rigor to mathematics. While mathematicians were proving things up until then for the most part, there was often an element of intuition (and sometimes hand-waving) going on, especially back in the 1600s. A discovery by Fourier that functions could be approximated by series of sines and cosines (later, Fourier analysis) made the mathematical community worried. With the foundations of calculus on such empirically-strong, but theoretically-shaky grounds, how could they be sure if these wild claims were really true? This quest to put Fourier analysis on solid ground led to a rigor revolution in mathematics. What we now call "analysis" (the more rigorous version of calculus) was developed, and many other discoveries were made as a result. The other revolution going on in the 19th century was in the field of abstract algebra. Many of the great algebraists lived during this time: Galois, Abel, Noether, and others. Noether actually lived about 80 years after Abel and Galois.
Towards the end of the 19th century, Cantor developed his theory of sets, which led to a second revolution in mathematics. Although they are ubiquitous now, sets were relatively unused at the time. The introduction of sets into mathematics, combined with the revolution in rigor discussed previously, had a huge impact on mathematics. Progress in nearly every field accelerated like crazy in the new set-based more rigorous framework. The 20th century brought a deluge of mathematical information unlike any time before it. Some of the key fields that were invented in the last century -- or really started to get going -- were topology, algebraic geometry, linear algebra (for computers), computational logic, model theory, set theory, graph theory, combinatorics, various applications of math to advanced physics like relativity theory, etc.
But these are just the tip of the iceberg. If you really want to know how far math has progressed in an intuitive sense, I invite you to open a recent math journal (or look at one online) and see how much you can understand. It's truly mind-blowing.
I hope this helped more than it hurt. I'm no historian, but I think I got at least the general shape of things right. If you want a more in-depth look, especially one that considers the more modern developments, try Kline's series of books:* link to the first one
Both "\" and "−" are common notations for set difference.
I suppose the reason "\" is commonly used instead of "−" is the same reason "∪" is used instead of "+" for the union of two sets. Just as union isn't really the same thing as addition, this thing we call "set difference" isn't really the same thing as subtraction.
I don't know the origins of the "\" symbol to mean set difference. Somebody should write an updated version of Florian Cajori's classic book A History of Mathematical Notation to bring it up to date with notations introduced in the 20th century.
It doesn't have much on visualization IIRC, but maybe this Introduction to Mathematical Thinking can be helpful to you. The professor also has a book by the same title and a blog.
I am going to guess this book
http://www.amazon.com/Mathematical-Universe-Alphabetical-Problems-Personalities/dp/0471176613
Try one (or a few) of these:
http://www.amazon.ca/Thinking-Mathematically-J-Mason/dp/0201102382
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/052171978X/
http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/
www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/
https://www.coursera.org/course/maththink
It may be a much older book, Bertrand Russell's philosophy of mathematics book probably has a lot of value. Russell was not only an important 20th century thinker, he was also an accomplished mathematician. Someone who's contributed so much to both fields, I would think, has something of value to say in this area.
USSR Olympiad Problem Book
Putnam & Beyond
Mathematical Gems I & II
Challenging Problems in Algebra
I am late to the party so nobody will probably read this, but just to settle it once and for all for everybody in this thread: every physicist and mathematician in the world pretty much agrees on who were the 'greatest of all time' in both fields. Here they are, respectively:
physics: Newton > Einstein > Maxwell
mathematics: Gauss > Euler > Archimedes (though the order of the last 2 may could be reversed perhaps)
(for what it's worth, I have an MSc. in both and am also an avid reader of the histories of both fields. For those interested, check out Struik for mathematics and Motz for physics)
---
Some interesting tidbits while I am getting all worked up about one of my greatest passions:
This book has a lot of what you're looking for.
Gödel's theorem is much more profound (and deeply linked to the halting problem):
Any self-consistent axiomatic system complex enough to reason about itself has a provably infinite number of axioms, i.e. statements which are obviously true but unprovable in the system.
Your (2) is a corollary of sorts, but it's not a choice, (1) is provably true. I recommend Meta Math!: The Quest for Omega which is very approachable if you've got some grasp of programming in scheme-like languages.
Cajori is the one.
http://www.amazon.com/History-Mathematical-Notations-Dover-Mathematics/dp/0486677664
William Dunham, an author I appreciate because he succeeds in popularizing mathematics without sacrificing rigorous exposition, wrote on the Baire Category theorem in The Calculus Gallery. I too was a bit confused as to the motivation of Baire's Category theorem until I went back and read his chapter. I don't have my copy handy at the moment but I recommend checking out that book for an expository and historical explanation.
A good book that covers Galois' story, and the development of group theory, is The Equation that Couldn't Be Solved by Mario Livio.
I'm in a similar boat with you. I went through calculus in high school, graduated university with a B.A. in music, but have recently taken a keen interest in developing an actual understanding of math.
Aside from music, I have a strong background in philosophy, and from philosophy, so do the natural sciences extend and I've taken advantage of that. Math was discovered through raw observation of the world and through the concourse of logic, and so I have designed for myself the study of math through the source works of where the math originated, for practical and ontological purposes. Here's a few books that I've picked up and began reading:
A History of Greek Mathematics, Vol. 1: From Thales to Euclid https://www.amazon.com/dp/0486240738/ref=cm_sw_r_cp_apa_RljGybYRSB723
The Mathematical Principles of Natural Philosophy: The Principia https://www.amazon.com/dp/1512245844/ref=cm_sw_r_cp_apa_AmjGyb14R4B2V
Euclid's Elements https://www.amazon.com/dp/1888009187/ref=cm_sw_r_cp_apa_7mjGybZ97DBR7
Introduction to Mathematical Philosophy https://www.amazon.com/dp/1420938401/ref=cm_sw_r_cp_apa_OnjGybQ0078ZX
The Fractal Geometry of Nature https://www.amazon.com/dp/0716711869/ref=cm_sw_r_cp_apa_lojGybPPY25P4
The study of equations and formulas had been unfulfilling and unengaging until I framed it with the historical context of the natural sciences. I'm still a novice to this approach, but I believe it to be of merit- Ive also see some indication (when researching my own self-study method) that this is more similar to the method which Waldorf schools teach math and science as opposed to the traditional American Public school classroom, which as I grow older and reflect upon the majority of my experiences in classrooms, were uninspired, with the exception of very few memorable educators.
You could even base your study on other, less abstract interests than the interest of learning mathematics, such as an interest in modern physics or economy (or Comp sci, anything that utilizes math). Using that interest as a guide, you would be more clear minded to reverse-engineer your own individually purposed self-study. Such a direction of interest would certainly help for you to be able to design your course and keep you engaged. I hate how I've worded most of this Frankenstein of a comment; it's unnecessarily verbose and unorganized, but it's late and I'm tired to I'm not gonna edit it, nevertheless, hopefully you'll get the point(s).
Anyway, I'm curious what other people have to say about this approach, and especially I am open for people to suggest in response here to additional and essential sourcebooks!
This reads like it was written by somebody who just had their mind blown (in fact he says so in the last paragraph). Which is understandable, but in my opinion one should wait to cool down a bit before rushing out to write a review. For what it's worth, I found the reviews on the Amazon page for the book more interesting and carefully-considered.
The reviews on Amazon also point out that the book is not just about the history of mathematics, but is also proposing an alternative mathematical algebra where negative times negative is negative. So the defenses that other people are making in this thread, that it's not a math book and shouldn't be criticized as such, don't hold water. On the other hand, criticizing the math before knowing what the axioms of the system are isn't valid either. −1 × −1 = 1 in a ring, yes, but the author can define alternative arithmetic operations on the integers that may not form a ring, but some kind of non-commutative algebra instead. In summary: STFU, guys.
I recently took the Math II. Compared to the Math HL syllabus, Math II is ridiculously easy. There are a couple of topics in Math II that are not covered by the IB but if you can take Math HL the SAT will seem like child's play.
What I did - and what I think is the best choice - is to get a good book on the SAT and study that over the summer. You can easily finish 2-3 chapters/day (studying like 1-2 hours) and be done in a month tops. As for the book I recommend Barron's SAT Math II book.
For a different take on this question, I recommend Burn Math Class.
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.
An Imaginary Tale, by Paul Nahin. It's about imaginary numbers and was written by an electrical engineer. I'm pretty sure it has a section about radios somewhere.
The Calculus Gallery - Masterpieces from Newton to Lebesgue by William Dunham it's like having the world's best math teacher lead you thru all the jaw-dropping amazing parts of the history of calculus; how at each stage someone figured out some bizarre counter-example which caused the next person to push calculus along further. With understandable equations!
Pick up Negative Math by Alberto Martinez. It turns out that historically, the idea of negative numbers was just about as controversial as that of imaginary numbers. Why should -1 × -1 = 1 instead of -1 × -1 = -1 for instance?
I'm reading Mathematical Thought from Ancient to Modern Times by Morris Kline, and I really like it. It's three volumes, so it's pretty thorough. I've only finished the first volume, but he does a really good job of explaining the way the Greeks thought about mathematics, compared to the modern Western way. It's not too technical to follow, but he tries to explain the important theorems and their proofs in some detail. Volume 3 goes up through Poincare and Goedel, and then stops, so you won't learn about later twentieth century developments from him. But he says it's too soon to know what's historically important there, anyway. I'm looking forward to the next two volumes.
For Geometry
https://www.amazon.com/Mathematical-History-Golden-Number-Mathematics/dp/0486400077
https://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859
For trig
https://www.amazon.com/Mathematics-Heavens-Earth-History-Trigonometry/dp/0691129738
Edit:
Boyer also has a history of calc only
https://www.amazon.com/History-Calculus-Conceptual-Development-Mathematics/dp/0486605094
And analytic geometry
https://www.amazon.com/History-Analytic-Geometry-Dover-Mathematics/dp/0486438325/ref=sr_1_1?s=books&ie=UTF8&qid=1540239797&sr=1-1&keywords=Boyer+geometry
And more practical for learning how to problem solve for a program is
https://www.amazon.com/Solve-Mathematical-Problems-Dover-Mathematics/dp/0486284336
I've been having the same issue! I felt discouraged because it felt like Magoosh assumed I had all these formulas and concepts already memorized and it was overwhelming to be thrown into difficult questions when I didn't even know how to divide exponents. I bought this book to refresh on basic algebra/geometry skills and so far I've felt a lot more confident about testing! https://www.amazon.com/gp/product/0544631021/ref=ppx_yo_dt_b_asin_title_o02_s00?ie=UTF8&psc=1 Also, just practicing the GRE style of questions really does help because the test will never ask you in the basic form we are used to being tested on in math class. It's important to practice GRE questions because that's what we're going to see on the test and learning the nuances of what's really being asked of you helps you answer questions quicker! Gregmat has some great videos on the Quant section. Good luck!
Smith's "A Source Book in Mathematics" (https://www.amazon.com/Source-Book-Mathematics-Dover-Books/dp/0486646904) is a solid collection of original works by the masters. It's important to read how the masters presented discoveries, along with textbook descriptions of the same. There's a source book by a guy named Struik too, but that one's harder to find.
This has little bearing with grade school math in the sense that they are puzzle books (and hence more focused on general problem solving than specific math concepts) but I really like Mathematical Circles and The Moscow Puzzles.
i have done a lot of research into this area. people in this thread are a bit shortsighted in my opinion. here are some references that do exactly what you ask and what they state can't be done:
also, the popular science book black hole by marcia bartusiak is a great account of some of the history of general relativity with respect to the acceptance of black holes. none of the books above are popular science though. they describe real physics in real ways.
there are more, but i think this is a good start. have fun!
If you're interested check out Mathematical Thought from Ancient to Modern Times by Kline or A History of Mathematics by Suzuki.
Hi,
I'm a TA in my school's CS theory course (a mixture of discrete math, and the automata, languages and complexity topics most CS theory courses cover).
As others have said, "theory" is pretty broad, so there are an awful lot of resources you could look at. As far as textbooks go, we use two - Sipser's Introduction to the Theory of Computation (which others have recommended), and the freely available textbook Mathematics for Computer Science, by Lehman, Leighton and Meyer - which concentrates more on the "discrete math" side of things. Both seem fine to me. Another discrete-math–focused set of notes is by James Aspnes (PDF here) and seems to have some good introductions to these topics.
If you feel that you're "terrible at studying for these types of courses", it might be worth stepping back a bit and trying to find some sort of an intro to university-level math that resonates for you. A few books I've recommended to people who said they were "terrible at uni-level math", but now find it quite interesting, are:
These aren't specific to computer science, but might be of help. Good luck!
I'm using barrons for math 2. https://www.amazon.com/Barrons-SAT-Subject-Test-Level/dp/1438007914
This looks like an interesting link. If you want more of this sort of stuff, you might want to check out a source book in mathematics. It has a lot of classic papers, advertisements, some patents and other things related to mathematics.
You can look up courses/lectures that use your text to see what %age exercises they require, what needs to be skipped/supplemented etc.
For a lot of applied topics, you could maybe get some kind of in person study group, or start a subreddit like /r/babyRudin or piazza or a google group, make it a social/cooperative thing
You can look at math study guides by Lara Alcock, Kevin Houston, Keith Devlin, lots of amazon reviews: https://www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634/
The kind of mathematics you learn should depend on your needs; like anything, it's a tool to solve problems, not an end in itself. Assuming you simply want to be competent at problem-solving and clever thinking, I suggest these books (or books like them):
One reason why I suggest these books is because (at least in the first one) all solutions are given in the back of the book; since you have no mentor/someone to help you out, I think it's best that you don't waste too much time 'struggling' in the beginning. Once you've seen and understood a few methods, then you can try doing problems on your own, but sometimes it's good to cheat in the beginning and just try to make sure you're capable of properly understanding the underlying logic/reasoning in someone else's solutions.
Also, keep in mind that the problems in these books might not look like what you recognize as math. However, if you can do these problems, then you'll easily be able to do the more computational kind of mathematics, but it's not true that being good at calculations/computation means that you will be good at mathematics or clever thinking.
Websites on problem solving: Project Euler, Maths Challenge or even SPOJ. Programming competition problems are a good source of math problems but they also include a lot of problems which require knowledge of data structures or algorithms. Therefore use a problem classifier tool for SPOJ (they are listed on the main page under tools) to find problems on combinatorics/number theory/ad-hoc techniques.
Books like this might help you.
For physics the only book that I can think of is this. This is one of those classic Russian problem texts, not so well known in the west, but very popular where I live (India).
Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.
You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.
Two other books you may be interested in instead, that teach the same kinds of things:
Introduction to Mathematical Thinking which he wrote to use in his Coursera course.
How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.
Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.
What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.
Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics without any freaky looking symbols, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/phil110-gmh/text/c01_3-99.pdf
Honestly if I were starting out I would love that last link, it looks fantastic actually.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
I found the ASVAB specific books don't help me at all. I only need to study math for the ASVAB so I got CliffsNotes Math Review For Standardized Tests and it's helped me so much! It's great for those taking a standardized test and didn't learn math properly and/or has been out of high school for a while (Me). It goes through a quick review of all common core math topics with practice problems. I actually enjoy cracking open the book and studying for an hour every day!
https://www.amazon.com/CliffsNotes-Math-Review-Standardized-Tests/dp/0544631021/ref=pd_lpo_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=S9192TDGGPS9REEATZRH
I believe the best first book on relativity is Lillian R. Lieber's The Einstein Theory of Relativity. It's written in a very clear style and presents all the necessary mathematics alongside lucid explanations of all the important concepts of both special and general relativity.
Most of the book only requires a familiarity with algebra and trigonometry, but she also introduces some calculus and mathematical objects called tensors in a very friendly way. There are excellent derivations and notes at the end of the book that walk the reader through some of the more technical aspects of the theory.
Morris Kline isn't a professional historian but he has a book on this that you might be interested in. IIRC, volume 2 has the most stuff on the history of calculus.
Go read Burn Math Class.
It's very fun and an excellent re-discovery of how math is supposed to be useful, not a burden. Great, great book.
It should be required reading for Calc 1 in all disciplines, if schools cared more about teaching their Freshmen properly.
I really enjoyed The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
790 - Biology M
800 - Math: Level 2 (do NOT take Level 1!)
As for resources:
Biology M - Barron's - considered the "Bible" -- has everything you need. A lot of people never even took Biology in High School yet and still scored very high just by using this book. For upcoming takers in the future, I also recommend the awaited College Board version.
Math L2 - Barron's - just so you know, Barron's can be overkill but I really wanted the perfect score so this is perfect. Princeton Review or Kaplan's should be good enough for 700-750+
An Imaginary Tale: The History of the Square Root of -1 was a great read.
For light, math-related reading I've always enjoyed semi-biographical books about mathematicians, because these books usually include a summary of their mathematical contributions without getting too technical or dry. And they always get me pumped to do more math. Here's books I'd recommend in that vein:
The Man Who Loved Only Numbers An Autobiography of Paul Erdos. This one's a really great read with lots of Number Theory and Graph Theory in it. Not to mention a heartwarming bio of Erdos. If you haven't read this yet, give it a go.
The Equation That Couldn't Be Solved - A book about the history of Group Theory and how Galois was able to prove the Abel-Ruffini Theorem. At times it's a little simple, but it's fun to read, and it gives some insight into elementary group theory.
Of Men and Mathematics - not too much actual math in this one, but it's a very solid cheap, quick read. Well worth the money/time.
This is a great book (i.e. easily accessible to the layman) written by Chaiten (who many have brought up in this thread).
it is my favorite math book by far and is directly related to the OPs question:
https://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/1400077974
Lillian Lieber. The Einstein Theory of Relativity
Victor Stenger's The Comprehensible Cosmos.
https://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/1400077974
I fall firmly on the discovered side of the debate although I have had this discussion many times before. I believe math exists independently of humanity and that math is simply a tool used to describe patterns in a logical and rigorous way, but that those patterns have always and will always be in the universe and did not need us in order to exist.The driving force in the history of mathematics has not been practical invention by someone determined to solve an existing problem but by investigation of someone into a pattern that they saw or suspected that they saw. Math ends up being useful to people who want to solve a real world problem, but very little math gets created to solve a real world problem. Mostly because the real world problem didn't exist before the math, or that some very clever person saw the math and realized it could be adapted to solve a longstanding problem.
A great book for non mathematicians covering the history of how math came to be what it is today, and all of the impressive fundamental changes that had to occur in the way we though about math would be The Language of Mathematics by Kevin Devlin, a professor of mathematics at Stanford University. My favorite bit from the book is that things that we think are very simple in fact took thousands of years for humans to abstract and notice the patterns. Egyptians for example basically knew the formula for the volume of a pyramid. However we have not a shred of a formula from that period. Instead they treated each set of dimensions a pyramid could have to be a separate problem and therefore failed to notice the formula that was right in front of their faces. It took the greeks (and a good deal of time) for someone to notice the pattern between all pyramid formulas and abstract it into a general solution to the pyramid volume problem.
A nice pair of books you might like is the two volume set Challenging Mathematical Problems with Elementary Solutions, by Yaglom and Yaglom. Here's the table of contents from volume 1 and the table of contents from volume 2, so you can see what areas the problems are drawn from.
I haven't looked at this book, but noticed it on the site while looking up the details on the above two books: The USSR Olympiad Problem Book. It's from the height of the cold war, when Russia was generally kicking ass in math, so I'd expect it to be pretty good.
Actually, there's about a bazillion books of problems at Dover, most quite affordable, so I'd recommend browsing around that site a bit.
Plenty of problem books have been put together from the International Mathematical Olympiad problems, such as this one. I think you can find most of these problems online if you prefer.
A book less of the "here are problems you can solve!" variety and more of the "here are some neat problems and their solutions!" is 100 Great Problems of Elementary Mathematics: Their History and Solution by Heinrich Dorrie. It covers everything from the Hermite-Lindemann theorem to calculating how to tack into a north wind as quickly as possible. It's just all over the place, which might be something you'd like given your desire for variety. Here's the table of contents.
Your adventure made me wonder what math they are talking about on reddit, since I am also using math at work that I never studied in college.
One person recommended this book, Burn Math Class which I guess focuses on the motivations behind the proofs rather than the proofs themselves. It's aimed at people that have given up on math.
I'll throw out some of my favorite books from my book shelf when it comes to Computer Science, User Experience, and Mathematics - all will be essential as you begin your journey into app development:
Universal Principles of Design
Dieter Rams: As Little Design as Possible
Rework by 37signals
Clean Code
The Art of Programming
The Mythical Man-Month
The Pragmatic Programmer
Design Patterns - "Gang of Four"
Programming Language Pragmatics
Compilers - "The Dragon Book"
The Language of Mathematics
A Mathematician's Lament
The Joy of x
Mathematics: Its Content, Methods, and Meaning
Introduction to Algorithms (MIT)
If time isn't a factor, and you're not needing to steamroll into this to make money, then I'd highly encourage you to start by using a lower-level programming language like C first - or, start from the database side of things and begin learning SQL and playing around with database development.
I feel like truly understanding data structures from the lowest level is one of the most important things you can do as a budding developer.
If you basically need to restart math learning, I recommend this book highly. https://www.amazon.com/CliffsNotes-Math-Review-Standardized-Tests/dp/0544631021/ref=sr_1_2?crid=A53LHLMXC9VM&keywords=math+review+for+standardized+tests&qid=1557788562&s=gateway&sprefix=math+for+standard%2Caps%2C158&sr=8-2
I took a course in it as part of my undergraduate math degree, which was about 40 years ago. This is the text I used.
I would also recommend reading about the philosophy of math, as is it discovered or invented? http://www.pbs.org/wgbh/nova/blogs/physics/2015/04/great-math-mystery/
Okay!
A good history book will probably be best, consider this one.
For other possibilities see:
https://www.amazon.com/Negative-Math-Mathematical-Rules-Positively/dp/0691123098/ref=sr_1_1?ie=UTF8&qid=1503043974&sr=8-1&keywords=negative+math
Not a doubt in my mind The Mathematical Universe is an excellent book for this. It ranges from basic trig about triangles, to integrals and derivatives. This isn't a book on learning math from, its a book on what math can do, and why we do math. Use it to introduce her to many ideas, as a whole of what the applications are. Since she's more into knowing about math, then actually reading texts and learning it, I really do recommend this book.
Not sure what this is but its $4 here.
Does he like to read? Maybe a book, something like this: http://www.amazon.com/The-Mathematical-Universe-Alphabetical-Personalities/dp/0471176613
look for this book and give it a try:
http://www.amazon.com/USSR-Olympiad-Problem-Book-Mathematics/dp/0486277097