50 Mathematical Ideas You Really Need to Know [Hardcover]

I have read and reviewed 50 Physics Ideas and 50 Philosophy ideas. I enjoyed both books and therefore looked forward to reading this one. Unfortunately, I was deeply disappointed with this book. This is probably mostly due to the fact that the subject does not lend itself to the format of this series (which allots only 4 pages per idea). Mathematical ideas require some development, which is completely missing here. A statement of each idea is given, but in my opinion not sufficiently explained before a few examples are given, and then on to another idea. I found many subjects to have been presented in a somewhat haphazard and incomplete manner. Instead of focusing on the underlying mathematics, too many of the chapters seem focused on "bar bet" problems, such as the "birthday" problem, with the underlying mathematics (in this case probability theory) given only incidental coverage. Very valuable space is taken up by mentioning, but not developing additional ideas in each section, while some ideas, such as the Logarithm are not even included at all. To site just one example, in the section on Chaos Theory there is mention of the Navier-Stokes equation, but it is not developed or even given. It is introduced because it is contains nonlinear terms (but there is no discussion as to what a nonlinear term is), but the fact that it is this nonlinear behavior is the root cause of chaotic solutions is not discussed. The idea of phase space is also mentioned in the same section, but is never defined or even described. If I had never heard of Chaos Theory I doubt that I would have come away from reading this book with a idea of what it is or why the mathematics behave as it does. This was not the case for the Physics and Philosophy Ideas. In the case of these books, the 4 pages allotted to each idea were enough to understand what the idea was and why it was important. If you want an overview of mathematical ideas I recommend Kline's Mathematics for the Non-mathematician. Reading it will be more work, but you will get much more out of it.

I am giving the book 3 stars because there are some reasonably presented sections in the book, but in my opinion not enough to warrant a higher rating.

This is the best short survey of mathematics I know of, and I think the format works very well (4 pages each for 50 key mathematical ideas). I personally read one idea per day for 50 days, and this book delivered a bright spot for every one of those days.

To address another reviewer's comments, I do agree that this book has lower and upper bounds to be aware of.

The lower bound is that readers should come to the book with at least a general familiarity with the subject of mathematics, including having at least heard of concepts like complex numbers, calculus, probability, chaos, abstract algebra, group theory, Fermat's last theorem, the Riemann hypothesis, etc. And certainly readers should come to the book with a genuine interest in mathematics. In other words, this isn't a book for readers with no background or interest in mathematics, nor readers with a fear of mathematics.

The upper bound is that the book doesn't (and can't) develop the mathematical ideas in step-by-step detail. Rather, the book goes into just enough detail to give a meaningful sense of what the ideas are about, and it does this quite well, with nice features like timelines, examples, historical asides, etc. This book isn't a mathematics textbook, nor does it purport to be, so it shouldn't be judged on that basis.

The only thing I really found lacking was that the book doesn't include suggestions for further reading. But this omission isn't enough to lower my rating from 5 stars, and I highly recommend the book to anyone looking for a short survey of mathematics. Tony Crilly provides a wonderful and panoramic guided tour of the subject, spanning from elementary to fairly advanced ideas, and does it in a way that both entertains and reveals the rich beauty of mathematics. For readers who take the tour and feel sorry to see it end, I suggest moving next to the massive and outstanding The Princeton Companion to Mathematics.

This is a concise book covering from ancient Greek Math to 21st century Modern Math. The author has smartly picked the most important 50 ideas, from Greek time, to modern Algebra, FLT, Riemann, Fractal, Genetic Math... it is surprisingly fun stuff to read, unlike other boring Math textbooks, yet it opens the reader's eyes to the beauty and wonderful Math world. For high-school math students going to the University, this is a good transition book, laying the foundation for them to grasp the abstract math ideas in the University, where unfortunately the Math professors would hardly tell the students the roots of these math ideas dated since 3,000 years ago.

Some interesting highlights:
1. Hardy-Weinberg Genetic Law: Cambridge Prof Hardy, as the greatest Pure mathematician in 20th century, prided Pure Math as being 'useless', yet he discovered independently the Genetic Law with Dr. Weinberg (Germany). He wrote the math proof at the back of an envelope after a cricket match in 1903. You can see Hardy's Pure math is not 'useless' at all - he proved the Genetic Law without being a Bio-Scientist, just by applying the beautiful Probability theory (the proof can be found in this tiny book).

2. Abstract Algebra: Since 825AD Arab Mathematician Al-khwarizimi introduced 'Al-jabr' dealing with numbers, Viète (France) in 1591 AD introduced symbols for known and unknown variables, Algebra was transformed into dealing with non-numbers 'Modern Algebra' by Emmy Noether (Germany) in 1920, with axiomatic structures and its 'isomorphism', etc. Bourbaki (France) in 1939 re-built the whole math using Set Axioms under rigourous structures.

Together with the other Twin book: "50 Physics Idea You Really Need to Know" (by Joanne Baker) form a great gift for your to-be-university children or friends. They will definitely be enthused by Science & Math and excel in these 2 subjects in the University.