Customer Reviews

The New Mathematics

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I can't believe this is out of print. I read this book first when I was about 13 or 14 and found it fascinating.

I reread it many years later and it was still interesting. The book presents an argument for why we need all those kinds of numbers (integers, reals, imaginary, complex...) and does so in a straightforward and elegant way. Make no mistake, it is NOT a mathematics textbook - there are no systematic proofs - but it presents solid reasons for adding different kinds of numbers.

The book starts with the counting numbers, then shows why we need zero, negative numbers, fractions (rational numbers), irrational numbers, imaginary numbers and complex numbers.

Hopefully it will be reprinted someday... I'll certainly buy a copy.

Most historians would say mathematics began with numbers 'and' geometry. I don't know how true this is beyond the fact that we have cultural fragments of numbers going back to tens of thousands of years ago(tally sticks); unless you want to call the cave paintings geometry(I think you can argue that a little bit - kindof the beginnings of geometry); one must be forced to conclude that the first matheamtics was numbers.

It's not that this book doesn't have any geometry; it's just that this book focuses on showing what advanced mathematics can be shown to a maybe pre-junior high schooler through numbers more than geometry. I read my dads copy when a junior high schooler; made me love mathematics; for this reason alone, it should be back in print. I actually got my dads copy stolen after being mugged! Lucky for me, I had bought a copy(a hard cover instead of the falling apart softcover from my father!) . . . through a library. At the time, I checked the book out; i came back and told them I lost it; they charged me five dollars! At that time, I held the book in such high regard!

Today, I still think that everyone should learn how the number systems can be derived from each other and how much more advanced mathematics relates to each number system. I've read E.T. Bell's "Development of Mathematics", all of Morris Kline's stuff. Mostly, I've read lots of history of mathematics; i guess I'm more of a mathematics historian than a mathematician. But, after reading books like Van Der Waeden's "Science Awakening" which is a much more hands on technical history of mathematics and learning about all the mathematics of Euclid's Elements(I have a copy and look up theorems as books show which theorems are generalizations of Euclid's elements theorems), Appollonius Conics, the collective works of Archimedes, and Ptolemy's Almagest, and John Stillwell's technical histories of mathematics, I see the interaction between geometry and numbers in the 'real' origins of numbers. Well, "The New Mathematics" does mention the pythagorean theorem; but, what about the phythagorean triples? It mentions the relations of abstract algebra concepts in relation to each higher number system; but, it doesn't say things like fields and groups come out of galois theory; rings comes from number theory. What I think I'm trying to say here is that I can see a sequel to this book making sence. Well, one could argue that John Stillwell's books can be used as a sequel to this book, "The New Mathematics." One thing I thought was missing from John Stillwell's books was what this book presents - the derivation of each number system and how the higher abstract algebraic concepts fit in. Cause, when you read the history of mathematics enough, and study algebra enough, you realize that the main problem in understanding classical algebra is knowing how the various numbers work and relate to one another. And, you learn that when the calculus was created, mathematicians found that they needed to define real numbers, rational numbers, and number systems better(see E.T. Bell's The Development of Mathematics for this). John Stillwell's works seems to miss this imo. But, this book stops short of what John Stillwell shows - things like the chord method of diophantus. I mean there's so much advanced mathematics right under your nose so to speak of even elementary mathematics - things like Euclid's algorith needed for abstract groups as oppossed to the congruence of integers concrete representation of groups as presented in "The New Mathematics". The congruence of integers is important. I'm just saying, one could introduce more advanced mathematics to lower levels than most, even the elementary mathematics teachers probably don't realize this, realize! Sorry to jumble probably these two last paragraphs together!

I want to say that I'm finding that through the Thomas Heath editions of the classical Greek mathematics works(for the most part four in total! Already mentioned above), and knowing my history of mathematics, one should study those for the best practice in classical mathematics; if you know your history of mathematics, you should be able to see the connections to the more advanced(not what the school teaches in those awefull school books!) real mathematics. Also, see John Stillwell again.

Best intro to the foundations of mathematics. I have ordered this books many times and given it away as gifts to students and friends. It opened my eyes to the wonderful world of abstract mathematics.

When the phrase, new math, was all the rage. Good for learning the vocabulary.

To add to the first review, my memory of the book is that it added elementary introductions to set theory, group theory, matrix algebra (I think; I also had a semester of M.A. in high school) and associated concepts such as commutivity, etc. It was very well-written, clear. I tried to look up someting in my copy today on groups and couldn't find it. Which is why I'm here.

I highly reccommend the book.