"This extraordinary work takes the reader on a long and
fascinating journey--from the dual invention of numbers
and language, through the major realms of arithmetic,
algebra, geometry, trigonometry, and calculus, to the
final destination of differential equations, with
excursions into mathematical logic, set theory, topology,
fractals, probability, and assorted other mathematical
byways. The book is unique among popular books on
mathematics in combining an engaging, easy-to-read
history of the subject with a comprehensive mathematical
If asked, I tend to direct people towards the following books:
Note: you need both the student manual (which most people don't know exists) and The Art Of Electronics.
To cover the maths background required, I recommend:
They are not cheap but worth it.
Oh and a calculator. Any old cheap scientific (Casio/TI/HP) will do as long as it doesn't make errors.
The big problem for me was the maths initially. It doesn't take long before you hit a brick wall at the age of 12. My 10 year old daughter is learning algebra and programming (in python!) though at school so things are looking up.
It's better than anything I've read from any mathematician. They seem to forget that people don't know what they are talking about to start with.
It's filled with a lot of history on why things are as they are and it builds up a substantial base of math knowledge from there. I can't comment on whether the additional background information would help someone who is math shy to "get it" but, from the parts I read, it certainly rounded out (and expanded) my knowledge.
I'm pretty sure the result would be the utter destruction of my productivity for a few years.
A couple of recommendations (not specific to just Calculus):
- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)
- Calculus (Apostle http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...).
- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.
- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html
- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)
- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.
- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/
I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.
Definitely not a quick summer read though IMO
Everything you need to know in the correct sequence.
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