Good introductory book on mathematical proofs?

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  • #1
Scrumhalf
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My 7th grader has enough mathematical background in algebra, geometry and trigonometry to start learning how to write out proofs. Are their good books that teach this step by step? I can certainly teach him myself with examples, but I figured there must be a systematic way to teach this.

I just searched on Amazon and found a few books with good reviews, but any recommendations would be great! Given the age of the student, the book should start off at a simple level and then go from there.

Thanks!
 

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  • #2
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I think proof writing and reading are best learned as you learn other math, not by reading a book dedicated spesifically to proof writing. I think it might be a good idea to try Serge Lang's book "Basic Mathematics".
https://www.amazon.com/gp/product/0387967877?ie=UTF8&tag=pfamazon01-20

Alternatively, if he/she is reasonably proficient in geometry, it might be a good idea to try reading Elements by Euclid, although it may still be too advanced, or too dry reading, as there is literally no exposition.
 
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  • #3
Scrumhalf
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OK.. thank you for that recommendation!
 
  • #4
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I agree. While there are some decent proof-writing books, it is better to learn by applying it to something interesting.

Some basic Discrete math textbooks contain a good introduction to writing proofs. I liked this one (and it essentially contains a partial solution manual):
https://www.amazon.com/dp/0131679953/?tag=pfamazon01-20&tag=pfamazon01-20
Ignore the reviews on Amazon.

Don't get the expensive new edition, though. Get a used older edition or an international edition:

https://www.amazon.com/dp/B008ITTTPC/?tag=pfamazon01-20

http://www.bookfinder.com/search/?keywords=0131679953&st=sh&ac=qr&submit=
 
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  • #5
Fredrik
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Alternatively, if he/she is reasonably proficient in geometry, it might be a good idea to try reading Elements by Euclid, although it may still be too advanced, or too dry reading, as there is literally no exposition.
That one is a little dated, don't you think? I haven't read it, but I'm sure someone has figured out a better way to do these things in the 2300 years since it was written.
 
  • #6
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That one is a little dated, don't you think? I haven't read it, but I'm sure someone has figured out a better way to do these things in the 2300 years since it was written.
I think the only outdated aspect of the early editions is the language, and a lack of rigor in some places, but there exist modern translations with expository notes, see for example http://farside.ph.utexas.edu/euclid.html [Broken].
The mathematics, however, is not outdated. In fact, the book was used as the primary geometry reference book long into the 1600's, and the constructive (non-trig based) geometry that is taught in high school today is usually a dumbed down version of Euclid.

EDIT: In books 9 and 11, Euclid uses an early form of integration called the "method of exhaustion". I think you can safely say that these books are pretty outdated.
 
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