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I want to learn how to read and write proofs. I only have basic pre-calculus skills. As I am preparing for my upcoming calculus 1 course, I wanted to understand questions like

Prove $\left|ab\right|=\left|a\right|\left|b\right|$ for any numbers $a$ and $b$.

These questions pop up in various calculus books and I need to understand them.

Any book recommendations?

Martin Sleziak
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Alexander John
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    Not a book recommendation, but if you answer a lot of the calculus questions here, your proof-writing skills will most likely improve. :D – Simply Beautiful Art Jan 01 '17 at 23:45
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    Here http://math.stackexchange.com/questions/7743/getting-better-at-proofs you can find several book recommendations to your question. – Cahn Jan 01 '17 at 23:53
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    For better or for worse, most introductory calculus classes in the US at least tend to be proof light and computation heavy. Learning proofs is definitely something you'll need if you plan on studying math further, but for the most part the proofs you should see in Calculus shouldn't be bad. – eepperly16 Jan 02 '17 at 00:40
  • Start on first order logic and set theory. – Henricus V. Jan 02 '17 at 04:08
  • When writing a proof, write in complete, grammatical sentences. Justify everything. Especially do not omit any "implies" or "iff". Practice this with proofs you already understand. Many students present a proof as a sequence of assertions with no "connective tissue" explaining how they are related. – DanielWainfleet Jan 02 '17 at 06:42
  • In addition to the great content linked by @MarvinF, I benefit most from the particular answer of Loop Space (when I remember to follow it). One thing missing, I would say, is building intuition about the topic through constructing examples. They can be numerical or qualitative, but developing some sort of intuition for a proof through constructing examples makes it 'stick' a lot better for me. A big part of the difficulty with understanding very abstract topics probably boils down from not being able to easily construct 'toy' examples to play with. – shredalert Jan 15 '17 at 12:57
  • You can probably find a few similar questions on this site. Two examples I was able to find: Book covering introduction to mathematical proofs and Is there anything like "How to prove this"? – Martin Sleziak Jan 15 '17 at 13:18
  • Does this answer your question? Basic book about mathematical proofs – user1110341 Oct 07 '23 at 09:18

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If you're just about to move into Calculus I, you don't need to worry much about the formal logic behind the mathematics. Proofs typically pop up first in a linear algebra or abstract algebra course sophomore or junior year at a university. Real analysis is where you need to worry about proving both integral and differential Calculus, but that's a class that you take after you finish the Calculus sequence anyway. Any facts in a basic single variable Calculus class should be easy to prove with information from your textbook, if you are even required to prove anything at all.

Sir_Math_Cat
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Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry, George F. Simmons

Pre-calculus Demystified 2/E, Rhonda Huettenmueller

TripleA
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