Starting Out with Proofs: Seeking Advice from Experienced Math Students

In summary, if you're bored with calculus and want to get a head start on real mathematics, you should read a mix of elegant and difficult proofs, practice proofs, and try to do a proof that's decently difficult.
  • #1
Gustafo
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I'm a junior in high school taking calculus 1 at a local junior college, and I am getting quite bored with how very simplistic calculus seems to be in calculus 1. To me it just feels like an expansion upon algebra and trig with a few new twists, but nothing that actually requires me to use my brain.

So what I wanted to do was get a head start doing some more thought provoking math, like proofs, since I want to take real analysis before I graduate high school anyway. I really am a beginner when it comes to proofs and similar mathematics, I honestly don't really know the first thing about how to do a proof. So I was wondering if any of you could tell me me some good videos or reading material to get me started as a beginner?

You help would be greatly appreciated. :)
 
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  • #3
Reading The Art of Problem Solving (there are two volumes) and doing the problems is a pretty good way to learn about proofs.
 
  • #4
A couple of small books that I've accumulated over the years are these two:
The Nuts and Bolts of Proofs, by Antonella Cupillari - http://www.alibris.com/search/books/qwork/4767666/used/The%20Nuts%20and%20Bolts%20of%20Proofs
How to Read and Do Proofs, by Daniel Solow - https://www.amazon.com/dp/0471510041/?tag=pfamazon01-20
 
  • #5
Thank you guys for all the great replies! I will definitely read these :)
 
  • #6
Hello,

my first attempts at proving theorems were really fruitful and engaging so I definitely think this is a good idea. I worked through the Dover book "Number Theory" by G.E Andrews. It starts off with some basic proofs by induction for how to sum different series of numbers. Basis representation, fundamental theorems of arithmetic and algebra, computational number theory etc etc.

I found it got quite hard very quickly, but the sense of reward was worth the effort :)
 
  • #7
Gustafo said:
I'm a junior in high school taking calculus 1 at a local junior college, and I am getting quite bored with how very simplistic calculus seems to be in calculus 1. To me it just feels like an expansion upon algebra and trig with a few new twists, but nothing that actually requires me to use my brain.

So what I wanted to do was get a head start doing some more thought provoking math, like proofs, since I want to take real analysis before I graduate high school anyway. I really am a beginner when it comes to proofs and similar mathematics, I honestly don't really know the first thing about how to do a proof. So I was wondering if any of you could tell me me some good videos or reading material to get me started as a beginner?

You help would be greatly appreciated. :)
Yay, juniors! :tongue:

To start out, read a mix of elegant and difficult proofs. Learn to anticipate what the writer of the proof will do, and begin to ponder what approach you might take to proving theorems before you read their proofs. Since you're just starting calculus, you might want to try some basic epsilon-delta proofs to get you started.

When you feel confident/comfortable with the math, try a proof or two. To start, do proofs that are probably easy. By probably easy, I mean that it's best to not start with the Riemann Hypothesis right at the start.

When you think you're ready to go further, reserve a weekend for yourself. Lock yourself in your room (Please do this within reason. If you feel hungry or your house is burning down, it is, in fact, okay to leave your room.) and proceed to prove something that is decently difficult. What I've done in the past is find a "pen pal" mathematician on the internet and challenge them to a "game" where we take turns trying to stump each other with intricate proofs. This, I think, is the BEST way to learn proofs.
 
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1. What is the purpose of "Starting Out with Proofs"?

The purpose of "Starting Out with Proofs" is to provide advice and tips from experienced math students to those who are just beginning to learn how to write proofs. It aims to help students understand the basics of proof-writing and improve their skills in this area.

2. Who can benefit from "Starting Out with Proofs"?

Anyone who is new to writing proofs can benefit from "Starting Out with Proofs". This includes students in high school or college who are taking a math course that requires them to write proofs, as well as individuals who are learning proof-writing on their own.

3. What topics are covered in "Starting Out with Proofs"?

"Starting Out with Proofs" covers a range of topics related to proof-writing, including understanding the structure of a proof, choosing appropriate proof techniques, and common mistakes to avoid. It also includes examples and practice problems to help reinforce concepts.

4. Is "Starting Out with Proofs" suitable for all levels of math?

Yes, "Starting Out with Proofs" is suitable for all levels of math. It covers basic proof-writing techniques that are applicable to all levels, from introductory courses to advanced topics in mathematics.

5. How can "Starting Out with Proofs" help improve my proof-writing skills?

"Starting Out with Proofs" provides advice and tips from experienced math students who have successfully mastered the art of proof-writing. By following their advice and practicing the techniques outlined in the guide, you can improve your proof-writing skills and become more confident in your abilities.

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