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#1 2014-05-01 08:48:54

PatternMan
Member
Registered: 2014-03-08
Posts: 171

How do write a proof?

I have started doing some questions that ask you to show or prove something. In algebra they ask me to prove that some types of numbers have a common pattern. You usually end up setting up an expression and equation then factoring. However there are geomtric questions that ask you to prove things about vectors, shapes, congruency, similar triangles etc. My problem is that usually there are multiple ways to do this. Actually I don't really know how to write the notation properly. I will be able to tell myself wether it is something but I'm not sure if the proof is rigorous enough. I want to know specifically what they want or not. Because there are several ways to prove and your method isn't always there as an answer.

Last edited by PatternMan (2014-05-01 08:49:29)


"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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#2 2014-05-01 08:52:29

ShivamS
Member
Registered: 2011-02-07
Posts: 3,465

Re: How do write a proof?

You do not need to worry about rigorously proving something right now. As long as it makes sense to you, it is fine.

However, I do recommend looking at
http://zimmer.csufresno.edu/~larryc/proofs/proofs.html

It will teach you several proof methods.

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#3 2014-05-01 09:16:30

PatternMan
Member
Registered: 2014-03-08
Posts: 171

Re: How do write a proof?

thank you! Also do you know of any link for writing geometric proofs? For sides, angles, vectors etc?

Last edited by PatternMan (2014-05-01 09:40:06)


"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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#4 2014-05-01 12:28:29

ShivamS
Member
Registered: 2011-02-07
Posts: 3,465

Re: How do write a proof?

Geometric proofs are more or less the same method, except you use the proper names for lines and all other geometric objects.

This is somewhat not needed, but if you want to, read/watch
http://www.sophia.org/tutorials/introduction-to-geometric-proof

Later, if you have time, read one of the best proof books
How to Prove It: A Structured Approach Daniel J. Velleman
You can read that right now (I recommend it - and even online) if you have time too.

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#5 2014-05-01 20:15:52

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,269

Re: How do write a proof?

hi PatternMan,

All mathematical theories start with a set of axioms.  Then a proof is used to generate theorems.  A proof is valid if other mathematicians can follow it and agree that the proof is rigorous.  But that last bit has a flexible, and wide interpretation.  In some areas of maths proofs are sketchy and big jumps may be made without criticism.  In others, every step must be made with careful use of the axioms.  The best example I have met for this was in propositional calculus.  The lecturer didn't talk in English; he talked in 'logic'; every sentence being a part of the theory.    It was certainly rigorous, but it was very hard to follow.

I did a search through past posts looking for 'proof' and there are loads to look at.  And they tend to have a common feature: someone asks for something to be proved; some thoughts follow; these gradually develop into a first attempt proof; someone comments and the proof is improved to cover the new points made.  When no one can find any more objections, it becomes a proof.

Andrew Wiles spent 7 years developing his proof of Fermat's last theorem.  He presented it during a series of lectures, finally ending with 'And hence, ..., Fermat's Last Theorem' ... Applause!  smile   The proof was reviewed and found to have some missing details.  It was some months before he filled in all the steps and his proof was accepted.

http://en.wikipedia.org/wiki/Wiles'_pro … .931995.29

So what I'm trying to say is there is no generally acceptable way to prove something.  If it's for an exam course, you'll be given some examples to show what is expected; otherwise it just depends on your audience and how much time you've got.  My favourite is just an animation showing Pythagoras ... it's on the forum somewhere but I cannot find it at the moment.  No words but a beautiful proof all the same.  smile

EDIT:  Found it.  http://www.mathisfunforum.com/viewtopic.php?id=18393

Bob

Last edited by bob bundy (2014-05-01 20:25:26)


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2014-05-05 08:08:50

PatternMan
Member
Registered: 2014-03-08
Posts: 171

Re: How do write a proof?

Hmm..... Since there is no generally accepted way to prove something, what is considered a bad proof? Or what types of proofs are likely not to be accepted?


"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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#7 2014-05-05 08:14:47

ShivamS
Member
Registered: 2011-02-07
Posts: 3,465

Re: How do write a proof?

Mistakes, mathematical fallacy, logical fallacy, very illegible/not organized proof etc.

You shouldn't worry too much about it right now though - proof writing comes with practice and even in university there are no hard and fast rules on what constitutes a good or bad proof.

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#8 2014-05-05 10:29:17

PatternMan
Member
Registered: 2014-03-08
Posts: 171

Re: How do write a proof?

Okay I wont worry right now but I want to start writing proofs in a year or so.


"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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#9 2014-05-05 12:28:26

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,365
Website

Re: How do write a proof?

PatternMan wrote:

Hmm..... Since there is no generally accepted way to prove something, what is considered a bad proof? Or what types of proofs are likely not to be accepted?

Anything that contains logical mistakes, lots of assumptions, etc.

Do not worry about proofs right now, you'll pick them up as you go through your in text proofs


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#10 2014-05-05 12:55:05

ShivamS
Member
Registered: 2011-02-07
Posts: 3,465

Re: How do write a proof?

Actually, the books he has ordered have a considerable amount of exercises which require proof. Of course, they need not be sophisticated as long as they are neat and do not contain any fallacy. In university, they may set some standards of course.

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