Math Methods: Proving Equations & Formulas

[Astronomer107]

Mathematical Proof?

Can anyone tell me how to prove things mathematically? I'm not sure you can because I'm convinced that proving equations and formulas is something that you just see how to do if you are intelligent (if that is the case, then I must be stupid). I have trouble with this in my Math Methods class. If you can help, thank you!

 

[selfAdjoint]

It's an art that can be taught, and is in graduate math departments. Largely it's taught by hands on proving.

Some are better at it than others. this is partly but not at all entirely based on intelligence. There's the usual force quality (passive intellect gets you nowhere) and a not very well understood quality called mathematical taste, which goes to the selection of problems to study. Leading people like Witten and Smale have enormous mathematical taste - seemingly everything they touch turns to gold, which suggests they have a talent for finding gold-turning topics in the big universe of all topics.

 

[PrudensOptimus]

Originally posted by Astronomer107
Can anyone tell me how to prove things mathematically? I'm not sure you can because I'm convinced that proving equations and formulas is something that you just see how to do if you are intelligent (if that is the case, then I must be stupid). I have trouble with this in my Math Methods class. If you can help, thank you!

Claim: Astronomer107 = Noob.

Proof: Let o = 0,

Astronomer107 = N00b = 0.

0 = 0.

QED.

 

[Lonewolf]

How constructive...I'm sure he learned a lot from that, Prudens.

 

[mathwonk]

the first step in learning to prove things mathematically is to discipline yourself to learn the definitions of the terms. proof means deriving the conclusion from the definition of the concept, or from another previously proved result.

The primary problem in doing a proof is not knowing the meaning of the words in the statement, i.e. the definitions.

Thus when asked to prove that every dweeb is a doofus, ask first for the precise definition of a dweeb. then ask for the definition of a doofus, then think about how to connect the two.

(By the way every mathematician may be a dweeb but not all mathematicians are doofuses. Thus in fact it is false that every dweeb is a doofus, assuming there exists at least one mathematician.)

If this makes sense to you you are on your way to proving theorems.

 

[Jin314159]

Read the math proofs in your textbook. Learning by example is the best method.

 

[homology]

There are a ton of good books usually called something like: Introduction to abstract mathematics, or Writing Proofs or something of the like. They all introduce sentential and predicate logic in order to demonstrate the logical structure of theorems and their proofs.

This is important since there isn't just one way to prove something, you could use a direct proof, a contrapositive proof (my favorite), proof by contradiction, or proof by induction.

Its also good to be come aware of the quantifers in a theorem you want to prove. Then you'll be able to screw around with the theorem (especially if proving it by contrapositive or contradiction) without bunging it up. You will have noticed by now that Analysis for example is loaded with quantifiers.

Kevin

 

[Zurtex]

Claim: Astronomer107 = Noob.

Proof: Let o = 0,

Astronomer107 = N00b = 0.

0 = 0.

QED.

Claim: 1a = 2a for all a.

Proof, let a = 0

1*0 = 0, 2*0 = 0

0=0

QED

Unfortunately multiplying 2 functions by 0 proves nothing. (Oh dear I'm starting to pun like my friend)

 

[mathwonk]

My favorite intro to proofs book in high school was Principles of Mathematics, by Allendoerfer and Oakley, of which used copies may exist. An excellent recent book also written for high school, but that I find appropriate for college intro to proof courses is Geometry, by Harold Jacobs.

They have light hearted stuff on proofs and logic from Lewis Carroll that kids enjoy. In thatm spirit I gave my class the sentence: "For every man there is a wopman who can love him" to negate. One answered with "there are some men no woman can love, and you got that right!"

My experience with the many recent books on proof writing for college, is pretty discouraging. I do not like most of them I have seen. If a few that people have used successfully were mentioned by name I would benefit.

 

[homology]

My experience with the many recent books on proof writing for college, is pretty discouraging. I do not like most of them I have seen. If a few that people have used successfully were mentioned by name I would benefit.

Well I used Proofs and Fundamentals by Ethan D. Bloch. I used this text at the same time that I took a course in logic out of Logic and Philosophy by Paul Tidman and Howard Kahane. I liked the combination quite a bit.

The tricky thing about proofs however is that a mentor is really necessary. A professor can catch little errors, help a student clean up their proofs, recommend better methods and a lot of other little stuff.

Kevin

 

[mathwonk]

thanks for the suggestions Kevin. I made a note of them.

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