Math Methods: Proving Equations & Formulas

In summary: L.H. Mathieu. It was a bit too formal for my taste, but it may have been more appropriate for someone with a stronger mathematical background. Proofs and Logic, by J.W. Gibbs and P.C. Wason. This is a classic and is still in use today. A first book that I would recommend for someone with an introduction to mathematics is The Geometry of Euclid, by David E. Smith. In summary, the problem with proving equations and formulas is that you need to understand the definitions of the terms. This is usually done by reading the math proofs in your textbook. Once you have a basic understanding of the terms, you can start
  • #1
Astronomer107
31
0
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!
 
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  • #2
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.
 
  • #3


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.
 
  • #4
How constructive...I'm sure he learned a lot from that, Prudens.
 
  • #5
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.
 
  • #6
Read the math proofs in your textbook. Learning by example is the best method.
 
  • #7
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
 
  • #8
PrudensOptimus said:
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)
 
  • #9
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.
 
  • #10
mathwonk said:
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
 
  • #11
thanks for the suggestions Kevin. I made a note of them.

roy
 

1. What are the different methods used to prove equations and formulas in math?

There are several methods used to prove equations and formulas in math, including direct proof, proof by contradiction, proof by induction, and proof by contrapositive. Each method involves a different approach to showing that an equation or formula is true.

2. Why is it important to prove equations and formulas in math?

Proving equations and formulas in math is important because it provides a logical and rigorous way to verify the truth of a statement. It also allows us to generalize patterns and relationships, making it easier to solve more complex problems.

3. How do you know which method to use when proving an equation or formula?

The method used to prove an equation or formula depends on the statement being proved and the structure of the problem. It is important to carefully analyze the problem and choose the most appropriate method for the given situation.

4. Can equations and formulas be proven wrong?

Yes, equations and formulas can be proven wrong. If a counterexample is found, it means that the statement is not always true and the equation or formula needs to be revised or reevaluated. However, a single counterexample does not necessarily disprove a statement, so it is important to thoroughly test and analyze before concluding that an equation or formula is incorrect.

5. Is it possible to prove an equation or formula using more than one method?

Yes, it is possible to use more than one method to prove an equation or formula. Different methods may provide different insights and strengthen the overall argument for the statement being proved.

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