Is this a Valid Method of Proving the Pythagorean Theorem?

[JG89]

I haven't taken any classes on proofs before, so I am wondering if the following is a correct method to prove something.

If I'm given the Pythagorean theorem, a^2 + b^2 = c^2, and asked to prove it, then I say there is a triangle, with hypotenuse, c, the horizontal leg of the triangle, a, and the vertical leg of the triangle, b.

Then CosX = a/c and SinX = b/c,

Therefore a = cCosX, b = cSinX and c = a/CosX

I can then substitute those into the theorem a^2 + b^2 = c^2 and get:

a^2/(CosX)^2 = (c^2)(CosX)^2 + (c^2)(SinX)^2

Simplifying:

a^2/(CosX)^2 = (c^2)(1 - (SinX)^2) + (c^2)(SinX)^2

a^2/(CosX)^2 = c^2 - (c^2)(SinX)^2 + (c^2)(SinX)^2

a^2/(CosX)^2 = c^2

Taking the square root of both sides:

a/CosX = c

Which is what I said c was equal to in the beginning. Basically, I assumed the premise, and since I arrived at a truth, I assumed the theorem is true.

I am pretty sure that the Cosine and Sine ratios are proved using Pythagoren theorem, so that automatically invalidates my proof, but is the method (assuming the premise then arriving at another basic truth) valid?

 

[uman]

How did you do this?

a^2/(CosX)^2 = c^2 - (c^2)(SinX)^2 + (c^2)(SinX)^2

a^2/(CosX)^2 = c^2

There are probably other methods to prove that $$sin^2(\theta)+cos^2(\theta) = 1$$, but the only way I've ever seen uses the Pythagorean theorem. So no, your proof is probably not valid depending on how you (or your book) prove this.

But as to your other question, yes, as long as each step is justified (in your case one is not), this is a valid method of proof.

 

[matt grime]

If I'm given the Pythagorean theorem, a^2 + b^2 = c^2, and asked to prove it, then I say there is a triangle, with hypotenuse, c, the horizontal leg of the triangle, a, and the vertical leg of the triangle, b.

Fine so far

Then CosX = a/c and SinX = b/c,

Therefore a = cCosX, b = cSinX and c = a/CosX

Yep, that's still fine (as long as you've not used pythagoras theorem to prove

cos^2 x+sin^2 x=1

that is.

I can then substitute those into the theorem a^2 + b^2 = c^2 and get:

This is _not_ fine. You cannot assume the premise, nor do you need to.

Basically, I assumed the premise, and since I arrived at a truth, I assumed the theorem is true.

I am pretty sure that the Cosine and Sine ratios are proved using Pythagoren theorem, so that automatically invalidates my proof, but is the method (assuming the premise then arriving at another basic truth) valid?

No. To answer your final question.

SImple example? Well, using your idea I can prove 1=-1. Assume it, square both sides, and I get a basic truth of 1=1.

Ignoring the validity of chosing sine and cosine (the identity for them can be proven in other ways), then just note that substituting sin X = b/c and cos X= a/c into

sin^2 X+cos^2 X = 1

proves the theorem.

 

[matt grime]

But as to your other question, yes, as long as each step is justified (in your case one is not), this is a valid method of proof.

No it isn't. See above for an elementary counter example.

In propositional logic (X=>Y) is true whenever X is false.

 

[Gib Z]

The funny thing here is that you can very easily make this a valid proof, just by writing your lines backwards =] Of course, as already stated, you must have an alternative proof to sin^2 x + cos^2 x = 1, but that's all.

 

[JG89]

The funny thing here is that you can very easily make this a valid proof, just by writing your lines backwards =] Of course, as already stated, you must have an alternative proof to sin^2 x + cos^2 x = 1, but that's all.

I suppose that even though the method I used is incorrect, it may be useful to help me see the actual proof.

Thanks a lot for the replies everyone.EDIT: When writing a proof, how far do you have to go? What I mean by this is, I initially wanted to (algebraically) prove that the hypotenuse is always the longest side of a right angle triangle. Would it be sufficient to say that it HAS to be the longest side right-angle triangle because if it were shorter than or equal in length to either of the legs, then it would be impossible for form a triangle?

 

[uman]

Oh yeah, wow, sorry, I didn't read that correctly. No, it wasn't valid.

/me feels stupid

 

[uman]

As an example of why it is incorrect, let's consider a similar "proof" that $$1=2$$.

Assume $$1=2$$. Then $$0*1 = 0*2$$ and so $$0=0$$, which is true. Hence $$1=2$$.

Or, assume pigs can fly. Since New York is always north of Phoenix, it follows that New York is north of Phoenix. Hence pigs can fly.

 

[matt grime]

EDIT: When writing a proof, how far do you have to go?

That entirely depends on what your purpose of writing the proof is. Is it to convince yourself? An examiner? Someone who can be presumed to have at least as much mathematical sophistication as yourself or less?

Essentially, though, a proof should consist of a series of logically related (and correct) deductions, inferences, etc, where any omitted steps will be automatically filled in by the reader without any problems. Exactly what you can assume of your reader depends on the situation as I intimated above.

 

[CRGreathouse]

EDIT: When writing a proof, how far do you have to go?

That entirely depends on what your purpose of writing the proof is. Is it to convince yourself? An examiner? Someone who can be presumed to have at least as much mathematical sophistication as yourself or less?

Essentially, though, a proof should consist of a series of logically related (and correct) deductions, inferences, etc, where any omitted steps will be automatically filled in by the reader without any problems. Exactly what you can assume of your reader depends on the situation as I intimated above.

That's a really good response.

 

[JG89]

Matt Grimes, I've seen from your posts around here that you're an excellent mathematician, so I hope you can help me out here.

I am currently a high-school student and want to major in pure mathematics. I will be starting university in September of 2009, not 2008. The following three courses must be taken in my first year:

Analysis I:

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor’s theorem; sequences and series; uniform convergence and power series.


Algebra I:

A theoretical approach to: vector spaces over arbitrary fields including C,Zp. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.


Algebra II:

A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra. Examples of symmetry groups and linear groups, stochastic matrices, matrix functions.


I'm assuming that Analysis I is Real Analysis? And that the two algebras make up Linear Algebra.

Now, for the past few months I've been studying stuff from Calculus and I'm currently learning improper integrals and I'm making great progress. One of my questions is this:

I was planning on teaching myself calculus until I've learned the taylor series, power series, etc because I see that those are needed in Real Analysis. However, will I learn these in Real Analysis anyway? So teaching myself it in Calculus first then teaching myself Real Analysis would be kind of redundant? Basically what I'm asking is, should I just go ahead and start learning the Real Analysis, or should I complete the Calculus first?

Another question I have is do you think it would be wise to teach myself the Linear Algebra while doing the Real Analysis as well or should I just focus on one subject?

My very last question is since I am going to be majoring in Pure Mathematics and for sure there will be a lot of proofs, should I borrow any books from the library on proof theory?

I don't want to bite off more than I can chew, but I am a fast learner and a bright kid, so I want to be as productive as possible in my studies.

All help would be appreciated, thanks.

 

[LukeD]

Analysis I:

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor’s theorem; sequences and series; uniform convergence and power series.
---
I'm assuming that Analysis I is Real Analysis? And that the two algebras make up Linear Algebra.

The two algebra courses do sound like they'd cover most of what you'll need to know about Linear Algebra.
The Analysis course however does not sound like it covers what is typically included in a Real Analysis course. A lot of colleges have a kind of introduction to semi-rigorous calculus where you'll learn about analysis as it pertains to the real numbers alone rather than how many of the concepts such as limits, open sets, etc. generalize to metric spaces and where you do not learn too many things about classification of Riemann integrable functions (to give some examples of how the courses would differ). It sounds like this is one of those classes. You will likely take another course that is a more formal and more general introduction to Real Analysis later.

 

[JG89]

The two algebra courses do sound like they'd cover most of what you'll need to know about Linear Algebra.
The Analysis course however does not sound like it covers what is typically included in a Real Analysis course. A lot of colleges have a kind of introduction to semi-rigorous calculus where you'll learn about analysis as it pertains to the real numbers alone rather than how many of the concepts such as limits, open sets, etc. generalize to metric spaces and where you do not learn too many things about classification of Riemann integrable functions (to give some examples of how the courses would differ). It sounds like this is one of those classes. You will likely take another course that is a more formal and more general introduction to Real Analysis later.

About the Analysis, should I still study Real Analysis or should I look for a book at my library on just "Analysis"?

 

[dx]

About the Analysis, should I still study Real Analysis or should I look for a book at my library on just "Analysis"?

Don't try to learn analysis before you know calculus well.

 

[matticus]

anyways i was thinking about how you would prove pythagorean theorem. if you use the complex plane, you could start at the origin and have a line going to the point a, a line going to the point bi, and then a line connecting them a - bi. and then you would have to show that a^2 + (bi)^2 = (a - bi)^2. It doesn't though. however, their real components are equal. any thoughts?

 

[LAVRANOS]

Calculus is based on analysis .analusis on a soft base can help you with calculus,althogh you can learn calculus on its own by simply learnig the formulas of integration and differentiation.Now hard core analysis deals with completeness,compactness,connectedness,continuity,closure etc etc ,whether is on vector,metric,topological etc etc spaces.You nmust realize that among all the courses that you will take in mathematics the most difficult and i dare say the most difficult of all the subjects in the campus ,is hard core analysis.We will talk latter about hard core analysis

 

[LAVRANOS]

This type of proof leaves no room for imagining things in mathematics or arguing for ever and ever or a teacher in every level to pass wrong proofs to the students because he likes to.For people involved in hard core analysis where a lot of quantification is involved that machinery can solve most of your problems. Of course it takes a little pactise but it is worth while

 

[LAVRANOS]

What Happened To Pages 3 And 4 They Vanish Into The Thin Air??

 

[matt grime]

They were moved to a separate thread - your posts had gained a momentum of their own. All your posts are still there, and you can continue your promulgation of whatever your theory is there.

 

[LAVRANOS]

PROMULGATION ? what's that

 

[Diffy]

promulgate |ˈpräməlˌgāt; prōˈməl-|
verb [ trans. ]
promote or make widely known (an idea or cause) : these objectives have to be promulgated within the organization. See note at announce .
• put (a law or decree) into effect by official proclamation : in January 1852, the new constitution was promulgated.

 

[LAVRANOS]

Thanks i never promulgate i rather urge or exhort in a common effort.

 

[Doodle Bob]

anyways i was thinking about how you would prove pythagorean theorem.

It's pretty easy to prove. here are 78 proofs: http://www.cut-the-knot.org/pythagoras/index.shtml

 

[LAVRANOS]

Vivian Shaw Groza in her book A survey of mathematics writes and i quote :Today trere exist several hundred proofs of this theorem .Elisha S. Loomis,in a volume called THE PYTHAGOREAN PROPOSITION,gives 367 different proofs,all classified by types.