Hello fellow Missourians. My name is Mike and this will be my first post on these forums. I did not attend MSA with any of you (I am a sophomore this year) but I attend Francis Howell High School near Saint Charles, MO.

Now that I have said that, let me get to the point of the post.

What exactly is mathematical proof and when is it used? I have proven basic geometric properties in school and have used the principle of mathematical induction to show some simple things like the sum formula for consecutive integers (n(n+1)/2) work, but I have no experience beyond that.

Looking at the fact that these two methods of proof are in completely different fields (or at least it seems to me), could it be true that there is not one kind of proof, but a whole range of different methods for different branches? If this is the case, could you give some examples of proof methods or famous examples in certain branches?

Let's keep it at that so as to not overload all of you with questions.

Thanks,
Mike Nolan

Welcome mpn - although we do not usually have non-MSA attendees at the board, we're always looking for enthusiasm in learning. If you are going to try for MSA this summer, best of luck.

A mathematical proof is only a means to prove a mathematical conjecture. There is no set way to prove something. In other words, you are correct in thinking there is more than one kind of proof.
To prove something, you shouldn't use a certain trick of proving necessarily, but rather do whatever works. It doesn't have to be (and often isn't) common.

There are methods of proofs however. One very well know and popular one is reductio ad absurdum. In English, that means roughly "reduced to the absurd". For this method, you start by assuming what you are trying to prove false and then show that it can't be that way.

A favorite of mine that is a reductio is Euler's proof that sqrt(2) is irrational. Yeeha Perfect and Pathological Mathematics class of MSA '02.
Here it is-

Assume sqrt(2) is rational.
Using the definition of a rational number, we can write sqrt(2) as p/q where p and q are integers. Thus sqrt(2) = p/q
Squaring both sides gives 2 = p^2 / q^2
2 * q^2 = p^2
This is where the thought part comes in:
Assume that p has n factors, and q has m factors. (remember prime factoring?)
We know then that p^2 has 2n factors and q^2 has 2m factors.
Put this into the equation 2 * q^2 = p^2 and you can see the problem.
From that, you have 2n + 1 factors equalling 2m factors. An number with an add number of factors cannot equal a number with an even number of factors. It is absurd :)
Thus the square root of 2 is irrational.

This same proof works for any prime number.

Fun stuff - I'm about to go watch Equilibrium now. If you have any questions, ask away.

Edit: phooie on Verdana and Invision - I can't use the nifty square root and squared characters... :(

So you are basically breaking down everything you have been told into its most basic elements, assuming only the most intuitive things are true?

I really like that sqrt(2) is irrational proof. I didn't think it would be so simple to prove!

Do you have any other favorites?

Something fun -

From a book I am reading:

"There is a story that one of his [Euclid] pupils complaining that learning theorems was pointless--they were of no practical value. Euclid commanded a slave to give the boy a coin so that he could make a profit in studying geometry."

I'm sure there are better things from this book (it's a great book), but I thought I'd throw something random out for people to read.

I was going to type out a longwinded (and multisyllable) response to the original topic, but this page says it a lot better than I could. Check out it and what it links to.

"Proof by contradiction: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false. "

Is this the same as what you showed me with sqrt(2) = irrational proof, Polarris? Switching the true and false in the previous statement seems to bring the definition of the reductio ad absurdum.

Actually their Proof by contradiction is essentially the same thing as a reductio, no switching required.

Also, we're really not assuming anything. That proof (and all others) use definitions of math to show something. You can't assume anything unproven in math. If it's already been thoroughly proven, then you can use kind of like an assumption.

heh, Nice idea Bo ;)

Fun Stuff Reply:
Patrick, did you enjoy Equilibrium? I found the plot delicious (and very Brave New World-like) and the samurai sword to the face scene wonderful.

There is, of course, metaphysical/ philosophical debate as to what qualifies as assume-able. Proofs can be valid when their assumptions are taken as absolutes (unless you're going to contradict them later to prove the opposite). For example, Euclid's axioms for geometry work quite well, but only when flat space is postulated. For curved space, one of the assumptions isn't true, and you can have triangles with more/ less than 180 degrees. Therefore, it's possible to have a set of axioms from which you can derive all kinds of stuff consistently, but it doesn't necessarily describe reality.
(plug for Wikipedia) The Wikipedia articles link to some good information on the topic... check it out.

To elaborate on/explain Bo's article, are you familiar with triangles in 3D space?
We're not talking about tetrahedrons or other hedron like things, but triangles that are drawn on a sphere.
If you draw a triangle on a sphere you can put one vertext on the top of the sphere and the other two further down at any distance apart you want. Your triangle could have angles with a sum upwards of 180 degrees. Is it really a triangle? Yes. Why? Because it has three sides. It's just a triangle that doesn't fit in 2D space.
Basically, the definitions can describe an object in general but the rules might not always work.
That's a pretty far cry from the original topic :)


[OT] Aubrey - Yes, I love that movie. That scene was crazy though - kinda made me want to jump up and start a mosh pit :P
Yes the plot was genius. Good plot + cool action = Great movie
There's math for you.

Interesting. I had seen pictures of 3D triangles in a trig book I got from the library, but I had to return it before I could get into that chapter.

I've stayed away from 3D mainly because the 3d coordinate plane doesn't make sense to me.

Why not?
It's just the cartesian plane with a Z-axis stuck through the middle.

Well, trying to draw 3D "stuff" by hand in one of those planes is a lot harder than 2D! Looking at an image, It makes sense now though.

Drawing is one thing; visualizing is another entirely