Math

[trolls' bane]

Okay, I feel that we (or at any rate, I) am in dire need of a Math thread. I seem to remember there being one, but the closest I could find was a Math Problem thread, which far from encompasses the rather broad scope that discussion of math requires.

I think to start, I would like it if someone could attempt to teach me how to do a formal proof, as I have learned little in that regard from Geometry.

[Gwaimir Windgem]

Do you mean, like in Euclid?

[trolls' bane]

Yes. Euclidean geometry. But it's proofs I need help with. The logic is there, but how does that tell you what steps to make (remember, logic is a double-edged sword: Dr. Pangloss told Candide to let someone drown because they bay they were in must have logically been created for the man to drown in).

[Curubethion]

Easy. In the left column, do the equations, etc. In the right column, give your reasons for doing it. I'll try to do it here, although vBulletin doesn't have columns...

1. a=x ----------------1. Given
2. a=a-----------------2. Reflexive Property
3. 2a=a+x--------------3. Addition
4. 2a-2x=a-x-----------4. Subtraction
5. 2(a-x)=(a-x)---------5. Associative Property
6. 2=1------------------6. Division

I don't think I got all the names right, but you get the idea. And I challenge you to find out what's wrong with this proof!

Can you give me an example problem?

[trolls' bane]

Well, I can't, because I can't post triangles or things like that.

I will maybe later. I have to go to the star party soon.

[Acalewia]

I never was good at math. After basic Algebra, it went over my head. It just looked like numbers and letters with signs and intergers.

[Count Comfect]

Originally Posted by Curubethion

And I challenge you to find out what's wrong with this proof!

a=x, so a-x=0. But you divided by a-x, thus dividing by 0 (not allowed).

That said, I never learned how to do formal proofs (alas).

[trolls' bane]

Originally Posted by Count Comfect

a=x, so a-x=0. But you divided by a-x, thus dividing by 0 (not allowed).

That said, I never learned how to do formal proofs (alas).

Originally Posted by My Calculator

2/(a-x)=ACK! ARE YOU OUT OF YOUR BEFRAKLED MIND!?!?

....

[Gwaimir Windgem]

Originally Posted by Count Comfect

a=x, so a-x=0. But you divided by a-x, thus dividing by 0 (not allowed).

That said, I never learned how to do formal proofs (alas).

Dammit! I was gonna say that, but you beat me to it.

[Beren3000]

Originally Posted by troll's bane

I would like it if someone could attempt to teach me how to do a formal proof

I'm not exactly sure if you're only talking about geometrical proofs; but here's a method for algebraic proofs:

Proof by Induction:

You start with a hypothesis, say, for instance:
2n + 1 is an odd number for all integers n.
To prove this:
First check that it holds true for n = 1 ==> 2(1) + 1 = 3 (Odd)
Then, assuming it holds true for n = n, try proving it for n = n + 1
2 (n + 1) +1 = 2n + 2 +1, but from our assumption 2n + 1 is odd so (2n +1 ) + 2 must also be add ==> proof done.

So basically:
1-check your hypothesis for n=1
2- assume it holds true for n = n and then prove it for n = n + 1

Note that this method only proves the hypothesis to be true for all positive integral values of n

Btw, kudos on starting a math thread It's nice to find someone else with a passion for math!

[juntel]

See this as a game, starting out at the bottom using only pre-defined objects and rules that govern how these object interact and play, building up other objects and rules about them, and like this on and on as if climbing a ladder towards a more complicated 'world'.

Proofs are the process of climbing these steps from what we know to what new property we want to dig up. Even if the conclusion you want to get at seems easy to understand as if it didn't need to be proven, you nevertheless need to be sure it can be obtained from simpler statements, ie that there is indeed a ladder that goes from the basic premises up to the conclusion.

You can learn to understand and create proofs by studying them closely, remembering that each step must be accounted for by earlier results. Think about Sherlock Holmes observing details and building up a case against an enemy, making sure that each step is right. If one step is wrong, then all that follows is nonsense (eg like the 'proof' above that '2=1'!)

A good example to study is a proof of the Pythagorean theorem. In fact there are many of them, some easier than others, some more interesting than others; some turn quickly into algebraic manipulations, others are almost purely geometric all the way. [And I've seen at least one that uses calculus (more than two millenia after pytha's proof!), to the point of asking oneself if that proof contains a vicious circle!]

Anyway, I'm not gonna teach you. Take your time, think about the necessity of each step for the particular ladder the proof is making you climb. Memorizing is not as important as understanding for this.

It's not necessarily easy though, and the steps to take don't always come out of the sky: thought and sweat must be applied. But when you have to give a proof of some math statement, you can always begin by: "Ok, what do I know right now?", and from what you know proceed to what you want to prove. When you've done that, you must then make sure that "what you know" is right, and especially -- and this can be quite important -- doesn't depend on what you wanted to prove in the first place (you don't want to place yourself in a vicious circle).

So, how would you begin to prove Pythagoras's theorem? Or the statement that there is an infinity of prime numbers? Or that there are no rational numbers for which the square is 2? What simpler statements can you make that could lead to these? Can you say "If I could only prove this little detail, then the rest would follow"? Little by little, pieces of the puzzle fall into place. Or not.
As in writing abilities can be aided by reading around what others have written, reading proofs and understanding the motivations behind the steps can also help you, little by little, to acquire some ease at doing your own stuff. Or not.

Just remember to have fun.

[Curubethion]

That's a really cool way to explain it!

[Beren3000]

I'm interested in Chaos Theory. I've done some background reading on it, but mainly in "popular science" books so I have yet to see a single equation or formula relating to it. If anyone knows about it or can direct me to a website about it I'd be grateful.

[trolls' bane]

Doesn't that have something to do with fractals?

[mithrand1r]

Originally Posted by trolls' bane

Doesn't that have something to do with fractals?

I think Fractals are somewhat related to Chaos Theory.

There are many sites on Math.
http://www.google.com/search?client=...=Google+Search

I like personally like
The Drexel Math Forum.

[Jonathan]

Chaos theory is certainly a fascinating branch of mathematics. Chaos is when you can't predict the behaviour of e.g. a mathematical function. A chaotic function is very sensitive to its initial input in that small changes in the input will result in dramatic and unpredictable differences in the output - the famous butterfly effect.

Originally Posted by Beren3000

I have yet to see a single equation or formula relating to it. If anyone knows about it or can direct me to a website about it I'd be grateful.

There are many equations that have to do with chaos theory.
In ecology, there's a (simplified) equation to predict the size of a biological population:

next year's population = R * this year's population * (1 - this year's population)

where R is the growth rate and the population is a number between 0 and 1. The equation is repeated over and over (next year's population becomes this year's population).

For low values of R, the population will eventually even out at a certain number, it will reach a steady state. No matter how many times you repeat the equation, you'll get the same answer all the time.

For high values of R, the equation doesn't reach a steady state. You will get different numbers for every time you repeat the equation and it won't be possible to predict the behaviour of the equation.

Here's a website for you Beren. That's where I got the equation above.
http://www.imho.com/grae/chaos/chaos.html

[trolls' bane]

Thanks. I will have to look into this one myself.

[Millane]

Originally Posted by trolls' bane

Okay, I feel that we (or at any rate, I) am in dire need of a Math thread. I seem to remember there being one, but the closest I could find was a Math Problem thread, which far from encompasses the rather broad scope that discussion of math requires.

I think to start, I would like it if someone could attempt to teach me how to do a formal proof, as I have learned little in that regard from Geometry.

please tell me that you need this help for some specific schooling purpose, and not for "fun"... People that find maths "fun" creep the **** out of me
people keep telling me to take a maths subject next year so i will have something practical at the end of my degree. well **** practicality if it means doing more maths

[klatukatt]

Math is fun. *creepy smile*

How you doin?

[shesabrandybuck]

If I did geometry then I would have to break the triangle into two right triangles, and I dont have enough information to do that
*curses trig*