Math topic Options

[Raijinili]

Math is not about figuring out specific numbers. Calculus, yes. Math, no.

For example, there's the math of logic. One of the theorems in math says that any theory which can say really good things (i.e. things that you know are true) about the natural numbers (1,2,3,...) has to be either wrong (IT PROVES EVERYTHING (even things which aren't true)) or it doesn't say everything.

Another weird thing is that you can't pick a random natural number (positive integer, i.e. 1,2,3,...) with an equal chance of getting any of them.

When you have a prime number (call it p), take any natural numbers (yeah, they're pretty important) and raise it to the p power, then divide by p and take the remainder, and you get that natural number.

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A real number is basically any number which can be written as a decimal number, positive or negative. It takes seven real numbers (no less) to define a cone in three-dimensional space. It takes four to define a sphere. Six to define a line.

It takes five real numbers to define an ellipse on a plane (flat two-dimensional space), while it takes only three to define a circle. It takes seven to define a cube, and nine to define a rectangular prism.

Since it takes three numbers to define a circle, you can associate each circle with a point in 3D space. The set of all points which would also be circles would look like a world in which the Earth is flat, infinite in all directions, and has an endless sky.

[Raijinili]

Paradoxes.

First, I have to explain a little bit about axioms. Axioms are things we assume to be true, because we can't prove them. We can only prove theorems using the axioms.

A consistent set of axioms is a set of axioms which doesn't prove anything that it also disproves, i.e. it doesn't prove two contradictory statements.

A theory is the collection of theorems that can be proved using a specific set of axioms.

Now that we're done with the definitions...

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Not all axioms look completely true. In fact, some axioms (i.e. assumptions) can be replaced with axioms that say completely different things and still come up with a consistent useful theory.

For example, many, many years ago, people believed in the existence of parallel lines (i.e. two lines don't meet each other) on two-dimensional space.

But we can make a perfectly good theory that says that there are no parallel lines (i.e. we can draw the lines on a ball, where a "line" is defined as a circle that goes completely around).

We can also talk about the possibility of three lines where two are parallel, but the third is parallel to the first and not parallel to the second (in the regular plane, we can't do that, since there we have that if two lines are parallel to the same line, they're parallel to each other). This kind of space is really weird, and I'd rather not talk about it. It would look sort of like paper which was wet on the outside and deformed more the further away it was from the center.

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Anyway, paradoxes. There's one axiom that's called the Axiom of Choice, which mathematicians used to think was provable but it turned out that it isn't. It's not important what it is (and it only deals with infinite things, so it's hard to imagine).

Assuming the axiom of choice led to a lot of nice results, such as the result that for any two sets, one set has more elements than the other, or they have the same number of elements. In other words, the size of one set is greater than, equal to, or less than the size of the other set.

Actually, that was the AoC paradox I wanted to talk about. I guess that's it.

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Here's another one:
I'm going to do a UBW today or tomorrow, but you won't know which one.
You're saying, "Well it can't be tomorrow, because then I'd know, so it must be today. But since I know that it's today, then you can't do a UBW today either! So you can't do a UBW today OR tomorrow."

I am going to prove you wrong.
This is the proof, though it's pretty lame as a UBW.
I have created over a thousand proofs.
My professors don't like them.
I leave too much out.
This probably won't have holes in it.
Yet this UBW will never be epic.
Well, here's to hoping, QUOD ERAT DEMONSTRANDUM.

This post has been edited by Raijinili: Nov 6 2008, 11:32 AM

[Walrus]

You are a very silly Rai.
You love Maths, and spew nonsense daily.
You have angered millions of Asaphs.
You never lose arguments.
You never start them.
You ignore insults to post topics.
You are always alone.
You should eat some cake, it's delicious.

[Raijinili]

Walrus is right.