Fun with infinity

What's the line of logic behind that? Also, just so I have clarity, what's the difference between natural and real numbers? Natural numbers are integers, whereas real numbers are any non-imaginative number, including decimals? (bleh I'm BSing, I don't really know the difference)

And Devil's Sunrise, what I mean by different degrees of infinity can be demonstrated by limit problems.

So say you have f(x) = x^2, and g(x) = x^3. Say we try to measure both f(x) and g(x) as x goes to infinity. They would both be infinite right? Say we define h(x) = f(x) / g(x). We try to measure h(x) as x goes to infinity. Does h(x) go to infinity, or does it go to zero? Since f(x) / g(x) = x^(-1), we can establish that we can pick apart different types of infinity.

infinity - infinity = undefined, since infinity "isn't an exact finite quantity".

Code:

Assume ∞ - ∞ = 0
∞ - ∞ + 1 = 1
Since ∞ + 1 = ∞,
∞ - ∞ = 1
1 = 0

Whoops!

The line of logic behind there being just as many even numbers as natural numbers:

We make a mapping of the natural numbers to the even numbers

1 -> 2
2 -> 4
3 -> 6
and so on.

Every natural number will have a corresponding even number associated with it. And since the natural numbers are countably infinite (<-its part of the definition for the set of natural numbers), then we can say that the natural numbers and even numbers have the same degree of infinity as each other.

And no, the two "types" of infinity are:

1) Countably infinite: rational numbers, natural numbers, integers
2) Uncountably infinite: Real numbers

Devil's Sunrise, the reason there are an infinite amount of irrational numbers is because they're what's needed to create the distinction between rational and real numbers.

Rational numbers (countable) +irrational numbers (uncountable) = real numbers (uncountable)