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## Can three infinities be greater than one?

The idea is very simple but the conclusion may redefine our very concept of infinity and quantity unless I am wrong.

There are infinite possible positions in one dimensional space. That is if we assume that a given position is infinitely small or that one dimensonal space stretches infinitely. Two dimensonal space offers more possibilities than one dimensonal space since two dimensional space has all possible positions of one dimensional space plus all possible positions that can occupy the second dimenson that was added. And we can keep adding dimensions thus creating more possible positions with each added infinity.

What seems to emerge is that as you add more dimensons you are adding more possible positions relative to the previous number of dimensons but you are not necessarily increasing the number of positions because the number will always be infinity. Thus you can add more to a quantity without increasing it.

Weird.

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## Nick West

Infinite can be different in rates of growth as well as densities. A line y = x grows at a rate of O(n) as it approaches infinity, but a curve y = x^2 grows at a rate of O(N^2) or exponentially faster as it approaches infinity. Rates of growth as you approach infinite (represented here by Big O Notation*) are important and they demonstrate how two different infinite quantities can behave differently. But, in the end, they both contain the same unending quantity of whatever.

Big O Notaton: http://en.wikipedia.org/wiki/Big_O_notation

## Krisztián Pintér 200+

## Nick West

My math knowledge is somewhat limited to what I've specialized in. As a programmer, I do work with growth rates as quantities approach infinite daily. I don't remember how to make my example clear as a mathematical representation, but obviously there's no one number that represents the distance between two random points. But there would be equations that demonstrate the probable difference when selecting from the two different pools of points.

## Krisztián Pintér 200+

## Nick West

This is making me wish I was back in college as a Math major. I forgot how much fun it can be.

## Budimir Zdravkovic 20+

I only had to take two years of calculus in college and that was also a long time ago so I apologize if my knowledge is rusty. In what I do I only need roughly 10% of the calculus I learned. .

## Nick West

A visual representation of a smaller scale could be thought of as a piece of paper with 64 equally spaced points in a line for a 1d space, then a piece of paper with a 8x8 square of points for the 2d space, then for the 3d space, you need a 4x4x4 cube of points. So when you measure the furthest straight line distance between the points in the different spaces, you get 64, 8 and 4 respectively. It's kind of like how the more you fold a piece of paper the "smaller" it gets.

But it was all wrong to show differences in infinite groups because as you approach infinity picking random points also creates a distance between those points that approaches infinity. I think it would actually prove the opposite, in that the different densities of infinite spaces would actually behave the same, in my example, as you approach infinity.

The order of growth stuff was correct though. It's helpful in programming so we can figure out if an algorithm is useful or not for very large inputs. If the order of growth is too great, when inputs are very large it would simply take too long, or too much memory, to use that algorithm.