first of all your post assumes there is a "smallest number" this isn't true in the reals. Secondly you don't understand what .99~ means when we write it. You picture it as a book someone writes 9's in forever. This is completely false it's a well known and defined series, which equals one. It has nothing to do with being close to anything. Lastly you are just making up how operations work, which is fine, but its a non-standard non-consistent algebra that you created so you might as well argue 1 = hotdog cause it has the same amount of meaning.

As a mathematics-inclined individual, when I reach level 30, I'd like to formally ask WhattayaBrian to participate in my .999...th ranked game.

I don't see how I invented rules for my algebra...

I used the fact that any numbers * infinity = infinity, which is commonly acknowledged (although exactly as true as 0.999~ = 1).
I also used the fact that since infinite exists, then 1/infinite must be equal to something. While calculus tells us that 1/infinite = 0, calculus is only used to calculate an approximate result that will be closer to a tangible result than is actually useful (AKA more precise than it makes a difference).

The only place where my maths (Real set math) and your maths (Integer set math) conflicts is that in my maths, there is an infinite number of decimals, so 0.999~*10-0.999~ = 9.000~1, since well, at the infinite + 1 decimal, there will be a difference, and if there isn't an infinite + 1 decimal, then you'd have to round it, since you couldn't express the real value in infinite decimals.

Honeslt,y the only reason 0.999~ = 1 is because we think of every single numbers as integer numbers. One could say that it's untrue, but what is Pi? What is e (euler's constant) let's say that we have a number that is .5 to the power of infinite... what is its value?

If you can TRULY give me the value of any of those, and not only give me an integer approximation, then, and only then, will I accept that 0.999~ = 1.

EDIT : To make a parallel, back when cavemen were doing maths, they had 1+1=2, and 2/2 = 1. Then appeared the 5, where 5/2 = 2+3... which isn't mathematics at all, because they weren't used to it. So, 0.333~*3 equals 0.999~ and not 1, no matter how used to it you are...

Pi is pi. Pi's transcendentality assures the lack of a repeating decimal expansion for pi. e is e. e's transcendentality assures the lack of a repeating decimal expansion for e. (1/2)^infinity is zero.

Done. Now accept it.

So basically, if you do an infinite root of 0, you get every single decimal numbers? I'm not quite sure I can buy this as more than an integer approximation of .5^infinite...
If you prefer to do it another way, find me an N number that is the result of an infinite number of square roots of .5, or basically the infiniteth root of .5

What? An infinite root? You mean like (1/2)^(1/infinity)? Or (1/2)^0? Or 1?

Yeah. The "infinitieth root" of any number (except for zero of course) is 1.

I meant infiniteth as (1/2)^(1/infinity), I simply didn't find a way at that moment to really put it into words, and I didn't think of using the fractional exponent...

The problem is that if you say that the "infiniteth root" of any numbers is 1, that means that while 1 to any powers is 1, 1^infinity = every signle numbers... that's why I say it's an "integer approximation".

1 is actually a number that is closer to the real value than could actually be useful... if our goal was anything but the truth...

---- EDIT ----
Now, this course is long past, and I can't remember the exact terms, but bear with me if I screwed over any of those terms, I'm talking in term of concepts, not vocabulary.

Basically, every sets has 2 operators, * and +. And for every sets, you have to define their operators behaviors.
For starters, you will always have a null value. An N, where X+N=N, and XN=N. For regular numbers, these null values are 0 (for +) and 1 (for *).
Next, you have symmetrical values, where X+S=N=0, and XS=N=1. These are -X, and 1/X.
There are also a lot of other rules that defines the sets (actually, not a lot, more like 3-5, but they're overly complex and basically all includes more than 1 meaning), but it can be said that all operations a succession of those 2. So basically, a Power is simply multiple multiplications.
Also, all operations has its "opposite", so if X*Y=R, then R/Y=R*(1/Y)=X, because of the symmetrical values.
From there, you can safely say that if X^Y=R, then R^(1/Y)=X

So, if X^(infinite) = 0, then 0^(1/infinite) = X, no matter X's value... or basically, X = {Real Numbers}
Or, if X^(1/infinite) = 1, then 1^(infinite) = X, no matter X's value... same conclusion

I'm really not sure what you're trying to say here. It sounds like you're trying to mix mysticism and ideas of "reality" into my mathematics, which quite frankly needs neither mysticism nor reality.

EDIT: Wrote the wrong form of you're. I'm just going to go shoot myself in the head now.

You're assuming the operation is invertable, and in this case it is not.

N * 0 = 0

You've lost N. You cannot "get back" to it through division. It is not invertable. 0/0 is invalid partly because of this. It has no strict value because it could be anything.

N ^ 0 = 1

You've lost N. You cannot "get back" to it through logarithms. It is not invertable. Log-base-0 is invalid because of this. It has no strict value because it could be anything.

These are also known as "projections", where you generally lose information, and sometimes entire dimensions. They are not invertable.

Edited in the basic foundations of Sets, which explains why every numbers would be equal to 1^(infinite). Also, mathematics is pretty much a part of mysticism (explaining the intangible) and reality (proving the tangible).

Valid point tho... I'll have to think that one through...

EDIT : Thought it through, and I think projection is a pretty good name for those... since there is basically a value for projections that aren't only operations used with their null value (those projections being basically operations with infinities), and we're coming back to the initial problem... there is no way to represent the infinity...

P.s. Screw you Bryan (and all other forumers) for stealing my sleep...