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if s (smallest number) is 1-0.9p, then an infinite number of p, or inf*p, = 1 since basically, the smallest possible number is 1 divided an infinite amount of times...

There can be no smallest real number. Just take s/2 and you have yourself a smaller one. Nor are there any infinitely small or large real numbers. This is known as the archimedean property.

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I used the fact that any numbers * infinity = infinity, which is commonly acknowledged (although exactly as true as 0.999~ = 1).

Infinity isn't a real number, so the operation x * infinity is not defined.

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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.

An infinite number of decimal places does not mean there's a decimal place "at infinity". The set of natural numbers is infinite. This is because, if you take any finite set of natural numbers, take the maximum element from that set, then add one to it, you get a new natural number. There are no infinities here, however. Every natural number is constructed by starting with 0 and adding 1 a finite number of times.

A decimal expansion is defined by assigning a digit to every natural number. i.e. the natural number 1 corresponds to the first decimal place. The nth natural number corresponds to the nth decimal place. Thus 0.99... is defined by assigning 9 to every natural number. But 0.00...1 is just bad notation, because there's no infinity in the set of naturals, so there's no place to assign the 1.

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just an application of Zeno's all over again. at no point is a FULL distance covered. only half of the remaining. it just gets to the point to where the difference is so irrelevant in terms of scale that the __approximation__ is enough.

Zeno was an ancient philosopher who attempted to support Parmenides' claim that the existence of motion is contradictory. If you accept his argument, then it follows that motion is impossible. As it turns out, it's quite easy to resolve this particular argument because he implicitly assumes that if you sum an infinite number of finite numbers, you'll end up with something infinite. This is false. Calculus proves it's false. Though it's not hard to see it intuitively: It should be clear that the sum

0.1 + 0.01 + 0.001 +... does not sum to infinity. In fact, it sums to exactly 0.11... = 1/9.

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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.

What is the difference between a number and its value?

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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

What you're describing doesn't apply to sets, it applies to fields. A field is a set together with two binary operations satisfying some strong properties. The set of real numbers is a field. In fact, it's the unique complete ordered field. However, if you construct a new set by taking all the real numbers, adding infinity and negative infinity as the supremum and infimum of the old real numbers, you lose a lot of structure. This set is known as the extended reals, and it isn't a field. Though as a sidenote, 1 = 0.99... still holds in this set.