This House Believes 0.99* Equals 1!

Following my opponent's declaration of intent (see http://www.debate.org... ), I hereby challenge him to deliver at least ONE logically, mathematically or philosophically acceptable argument for his outlandish claim that 0,99*=1.I say the two are not the same, and will prove so below. In order to win, my opponent has to disprove my argument and deliver one sound argument of his own, as disproving my proof is not enough to back up his claim. In this case, as it is apparently put in this site's jargon, burden of proof is shared.My case:The very idea of REAL numbers is that they are indeed continuous. That is why the rational numbers are superseded by the idea of real numbers by adding the irrational numbers (like pi or the square root of 2) which can not be described by a fraction.Between any two rational numbers there's an actual infinite count of irrational numbers continuously filling up the void between fractions.How do we achieve this continuity then?Obviously, there must be a concept of real neighboring numbers. This poses a dilemma.If two neighboring real numbers have actually no distance between them, they must be considered identical. If any two neighboring numbers are identical and we extend this to the infinity of all neighboring numbers, all numbers must be identical, which means there are no numbers at all. Only one number would remain.If there is a distance, how large is it?The answer is obvious: the difference is infinitesimally small/infinitely small.This is very hard to understand for many, and hence there's a misconception.Actually, there's no better example to illustrate this than the misconception of 0.99*=1.0.99* means that we have an infinite number of decimal places with a value of 9.Now, let's focus on the distance between 0.99* and 1.Obviously, the number we're looking for has an infinite amount of decimals with the value of 0, except for the last decimal, which must be 1.1-0.9=0.11-0.99=0.011-0.999=0.0011-0.9999999999=0.0000000001Now, people keep telling me that this number does not exist, because we never reach the 1, as there are INFINITE 0's in the way.But look more closely: the 1 is in the same decimal place as the last 9, always. This means there's INFINITE MINUS ONE 0's in the way only!Which means as long as you count the 9's on the one side, you have to accept the 1 at the end of the chain of 0's on the other side of the equation, or the whole problem does not make any sense to begin with. This turns the whole "pro" side of this debate into one giant straw man argument.1-0.9=0.01 is just plain wrong. Same goes in extension for any added decimal places. But that is not what people who see a difference between 0.99* and 1 have ever claimed in the first place! It's only what the pro side makes it out to be - wrongfully, one must add.What people who claim that 0.99*=1 do is to fool people with a double standard for infinity. They claim that "infinite" does away with differences and decimal places. Which is simply not how infinity works. Either we adhere to the same standards on both sides of the equation or we don't. If there is 0.99*, then so is the number that would have to be called 0.0*1 or 0.00...1.All claims that 0.99*=1 are based on the misconception that what cannot be told apart must be identical. But it is meanwhile common knowledge that indiscriminable is not identical.As renowned physicist Dr. John Denker so aptly puts it:"I am going to make an important distinction between the concepts of- identical, which means intrinsically identical, versus- indistinguishable.As a macroscopic illustration, consider two cars. I choose to call them intrinsically identical whenever they have the same make and model, same color, et cetera. Nevertheless you might be able to distinguish them on other grounds. For example you might be able to distinguish them on the basis of position: a car ’over here’ is distinguishable from a car ’over there’."http://www.av8n.com...The incomprehensibility of the concept of "infinity" adds to the confusion. "Suppose that you are presented with two black sticks lying half a meter in front of you on a white background, suppose moreover that their appearance is indiscriminable to you. But if you then go nearer to them and measure them with a ruler, you may discover that one of them is 20 cm long, while the other is 20.1 cm long. This would not be a surprise: we commonly accept that our perceptual abilities are not so fineR08;grained as to detect any slight difference in the physical bases which elicit our perceptions."http://www.academia.edu...Clearly, we have an extreme case of the same matter here. We have an infinite distance (to the last digit of an irrational number) and the smallest conceivable difference (between to neighbouring real numbers). The misconception does, as the above quote puts it, not come as a surprise. It's still plain wrong:Saying that a difference actually ceases to exist only because it happens beyond an infinite distance just doesn't make any sense.It is only natural that we don't CARE whether a hair on the head of a bank clerk in a remote town on the other side of the planet is white. That doesn't mean that the man does not have a white hair. Infinity does not mean that things don't happen, it only means it's so far away we can't MEASURE it. Two grains of sand on a Mars don't become IDENTICAL only because we can't reach them or tell them apart. They remain two distinct grains of sand. So, why would two numbers that differ in a single decimal place REALLY far away have to be considered identical? What far-fetched reason other than neglect could lead to such inane conclusions? Reality does not care a lot about our limited perception.So the answer is: 0.99* is the direct real neighbour to 1, both are indistinguishable in the set of rational numbers, because 0.99* is not actually a rational number, it is just taught at school as being identical in order to help undergraduate pupils fill the gap between rational and real numbers. Anyone above that level of maths should be able to understand that those two numbers cannot be identical due to the simple matter of decimal places, as shown above. What does this mean in application?We can still calculate day-to-day math problems with 0.99*=1. The difference is so small we can't measure it, after all.But it would be a critical failure to assume that this is "true" for all branches of mathematics.A probability of 1 means something is absolutely certain and in no way a matter of chance. Not a single exception to an event with a probability of 1 can be found, it MUST happen without a fault. Something with a probability of 0.99* however does know a SINGLE exception, ever, in all of eternity.In a way we may owe our existence to this fact with a certain probability, as the singularity some claim to be responsible for the creation of the big bang would have been this one-off chance, the one moment that made it all count.This is why this debate is considered a philosophical debate by me, not a scientific one.All in all, it just boils down to the question whether you accept that two things that are very much alike become the same thing. This may be a philosophy for some. It is, however, not logically sound.And in no way can a difference in decimal places be explained away by infinity.I leave this stage to my opponent, should he choose to accept.Thanks to all who read this.