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Proving 0.999... Is Equal To 1
eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."
No, he just dismissed BAD mathematics.
.9 = 10(.1) which is false.
I'm sorry, but as much as anybody want's to multiply an infinity repeating fraction by a limited decimal, you can't. it would be the equivalent to adding an int to a float. as much as you try to claim the precision of your float, you can only perform basic operations up to your decimal limit.
thus: if a=(1/9)
10a can't happen: you need to multiply by a float. and saying 10.0 is a float, means that
This just goes to show that people don't really know what numbers are, at least when they are infinite decimal numbers.
As far as I'm concerned, infinite anything doesn't even exist. These are just the math rules we have invented because the answers suit our needs. I could easily invent a math system where 0.999... == pi, and as long as the other rules correlate it's 100% valid. We define the rules about infinite series because the results fit with our other rules. All of our math rules have to fit together or else it's not really a "system" of mathematics. Just like how we say that pi is the ratio of circumference to diameter of a circle. Circles don't even exist. Where is a circle? Can you show me a circle? Maybe in terms of our mathematics system you can define something called a "circle" using a point and a radius, but it only exists in the context of that mathematics system. A point and radius in another mathematics system could end up being a square shape. Everything we do with numbers is purely fiction, a set of rules we invented to describe certain things. And ultimately, everything comes down to an approximation when you try to change contexts from paper math to the real world. Have you ever experienced anything "infinite" that wasn't just on paper described in our set of math rules that we invented? You can create all the complicated infinite equations you want in that system, but it means nothing because you can't make it real. You can use it to approximate real things, and that may be useful to you, but none of it really exists. So it doesn't even matter if 0.999...==1 in our system because once you move into the context of the real world it's going to become an approximation anyway and all that infinite precision that you thought existed ceases to exist...
So 10 multiplied by (1/9) would be the same as (1/9) + (1/9) + ... + (1/9) ten times. Or are you saying that we can't add either?
No, you can't. 1/9 cannot be added to 1/9. you can't add 1 to infinity, you can't multiply infinity by two.
you can add an approximation of 1/9 to an approximation of 1/9, but like I already mentioned earlier: it's like people claiming to use the value of pi in math: you can't! you can only use a definable approximation of pi.
like any sane person, people know that pi != 3.14, just like pi != 3.141592
pi IS however roughly equal to 3.14, and slightly more equal to 3.141592 but never does it equal it.
the math that we know is based on theory. that theory comes from real principals: the idea of 1/9'th comes from the idea that we can separate one object into nine equal parts.
.999... != 1, but rather 0.333+0.333+0.334 = 1
and 0.333+0.333+0.333 = .999
now the trick that get's messy, is that we have yet to determine IF there is a fundamental particle. if there is: then there is no infinity. there's just a f'in HUGE number of decimal places, and absolute numbers. if there isin't (and particles just keep getting smaller and smaller) then the math we have fails to describe them.
if you take an object and divide it into nine "equal" parts, you and I both know that you have eight parts that may be about equal, and one that's slightly bigger/smaller, or we have nine equal parts that have equal numbers of (mass/volume/whatever) but they each have a finite amount of each. thus we can determine that
you can only do classical math if you know the precision of your calculation. that's a limit we have, and depending on how the universe works, it might be correct.