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Reminds me of the paradoxes of Zeno [1], especially the paradox of Achilles and the tortoise. At least one can simply prove that 0.999... = 1 without much hard work. Maybe less controversial than the following: 1 + 2 + 3 + ... [somehow] = -1/12 {{Riemann's zeta(-1)?}} 1 + 2 + 4 + 8 + 16 + ... [somehow] = -1 As well as the weird prime product (Product of 1/(1-(p^-2)) for p prime) and the sum of x^-2 from x=1 to [ sigh…
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You can multiply 1/3 by 3 and not get 1.
I have a pizza. I divide it into three parts. You'd be asserting that if I eat the three parts I have not eaten the whole pizza. I'm unconvinced.
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> if you were doubtful that 0.999... = 1, then you should also be doubtful that 0.333.. = 1/3 I disagree. Any middle school student can calculate 1/3 to be 0.33333... using long division, but there's no immediately obvious way to go from 1 (or 1/1) to 0.9999...
> Any middle school student can calculate 1/3 to be 0.33333... using long division, ... ...the same way That Chuck Norris can count to infinity... twice!
I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0. I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.
What is the difference between them?
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. T…
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The crux of the matter is that you have to define what things like "0.3333..." mean in the first place. Any reasonable definition of it as a representation of a real number is going to lead to it being equal to 1/3. If you want to redefine it explicitly as not a real number, you can do that, and maybe even get to some amusing math that way, but you're no longer talking the same language as the rest of the world.
>"but you're no longer talking the same language as the rest of the world" yes, in the standard real numbers 1 = 0.999.., but people have dealt with numbers like "pi" and "sqrt(2)" before the standard real numbers were defined. Hence the question, if we define such a system such as 0.333... != 1/3. what are the consequences? by 0.3333... I mean a countably infinite sequence of 3s.
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> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…
Yes that proof depends upon the representation in text of rational numbers (a dot and a series of digits). Try it in hexadecimal - it becomes opaque nonsense. Without some mathematical basis for 0.9... X 10 being something, there's a dangerous dependency on the representation that makes many folks uneasy.
The rules are essentially the same.
How about the expression: 0.9999... And consider that if a < b then a != b.
Both are false. 0.99999... is not less than 1. It is the same as 1.
It is because the "limit" in
0.999... = lim[eps->0] 1-eps
UPDATE: So instead of interpreting the expression as:
(lim[eps->0] 1-eps) lim[eps->0] ((1-eps) I don't think that 0.999... = 1 is actually provable. I think this and all of calculus is actually axiomatic, which has the following axiom: Given ε = 1/∞ then: ε = 0 Am I wrong in thinking this way? It seems as though there's no way to actually truly prove that an infinite series converging towards zero actually hits zero (from a constructivist pov)