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## Does 0.9999(repeating) = 1?

Yes 705 56.9%
No 578 46.65%
Voters 1239 .

### Does 0.9999(repeating) = 1?

Worstcase

Senior Member

Quote:
Shyplix:
Dividing by infinity is as dumb as dividing by 0.

No it isn't. There are varying degrees of infinity, you know -- Infinity*1 and Infinity*2 are not the same. They are both Infinite yes, but the second one is a larger Infinite than the first. In fact, there are an infinite number of infinities.

Kaolla

Senior Member

Quote:
Imogen Poots:
So you are saying that because the process of deriving at the derivative involves limit and infinity, your speed is actually...not exactly 2?

i've said it before, what's with people taking the limit of an infinite function and assuming they have anything other than an approximation?

hi i'm going to LIMIT my function to 10 decimal places!

.999999999 = 1

wow this statement is FALSE

OrvilleRB

Member

Quote:
Kodoku:
"Not rational" just means it cannot be expressed as a quotient of integers. Ill-defined means it doesn't exist. The "number" 0.00...1 with an infinite number of zeroes inbetween the zeroes and the 1 is about as well defined as the integral of GIRAFFE, or the square root of your mom.

I don't think that really addresses whether it's sufficiently well defined or not, but I'm more curious to see if you have comment regarding the remainder of that post:

Quote:
OrvilleRB:

The proof people have offered relies upon a faulty premise: addition and subtraction of irrational numbers, after multiplying them, behave like rational numbers (i.e., 9.999~ - 0.999~ = 9)

That is not correct.

Thought Experiment 1:
Imagine two asymtotes, both approaching 1 on the y-axis from below. Their value approaches 1 as x -> infinity. They can be described as 0.9999~ as x approaches infinity.

However, the asymptotes can have different rates of approach. When you subtract one from the other, they have a non-zero value at all points.

Thought Experiment 2:
Now, imagine two asymptotes directly on top of one another. Multiply one by 10. You have shifted the value that it approaches as x-> infinity to 9.999~. Okay, so according to the proof others have offered, you subtract 9. That makes your value at infinity equal.

But keep in mind - when you multiplied by 10, you ALSO changed the rate at which you approach the asymptote. When you subtract one from the other, as in thought experiment 1, you have a non-zero value at all points. They have the same value "at" infinity, but at any given value x < infinity, the y values between the two are NOT the same. That is the faulty premise that the proof others are offering rely upon. They all assume x = infinity.

TL;DR: The proof people are offering relies on a faulty premise - when you multiply by 10, you change the rate at which 0.9999~ approaches infinity.

Kaolla

Senior Member

Quote:
Kodoku:
But the infinite sum 1/2 + 1/4 + 1/8 + ... DOES exist because the partial sums
1/2
1/2 + 1/4
1/2 + 1/4 + 1/8
...

converge to 1. No infinity involved.

hello mister limit

Kodoku

Senior Member

Quote:
i've said it before, what's with people taking the limit of an infinite function and assuming they have anything other than an approximation?

hi i'm going to LIMIT my function to 10 decimal places!

.999999999 = 1

wow this statement is FALSE
If you don't even know what a limit is and are unwilling to learn, then neither you nor the people you're talking to have anything to gain from an exchange of words with you.

Quote:
I don't think that really addresses whether it's sufficiently well defined or not, but I'm more curious to see if you have comment regarding the remainder of that post:
The one in your expansion is not actually in your expansion. Your notation does not make sense.

As for the remainder of your post, no assumption about whether or not 0.99.. is rational is made in the calculation. Multiplication by 10 works the same way whether your number is rational or irrational. Though as it turns out, 0.99... IS rational so the point is moot anyway.

Quote:
hello mister limit

hi?

Infinite summations, when they exist, are limits.

Senior Member

Quote:
Kaolla:
yes they are, that's why it's called Xeno's Paradox

A: It's Zeno.
C: They are based on illogical assumptions which, once undermined, allow logic to work just fine. If you have a JSTOR account I can give you refs.
D: You're still envisioning a repeating decimal as a process, which is silly.

Kaolla

Senior Member

Quote:
Kodoku:
Infinite summations, when they exist, are limits.

thank you for proving my point. limit = approximation

Kaolla

Senior Member

Quote:
They are based on illogical assumptions which, once undermined, allow logic to work just fine.

wrong, they are not illogical. the math is wrong

it's that simple

you make the assumption that infinity can never be reached, then you propose math to where you need to use an infinite function to calculate the result. then you sit back and scratch your head.

you need to pick one or the other, you can't have your infinity both ways. it either A. is infinite and can't be reached or B. you make the math work by making an approximation, not an equivalency

Senior Member

Quote:
Kaolla:
wrong, they are not illogical. the math is wrong

it's that simple

you make the assumption that infinity can never be reached, then you propose math to where you need to use an infinite function to calculate the result. then you sit back and scratch your head.

you need to pick one or the other, you can't have your infinity both ways. it either A. is infinite and can't be reached or B. you make the math work by making an approximation, not an equivalency

If that's what you think, fine. Zeno's paradoxes rely on a repeated process, however, so even if you take them as valid they are still not applicable. A repeating decimal is not a process. Stop trying to find the last nine.

Imogen Poots

Senior Member

Quote:
Kaolla:
sure you can. if you're going to allow the infinite sum of anything you're going to have to come to terms that it is a number

there is no half-baked pseudo math that allows you to say "infinite repeating decimal 9's" then allows you to turn around and say "infinity is not a number"

if .999 is a number then infinity is a number. there's no two ways about it

Infinity is not a real number (only part of the extended real number line); 0.999... infinitely repeating is a real number.

Every mathematics professor will tell you that you are wrong if you disagree with either one of the above. Similarly, products of imaginary numbers can be real numbers. Like imaginary numbers, infinity is an abstract useful in deriving real number consequences. Just because your real number results are derived from processes involving infinity and imaginary numbers doesn't make infinity and imaginary numbers real numbers.