/sci/ - Science & Math

>>8009228
>so hence, 1 + 1 is not 2 but actually 1.9999....8
Might want to check that reasoning again there, OP. You said that 0.999... = 1, (implying infinite repetitions of the digit 9), not 0.9999999 = 1 (implying a finite number of 9s, therefore rendering the statement false)

>>8009228
Wrong,

.99999... = 1 - lim(x -> 0) x

also, how did you round down to 8 in the final digit if it is .999 repeating? absolutely retarded bait, didn't even try.

>>8009228
As usual, the controversy stems from non-unambiguous notation. If you assume [math]0.99999...[/math] to be the limit of the series it represents, then the equality is correct. Though, if you interpret [math]0.99999...[/math] as an arbitrary partial sum of the same series, then it's obviously (arbitrarily) smaller than [math]1[/math].

>>8009297
>>8009352

Using limits when 0.99999... is just a fraction.

>>8009354
>is just a fraction

looks more like a repeating decimal to me, how about you try to write it as a ratio of numbers, friend?

>>8009419
not him but 3/3
which is one of the many reasons why 0.999... = 1

>>8009419
0.9999... = 0.9999.../0.999999....

>>8009352
>Though, if you interpret 0.99999... as an arbitrary partial sum of the same series, then it's obviously (arbitrarily) smaller than 1
But that's retarded.

>>8009446
Oh really buddy?

>>8009454
This is also wrong,

.999.../.999... is 1.

However, I asked you to write .999 as a ratio of numbers which you didn't.

>>8009446
except 3/3 is 1, that's how multiplication works friend

>>8009454
>>8009446
Just for example, you can write 999,999/1,000,000 which equals .99999... however this means .999 does not equal 1, it equals 999999/1000000. to write .999... as a ratio of numbers it will never be equal to 1

>>8009446
That's a circular argument you're trying to make there. By saying that "0.9999..." is equivalent to "3/3" you are somehow preassuming that "0.9999..." is also equal to "1". As a matter of fact, the equality between "0.9999..." and "3/3" (or any other fraction) implies that it's equal to "1" in the first place.

What "0.9999..." represents to most people is the [math]\sum_{n=1}^\infty \frac{9}{10^n}[/math]. As >>8009352 said above, the quarrels that routinely arise from presenting the equation from OP's picture boil down to whether you consider "0.9999..." to represent the limit of the series (which is indeed equal to 1), or just a partial sum of that series for an arbitrary [math]n[/math] (and such a partial sum is always _smaller_ than 1 - by how much, it depends on your choice of [math]n[/math]).

If that's not enough, you can bring surreal/hyperreal numbers into play which allow the explicit consideration of infinitesimal quantities.

>>8009477
>or just a partial sum of that series for an arbitrary n
Why would it ever mean that? You're saying that 0.999... is either:
a) A partial sum for any particular value of n, so that the single symbol 0.999... represents 0.9, 0.9999, 0.9999999, and infinitely other numbers (which is monstrously stupid), or
b) That the decimal expansion terminates, and so you're not even talking about the same thing.

I think you're trying to use words you don't understand, and you sound like a moron. Next you'll say "3/3 is equal to 1 if you mean dividing 3 by 3, but not if you mean dividing 3 by 3 an arbitrary number of times."

>>8009491
I'm not that guy but you seem like the retard to me, what's your definition of .999... then? Also he never said that was the definition of it, he gave it as an example of where two people could disagree on this topic.

>>8009511
It means the same thing as any other decimal expansion. [math]a_0.a_1 a_2 a_3...[/math] is the number [math]\sum_{k=0}^{\infty} a_k 10^{-k}[/math]. And before someone comes and complains about it "just being a limit," the limit of a convergent sequence of real numbers is itself a real number, and in this case that number is 1.

There is a widely accepted and universally used definition of decimal (or any other base -- decimal expansions really have almost nothing to do with the numbers themselves) expansions of numbers. As such, there is no reason for two people to disagree. The whole point of mathematics is that these things do have rigorous definitions, and that you can't ignore them. The people who think you get to have an opinion on whether 0.999...=1 are the same people who argue quantum mechanics is wrong based off of an article in a magazine. Not to mention, as I said before, his alternate suggestions aren't even sensible.

>>8009491
Whenever people deny 0.99999... being equal to 1, they argue that the left side must be smaller than the right side. The only way for it to be true is to either
a) interpret "0.99999..." as not the limit, but a partial sum for an arbitrarily large n, or
b) allow the use of infinitesimals, and say the the left side is infinitesimally smaller than the right side.

(If we do consider infinitesimals, a) and b) as given above are actually equivalent anyway).

This is the only explanation to the fact that the expression is controversial in the first place, and it is the possible ambiguity of "0.99999..." that can attribute at all to such a controversy. If the notation "0.99999..." is to be considered representing a quantity smaller than 1, then it's either infinitesimally so (if we consider infinitesimals in the first place), and if not, then it MUST be just a partial sum. Otherwise it's just equal to 1, and there's no controversy in the first place.

If you believe there might be yet another reason for the "0.99999... = 1" expression raising controversy, then please give an example.

>you sound like a moron
Thank you so much for helping keep the argument civilized.

>>8009475
999,999/1,000,000 = 0.999999

Those are not repeating, you retarded moron.

This is a bait thread made by a highschool drop out, nothing to see here.

>>8009511
>Also he never said that was the definition of it, he gave it as an example of where two people could disagree on this topic.
Yes, that's pretty much what I intended to say, thank you for expressing it very concisely (I tried explaining it again in >>8009535, but that turned out even more long-winded).

>>8009543
>This is a bait thread made by a highschool drop out, nothing to see here.

Then how about stepping up your game and, for instance, visiting this thread: >>8009271

>>8009228
fucking autistic piece of shit :)
i'll bite though:
you are trying to define a number which is equal to 1.999.....8 . You're saying here:
A 1 followed by LITERALLY INFINITELY MANY nines, and then an 8. that's like saying God made the earth, because there is something """""" else""""" beyond the visible horizon.`

And what do we say to relligious fanatics?
>>>/x/

>>8009228
notational variant

>>8009228
>1.9999....8
w8 m8 it doesn't termin8

>>8009894
This meme never gets old. I love it.

>>8009228
Your error is writing 1.9999....8

It's an infinite string of 9s you are adding so there is no final 8. If you try to add up 1.999.... using the algorithm you were taught in second grade, the first step is find the last digit, which you can never do, you just keep reading the number forever and do not ever return.

>>8009228
Very Important Mathematician here: you solved it! Still can't believe it!

You'll get your Field Medal and Nooble Price right after writing that infinite 0.9... number down, followed by the insightful final 8.

Congrats!

[math] \displaystyle
1 = \frac {3}{3} = 3 \cdot \frac {1}{3} = 3 \cdot 0.333... = 0.999...
[/math]

>>8009228

No, that just means you don't understand what "repeating" means.

If you ever actually get to an 8, then it's not really repeating, but is rather an actual, finite number.

>>8009228
interesting. So what is 1.999...8 + 2.393309....6? I am very curious about your arithmetic

>>8010646
4.392309:::14

>>8009228
1+1=1.9999999....

What all these 0.99999... Cucks dont get is that. '0.9999....' with infinate 9s isnt by itself even a number . what it means is the limit as you keep adding 9s. An expression which is equals exactly 1.

>>8010607
0.33333... might be viewed as only an approximation of 1/3 (as it cannot be written exactly as a decimal fraction), thus 0.9999... would analogically be just an approximation of 1.

As it was said countless times before, any disagreement boils down to what is understood by a decimal fraction with an ellipsis at the end.

>>8010764
>1/3 isn't 0.333...

k