/sci/ - Science & Math

[math]0.999...[/math] is the limit of summing [eqn]9*10^{-x}[/eqn] as [math]x[/math] goes to infinity.
[math]1.000...[/math] is the limit of summing [eqn]1+0*10^{-x}[/eqn] as [math]x[/math] goes to infinity.
Both of these limits evaluate to the same point on the number line, which is a number. However, [math]0.999...[/math], [math]1.000...[/math], [math]0.9[/math] and even [math]1[/math] are not numbers. The numeral system, regardless of base, is merely a human invention to help understand the numbers, but don't mistake some symbols like [math]0[/math] and [math]...[/math] for things that actually exist, because they're just models of what's really going on.
Additionally, I've seen the topic of surreal numbers and infinitesimals brought up in arguments like this. These are optional, in that the version(s) of math where they exist and the version(s) where they don't are both equally valid. Although I've said at the start of this post that both limits evaluate to the same number, keep in mind that I'm talking about limits, which are a different thing to numbers and have their own rules. You can have 0.999... and 1.000... represent different numbers if you use a version of math where infinitesimals exist, but even then both limits will evaluate to the same number, because limits are a different thing with complicated implications.
Wanna learn more? Go study real analysis, properly.

Not even aigarothmic disappears. Lmfao

>>11583578
reminder that Trump/gop saw it unfolding on the news for 3 months but did nothing
https://youtu.be/xfPk1HIBLfM
https://youtu.be/bkMwvmJLnc0
https://youtu.be/UHjJydZO99A

>>11583578
>TedEd
>Zeno
Not one second.

>>11583578
Thinking 0.9999...=1 is correlated with schizophrenia, and schizophrenia is correlated with low intelligence.

>>11583704
The LIMIT is 1, but the number NEVER reaches one. So 0.X is NEVER one. How many times do you schizo retards need to be told this?

>>11585019
This.

>>11585062
lol what a fucking idiot

>>11585019
>the number NEVER reaches
the number *is* the limit, once there are infinite additions, the result is exact

>>11585019
What do you mean, it never "reaches" 1?
If it is less than 1, what is a number between it and 1?

>>11585062
To elaborate further on the core of the problem in the minds of those who believe less than 1 is 1:

The plain truth is the following.

Zero point followed by an infinite amount of nines will never be the same as 1. It will /always/ be less than 1. No matter how many 9s are used; even when the amount of 9s are infinite.

The inability some people who view themselves as being "scientific" have in comprehending that ".999..." isn't the same as "1" - it's completely connected to their inability to grasp eternity and infinity. Their minds have basically been programmed to believe eternity and infinity are impossibilities. Over the course of many generations of indoctrination into an alien worldview, their parents and their parents etc gradually "learned" to be unable to comprehend it and this "learned disability" was inherited, and encouraged/fueled by various external factors from the echoing modern culture.

This difficulty they have with eternity/infinity shows up in many different fields, from math to astrophysics.

This mental handicap is inherited directly from the (((Abrahamic))) religions, more specifically (((Christianity))) for us Westernerns. In it's origin, the inability to understand infinity and eternity is 100% Judaic in thought/philosophy. In contrast, the non-Jewish man; the Pagan man, at least the /European/ Pagan man, never had any problem with infinity and eternity. (((Christianity))) introduced into the minds of people the idea of life and the world/universe being linear, starting from point A and ending at a point B, whereas in the Native European worldview everything is an infinite circle.

That's why many people today can't understand that .999 repeating forever will never reach 1 - they refuse to accept the idea of an infinite/eternal repetition. Saying "it's 1" is their method of escaping from the uncomfortable (and to them insurmountable) challenge which the concept of infinity/eternity is to their Judaically-induced mental disease.

>0.999... never reaches 1
>It means that if I take one step, I have to cross 0.999... of my step
>It means that I will never reach 1 step
>I'm stuck forever
Retard.

>>11585172
based retard

>>11585110
An amount that's less than "1" isn't the same as "1".
"0.95" is easy for everyone to understand that it's less than "1".
"0.99" is also easy for everyone to understand that it's less than "1".
"0.999..." (an infinite amount of 9s) is likewise less than "1", but difficulty for some people to grasp because they dislike the idea of infinity.

>what 1 minus 0 . an infinite amount of 9s?
Answer: 0.x1 where x is an infinite amount of 0s.

>>11585178
dumb frogposter
> infinite amount of 9s) is likewise less
no it isn't
finite isn't infinite, that's why you can do a bijection between just the even integers and N

>>11585178
>0.x1 where x is an infinite amount of 0s

0.1=10^-1
0.01=10^-2
0.001=10^-3
:
0.0...1=10^-inf=0

>>11583578
A real number defined by the cauchy sequence [math]x_0, x_1, ...[/math] is said to be equal to another real number defined by the cauchy sequence [math]y_0, y_1...[/math] iff for every positive real [math]\epsilon[/math] there exists a [math]N \in \mathbb{N}[/math] such that for every [math]m, n > N[/math],
[eqn]
|x_m - y_n| < \epsilon.
[/eqn]

Let 1 be defined by the cauchy sequence [math] 1, 1, 1, ...[/math]. Let 0.999... be defined by the cauchy sequence [math] 0.9, 0.99, 0.999, ...[/math].

Note that
[eqn]
0.\underbrace{999...9}_\text{$N$ times} = 9 \sum^N_{n=1} \left( \frac{1}{10}\right)^n = 1 - \frac{1}{10^n}.
[/eqn]
The first equality is trivial, the second equality follows from the sum of a finite geometric series.
Therefore, applying the definition of equality above, we find for the [math]m[/math]th term of the sequence defining 1, and the [math]m[/math]th term of the sequence defining 0.999... is
[eqn]
\frac{1}{10^n} < \epsilon
[/eqn]
Now for any positive real [math]\epsilon>0[/math], we can always pick the smallest natural number [math]N[/math] greater than or equal to [math] = - \left\lceil \frac{\log(\epsilon/2)}{\log 10} \right\rceil [/math] in which case we find the left side of the equality above becomes
[eqn]
\frac{\epsilon}{2} < \epsilon
[/eqn]
which is true for all [math]\epsilon>0[/math].

Therefore [math]0.999... = 1[/math].

>>11583578
https://www.youtube.com/watch?v=kTVRopTVjpQ

>>11585110
>>11585087
Search up what a limit is

>>11585282
I know what a limit is and the limit of the sequence 0.9, 0.99, 0.999, ... (aka 0.999....) is 1.

>>11585288
>I know what a limit
I don't think you do

>>11583578
1 and 0.(9) ARE DIFFERENT.
But they are INDISTINGUISHABLE for arithmetic operations.

1) for any e: 1 - 0.(9) > e. (e is arbitrary small)
2) e + e = e by (1) definition
3) 1 + 1 =
0.(9) + e + 0.(9) + e =
0.(9) + 0.(9) + e =
0.(9) + 1 ->
0.(9) = 1
4) you can apply the same logic for any arithmetic operation
5) so yeah you can use 0.(9) instead of 1.

>>11585254
One of the few people with a functioning brain in this board.

>>11585328
Point nine repeating and 1 are "different" in the same way that 5/5 and one are different: it's a different representation of the SAME Real Number.

>>11585328
>ARE DIFFERENT.
>But they are INDISTINGUISHABLE
lol no

>>11585336
e is infinitely small, but not 0.
so 0.(9) + e = 1 and 0.(9) != 1,

>>11585328
>1) for any e: 1 - 0.(9) > e. (e is arbitrary small)
did you mean 1 - 0.(9) < e ?

>>11585336
lmao how can 1/2 and 3/6 be the same number
the numbers are all different LMAO!
[math]\text{ }[/math]

>>11585324
ok retard

>>11585324
I don't think you know what 0.999... is

>>11583578
people will argue about anything. this thread pops up way to often and is stupid as can be.

>>11585192
Infinity is eternal. Infinity is not finite. An infinite amount of nines after zero point will never become a full 1 because it will forever be less than 1. If you cannot grasp this you are mentally damaged.

>>11585522
This. Nothing else needs to be said on this topic.

Cause it doesn't

>>11585522
>will never become
0.999... is static, the length is aleph_0 from the get go.
Your naive cartoon vision of a diesel engine chugging along is ridiculous. Embarrassing even.