/sci/ - Science & Math

[Anonymous No.5836519]

>implying 0.999... = 1

Most retarded troll I've ever seen. That's like claiming 1345313 = 78946798. They already differ in the first digit.

[Anonymous No.5836523]

>>5836519
you dont know what an = sign is do you....

[Anonymous No.5836528]

>>5836519
numbers can be represented in different ways. 1=.9999....=3/3=(2-1)=x(1/x).

[Anonymous No.5836532]

>>5836528
>x(1/x)
lold

[Anonymous No.5836549]

We say that two real numbers <span class="math">x,y \in \mathbb{R}[/spoiler] are equal (denoted x = y) if x ~ y, where ~ is the equivalence relation defined on the completion of the rational numbers with respect to the Euclidean metric as follows: x ~ y if
<div class="math">\lim_{n \rightarrow \infty} x_n - y_n = 0</div>
where <span class="math">{x_n} = x[/spoiler] and <span class="math">{y_n} = y[/spoiler] are representative Cauchy sequences of the elements x and y in the aforementioned completion of <span class="math">\mathbb{Q}[/spoiler].

Thus it is clear that 0.999... = 1 using standard technique involving geometric series and the definition of x = y given above.

[Anonymous No.5836698]

>>5836549
In layman's terms, if the difference between two numbers is zero, they are the same number. The difference between .999... and 1 is 0.000... so they're the same number.

[Anonymous No.5836702]

>>5836549
Okay, input 1 into all of those variables

Okay?

Now input 0.9999... and go drown in a puddle

[Pedro No.5836703]

>>5836532
why would you lol?
(1/x)x is defined to equal one
www.wikipedo/multiplicationalinverse.com
idiot

[Anonymous No.5836705]

>>5836703
yeah but x(1/x) is effectively saying x0/x by the properties of 1

just lold, w/e

[Anonymous No.5836707]

>>5836702
If you did not understand that post then you have no authority speaking of "equal" numbers.

[sage No.5836714]

The problem of whether or not <span class="math">0.999\cdots[/spoiler] equals <span class="math">1[/spoiler] begins with:
<span class="math">0.999.... = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots[/spoiler]
Proof:
When written in sigma notation we get:
<div class="math">\sum_{m=0}^\infty \frac{9}{10^m} = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots </div>
Now in order to prove that <span class="math"> 0.999 \cdots = 1[/spoiler] we begin by using the formula for the sum of a geometric series:
<div class="math">\lim_{n \to \infty}\sum_{m=0}^n \frac{9}{10^m} = \lim_{n \to \infty}\left( \frac{\frac{9}{10}(1 - \frac{1}{10}^{n+1})}{1 - \frac{1}{10}} \right)</div>
We are now left with:
<div class="math">\lim_{n \to \infty}\left( \frac{\frac{9}{10}(1 - \frac{1}{10}^{n+1})}{1 - \frac{1}{10}} \right)</div>
Which equals:
<div class="math">\lim_{n \to \infty}\frac{\frac{9}{10}}{1 - \frac{1}{10}}-\lim_{n \to \infty}\frac{\frac{9}{10}(\frac{1}{10}^{n+1})}{1 - \frac{1}{10}}</div>
And when solved equals <span class="math">1-0[/spoiler] or <span class="math">1[/spoiler]

Therefore,
<div class="math">0.999 \cdots = 1</div>

<div class="math">Q.E.D</div>

[Anonymous No.5836738]

>>5836714

You must be extremely bored.

[Anonymous No.5836743]

>>5836714
it begins with 1/1, which is as to say that xyz is immediate itself and implicatively deviding & or other to itself

[sage No.5836741]

>>5836738

[Anonymous No.5836748]

>>5836741
ogawd likesrslegitactually
somepeople.jpg

[Anonymous No.5836752]

>>5836714
>Proof:
That is not a fucking proof

To use that formula for geometric series you need to know that the sum converges which you need to know that the decimal construction of the reals satisfies the real axioms which you need to <span class="math"> \bf{ DEFINE}[/spoiler] .999.... = 1

[sage No.5836758]

>>5836752
lel

This is enough fucking proof for an internet argument. What the fuck am I doing? Writing a dissertation?

[Anonymous No.5836761]

>>5836714
with m starting at zero your first integer is 9

[Anonymous No.5836769]

>>5836707
Jesus christ you think theres such a thing as > - <(flat) equal?!!! THE EQUAL SIGN >>>>>>>>> =
> >>5836620

[sage No.5836771]

>>5836761
True, I spent only a few minutes typing it. Should be:

<div class="math">\sum_{m=1}^\infty \frac{9}{10^m}</div>

[Anonymous No.5836778]

>>5836758
Doing things without knowing the assumption is the single worst thing you can do in mathematics. Go back to the physics lounge with all the other cunts that don't care for rigor.

[Anonymous No.5836787]

>>5836771
k now that's > dimnuitive infinitive but not correcting in the notation by just using /8.... those kinds of structures are so much more than you'd figure though

[sage No.5836788]

>>5836778
I concur, however I do not feel like making a completely legitimate proof. Give me some sort of contact information and I will send you a full proof.

[Anonymous No.5836789]

>>5836771
k now that's > dimnuitive infinitive but not correcting in the notation by just using /8.... those kinds of structures you have there are so much more than you'd figure though

[Anonymous No.5836886]

>>5836788
But there is no proof. Like I said, it's purely by definition. Similarly you can't prove there exist a null set by just writing {}.

[Anonymous No.5837012]

>>5836769
The equal sign needs a proper definition as to what it means. It is not enough to say, "x = y if x looks like y." For example, we say that 1/2 = 2/4, yet they don't "look" like each other. The same is occurring here with .999... = 1. They don't look the same but they are equal according to the standard definition, seen here >>5836549 .

For further details, see
http://en.wikipedia.org/wiki/Cauchy_sequence
http://en.wikipedia.org/wiki/Equivalence_relation
http://en.wikipedia.org/wiki/Complete_metric_space#Completion
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences