/sci/ - Science & Math

[Anonymous No.3957241]

Hey guys, I want to ask a stupid question. .999... is equal to 1, which can be demonstrated in a number of ways, the most common one I know of being something along the lines of 1/3 = .333..., .333... * 3 = .999..., 1/3 *3 = 1, therefore .999... = 1. I have a quibble with this, but I'm willing to trust better scientific minds in a basic sense.

Now the reason WHY .999...= 1, at least as it was explained to me, is that there is no way to produce an infinitesimally small difference between the two (basically, that .00000...00001 is not a number).

By that reasoning, is it also logical to say that .333... is equal to .4? As such, does .4 = 1/3? Could someone tell me what, if anything, I'm doing wrong?

[Anonymous No.3957266]

.9999999... + 1/inf = 1

[Anonymous No.3957258]

.4 - .333... =.0666...

[Anonymous No.3957291]

0.9 recurring is = 1
But only in the real number system.
Heard of hyperreals?
http://en.wikipedia.org/wiki/0.999...#In_alternative_number_systems

"
Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. For example, the dual numbers include a new infinitesimal element ε, analogous to the imaginary unit i in the complex numbers except that ε2 = 0. The resulting structure is useful in automatic differentiation. The dual numbers can be given a lexicographic order, in which case the multiples of ε become non-Archimedean elements.[48] Note however that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.
"


Despite the fact that wiki isn't the best source, it's a good read none the less.

[Anonymous No.3957360]

x = 0.999...
10x = 9.999...
10x - x = 9
9x = 9
x = 1

[EK !EKFQOBUFnQ No.3957373]

[Anonymous No.3957490]

If an object is .999... complete, would it be whole?

[Anonymous No.3957499]

No, but .3999...=.4

[Anonymous No.3957540]

0.9999... is _not_ a number, it is a notation which denotes 1

[Anonymous No.3957863]

0.999999999999... = 1 - 1/10^H
\----------H---------/

Now, as H approaches infinity, lim H->inf 1/10^H = 0

So 0.999999999...= 1 - 0 = 1

[Anonymous No.3957892]

>>3957241
You are either an idiot or a half decent troll. But no, .33333... is not equal to .4
>but
.399999999... is.

[Anonymous No.3957905]

>>3957540
>0.9999... is _not_ a number
0.999... is a number, just as much as .111... .222... and .333... are.
<span class="math">\lim_{x\to0} \frac{sin(x)}{x}[/spoiler], 0.999... and <span class="math">\frac{4}{4}[/spoiler] all look different, but that doesn't mean they aren't all equal to 1.

[Anonymous No.3957919]

0.999999999999 = 1

This is why mathematicians are ridiculized by engineers and physicians.

[Anonymous No.3957921]

>>3957905
>my limit expression won't correctly line up
Oh, that's some bullshit right there.

[Anonymous No.3957926]

>physicians

[Anonymous No.3957932]

OP, think about what you've said. Now for my obligatory copy pasta.


First a note on notation here, I'm not texing this up, so or = union, and = intersection, !0 = the empty set, Q = the rationals, R = the reals

Let's start by defining a Dedekind section then and the real numbers then, so we all know what we're talking about.

A Dedekind section of the rationals is an ordered partition (L, U) of Q such that L or U = Q and L and U = !0 such that if x is in L and y is in U then x < y and L has no greatest element. Since L uniquely determines U and vice versa let's just refer to this to this as L in place of (L, U)

Now we define the set of real numbers as the set of all Dedekind sections of Q. We can show (I'm not going to) that this is indeed what we mean when we speak naively of the set of real numbers.

Now all that remains to be seen for the proof that .999... = 1 is that the Dedekind section defined by one is exactly that section defined by the other, then they are the same number by definition.

Consider the Dedekind section defining 0.999... this is precisely the set of rational numbers p such that p<0.9^n for any natural number n (so those less than 0 or less than 0.9 or less than 0.99 etc) that is p is less than 1-(1/10)^n for some n. Every element of 0.999... is also an element of 1 then (since 1 > 1-(1/10)^n for any n)

[Anonymous No.3957935]

>>3957926
How could you include "physicians" but leave out "ridiculized?"

[Anonymous No.3957938]

>>3957935

It's defined.

[Anonymous No.3957941]

>>3957932
Now for the converse

Consider the Dedekind section defining 1. This is precicely the set of rational numbers a/b such that a/b < 1. But then a/b < 1-(1/10)^n (I will show why this is so in a moment, see ***) so then every element of 1 is also an element of 0.999... But then 1 and .999... contain the same elements, and thus are equal by definition.

***
Since a/b < 1, we consider the minimum a/b (so for example 2d/2c = d/c, its the d/c we're interested in) and a is strictly less than b, clearly since otherwise a/b would not be less than 1. Now consider the LARGEST of these for each b, that is a = b-1, so we have (b-1)/b < 1, but (b-1)/b + (1/b) = 1, and its certainly true that there is some number of the form 10^n which is larger than b, and thus (1/10)^n is less than (1/b), so (b-1)/b + (1/10)^n < 1, but this was for the largest a for a given b, and is true for any b, so then its true for all a, b.

Please take the time to read this whether you believe 0.999... = 1 or not, this is the proof from the definitions, this is why it is true.

[Anonymous No.3957965]

This is the simplest way I can put it OP, pay attention.

Let us assume 0.999... and 1 are not equal.

This necessarily means that there must be a number between 0.999... and 1.

But there is no such number, therefore the assumption that 0.999... and 1 are not equal is incorrect.

[Anonymous No.3957980]

>>3957965
so is this the same for 3.9999999999999999...and 4?

[Anonymous No.3957984]

I remember a math teacher at my university that believed that 0.999... wasn't equal to 1.

That day I realized how shitty the educational system at my country really was.

[Anonymous No.3957987]

>>3957980

Yes.

[Anonymous No.3958004]

>>3957984
You can't say something like that without telling us which shitty country you're from.

Also, he could have been a Computer Science professor at heart and counted in base 16. Then, 0.999... doesn't equal 1.

[Anonymous No.3958006]

anyone wanna explain why this matters in any situation in real life?

[Anonymous No.3958014]

Prove it with a limit using a geometric series then fucking use an axiom.

[Anonymous No.3958020]

>>3958004

Chile.

Still, after we pointed it out, the teacher checked and apologized to the class the next day.

[Anonymous No.3958029]

>>3958006

If computers couldn't tell 1=0.999... we would not be having this conversation over the internet.

[Anonymous No.3958039]

>>3958029
source

[Anonymous No.3958053]

>>3958029
wrong, computers can represent the number 1 fine.
it's numbers like 1/5 that get them.

[Anonymous No.3958085]

>>3958020

Im from chile too, indeed our education sucks