this post was submitted on 27 Jun 2024
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[–] [email protected] 96 points 5 months ago* (last edited 5 months ago) (7 children)

x=.9999...

10x=9.9999...

Subtract x from both sides

9x=9

x=1

There it is, folks.

[–] [email protected] 70 points 5 months ago* (last edited 5 months ago) (4 children)

Somehow I have the feeling that this is not going to convince people who think that 0.9999... /= 1, but only make them madder.

Personally I like to point to the difference, or rather non-difference, between 0.333... and ⅓, then ask them what multiplying each by 3 is.

[–] [email protected] 1 points 1 month ago

The thing is 0.333... And 1/3 represent the same thing. Base 10 struggles to represent the thirds in decimal form. You get other decimal issues like this in other base formats too

(I think, if I remember correctly. Lol)

[–] [email protected] 1 points 5 months ago

Oh shit, don't think I saw that before. That makes it intuitive as hell.

[–] [email protected] 0 points 5 months ago (1 children)

Cut a banana into thirds and you lose material from cutting it hence .9999

[–] [email protected] 3 points 4 months ago

That’s not how fractions and math work though.

[–] [email protected] 48 points 5 months ago (1 children)

I was taught that if 0.9999... didn't equal 1 there would have to be a number that exists between the two. Since there isn't, then 0.9999...=1

[–] [email protected] 3 points 5 months ago

Not even a number between, but there is no distance between the two. There is no value X for 1-x = 0.9~

We can't notate 0.0~ ....01 in any way.

[–] [email protected] 23 points 5 months ago (2 children)

Divide 1 by 3: 1÷3=0.3333...

Multiply the result by 3 reverting the operation: 0.3333... x 3 = 0.9999.... or just 1

0.9999... = 1

[–] [email protected] 5 points 4 months ago* (last edited 4 months ago) (1 children)

Unfortunately not an ideal proof.

It makes certain assumptions:

  1. That a number 0.999... exists and is well-defined
  2. That multiplication and subtraction for this number work as expected

Similarly, I could prove that the number which consists of infinite 9's to the left of the decimal separator is equal to -1:

...999.0 = x
...990.0 = 10x

Calculate x - 10x:

x - 10x = ...999.0 - ...990.0
-9x = 9
x = -1

And while this is true for 10-adic numbers, it is certainly not true for the real numbers.

[–] [email protected] 1 points 4 months ago* (last edited 4 months ago) (1 children)

While I agree that my proof is blunt, yours doesn't prove that .999... is equal to -1. With your assumption, the infinite 9's behave like they're finite, adding the 0 to the end, and you forgot to move the decimal point in the beginning of the number when you multiplied by 10.

x=0.999...999

10x=9.999...990 assuming infinite decimals behave like finite ones.

Now x - 10x = 0.999...999 - 9.999...990

-9x = -9.000...009

x = 1.000...001

Thus, adding or subtracting the infinitesimal makes no difference, meaning it behaves like 0.

Edit: Having written all this I realised that you probably meant the infinitely large number consisting of only 9's, but with infinity you can't really prove anything like this. You can't have one infinite number being 10 times larger than another. It's like assuming division by 0 is well defined.

0a=0b, thus

a=b, meaning of course your ...999 can equal -1.

Edit again: what my proof shows is that even if you assume that .000...001≠0, doing regular algebra makes it behave like 0 anyway. Your proof shows that you can't to regular maths with infinite numbers, which wasn't in question. Infinity exists, the infinitesimal does not.

[–] [email protected] 2 points 4 months ago

Yes, but similar flaws exist for your proof.

The algebraic proof that 0.999... = 1 must first prove why you can assign 0.999... to x.

My "proof" abuses algebraic notation like this - you cannot assign infinity to a variable. After that, regular algebraic rules become meaningless.

The proper proof would use the definition that the value of a limit approaching another value is exactly that value. For any epsilon > 0, 0.999.. will be within the epsilon environment of 1 (= the interval 1 ± epsilon), therefore 0.999... is 1.

[–] [email protected] 2 points 4 months ago (1 children)

The explanation I've seen is that ... is notation for something that can be otherwise represented as sums of infinite series.

In the case of 0.999..., it can be shown to converge toward 1 with the convergence rule for geometric series.

If |r| < 1, then:

ar + ar² + ar³ + ... = ar / (1 - r)

Thus:

0.999... = 9(1/10) + 9(1/10)² + 9(1/10)³ + ...

= 9(1/10) / (1 - 1/10)

= (9/10) / (9/10)

= 1

Just for fun, let's try 0.424242...

0.424242... = 42(1/100) + 42(1/100)² + 42(1/100)³

= 42(1/100) / (1 - 1/100)

= (42/100) / (99/100)

= 42/99

= 0.424242...

So there you go, nothing gained from that other than seeing that 0.999... is distinct from other known patterns of repeating numbers after the decimal point.

[–] [email protected] 1 points 4 months ago* (last edited 4 months ago) (1 children)

The explanation I've seen is that ... is notation for something that can be otherwise represented as sums of infinite series

The ellipsis notation generally refers to repetition of a pattern. Either ad infinitum, or up to some terminus. In this case we have a non-terminating decimal.

In the case of 0.999..., it can be shown to converge toward 1

0.999... is a real number, and not any object that can be said to converge. It is exactly 1.

So there you go, nothing gained from that other than seeing that 0.999... is distinct from other known patterns of repeating numbers after the decimal point

In what way is it distinct?
And what is a 'repeating number'? Did you mean 'repeating decimal'?

[–] [email protected] 1 points 4 months ago* (last edited 4 months ago) (1 children)

The ellipsis notation generally refers to repetition of a pattern.

Ok. In mathematical notation/context, it is more specific, as I outlined.

This technicality is often brushed over or over simplified by math teachers and courses until or unless you take some more advanced courses.

Context matters, here's an example:

Generally, pdf denotes the file format specific to adobe reader, while in the context of many modern online videos/discussions, it has become a colloquialism to be able to discuss (accused or confirmed) pedophiles and be able to avoid censorship or demonetization.

0.999... is a real number, and not any object that can be said to converge. It is exactly 1.

Ok. Never said 0.999... is not a real number. Yep, it is exactly 1 because solving the equation it truly represents, a geometric series, results in 1. This solution is obtained using what is called the convergence theorem or rule, as I outlined.

In what way is it distinct?

0.424242... solved via the convergence theorem simply results in itself, as represented in mathematical nomenclature.

0.999... does not again result in 0.999..., but results to 1, a notably different representation that causes the entire discussion in this thread.

And what is a 'repeating number'? Did you mean 'repeating decimal'?

I meant what I said: "know patterns of repeating numbers after the decimal point."

Perhaps I should have also clarified known finite patterns to further emphasize the difference between rational and irrational numbers.

EDIT: You used a valid and even more mathematically esoteric method to demonstrate the same thing I demonstrated elsewhere in this thread, I have no idea why you are taking issue with what I've said.

[–] [email protected] 1 points 4 months ago (1 children)

Ok. In mathematical notation/context, it is more specific, as I outlined.

It is not. You will routinely find it used in cases where your explanation does not apply, such as to denote the contents of a matrix.

Furthermore, we can define real numbers without defining series. In such contexts, your explanation also doesn't work until we do defines series of rational numbers.

Ok. Never said 0.999... is not a real number

In which case it cannot converge to anything on account of it not being a function or any other things that can be said to converge.

because solving the equation it truly represents, a geometric series, results in 1

A series is not an equation.

This solution is obtained using what is called the convergence theorem

What theorem? I have never heard of 'the convergence theorem'.

0.424242... solved via the convergence theorem simply results in itself

What do you mean by 'solving' a real number?

0.999... does not again result in 0.999..., but results to 1

In what way does it not 'result in 0.999...' when 0.999... = 1?

You seem to not understand what decimals are, because while decimals (which are representations of real numbers) '0.999...' and '1' are different, they both refer to the same real number. We can use expressions '0.999...' and '1' interchangeably in the context of base 10. In other bases, we can easily also find similar pairs of digital representations that refer to the same numbers.

I meant what I said: "know patterns of repeating numbers after the decimal point."

What we have after the decimal point are digits. OTOH, sure, we can treat them as numbers, but still, this is not a common terminology. Furthermore, 'repeating number' is not a term in any sort of commonly-used terminology in this context.

The actual term that you were looking for is 'repeating decimal'.

Perhaps I should have also clarified known finite patterns to further emphasize the difference between rational and irrational numbers

No irrational number can be represented by a repeating decimal.

[–] [email protected] 1 points 4 months ago (1 children)

https://www2.kenyon.edu/Depts/Math/Paquin/GeomSeriesCalcB.pdf

Here's a standard introduction to the concept of the Convergence/Divergence Theorem of Geometric Series, starts on page 2.

Its quite common for this to be referred to as the convergence test or rule or theorem by teachers and TA's.

[–] [email protected] 1 points 4 months ago* (last edited 4 months ago)

Now, ask yourself this question, 'is 0.999..., or any real number for that matter, a series?'. The answer to that question is 'no'.

You seem to be extremely confused, and think that the terms 'series' and 'the sum of a series' mean the same thing. They do not. 0.999... is the sum of the series 9/10+9/100+9/1000+..., and not a series itself.

EDIT: Also, the author does abuse the notations somewhat when she says '1+1/2+1/4 = 2' is a geometric series, as the geometric series 1+1/2+1/4+... does not equal 2, because a series is either just a formal sum, a sequence of its terms, or, in German math traditions, a sequence of its partial sums. It is the sum of the series 1+1/2+1/4+... that is equal to 2. The confusion is made worse by the fact that sums of series and the series themselves are often denoted in the same way. However, again, those are different things.
Would you mind providing a snippet with the definition of the term 'series' that she provides?

EDIT 2: Notably, that document has no theorem that is called 'convergence theorem' or 'the convergence theorem'. The only theorem that is present there is the one on convergence and divergence of geometric series.