u/Sea_Handle_994

The latest deranged talking point bouncing around in this subreddit seems to be the following idea:

>The "0." prefix of 0.999... guarantees magnitude less than 1.

This is such a deeply confused idea. The prefix "0." is a prefix of the *numeral* "0.999...", not the *number* that the numeral represents. Numerals have prefixes, not numbers.

It's not true in general that the way the number is *written* implies anything at all about what the number *is*. This would be obvious to anyone who understands the difference between a number and a numeral. Talk to an elementary/primary school teacher if you're struggling with this incredibly difficult concept.

Some have suggested that u/SouthPark_Piano might be using a different number system when claiming that "0.999... ≠ 1." I understand this perspective, but I'm personally skeptical, so I figured it would be appropriate to ask directly.

u/SouthPark_Piano, are you using a different number system?

This extremely simple question is intended for u/SouthPark_Piano or anyone else who thinks that "0.999... = 1" is "silly" or a "lie."

Infinitely many nines

Positioned and aligned

Form a number greater than

The confines of your mind

Just a geometric sum

Your dogma doth proclaim?

Open your mind further

Or we'd have to do the same

Though our minds are all made up

A never-ending bliss

I can't help but wonder how

To find a sum like this

1.1 - 0.11 + 0.011 - 0.0011 + ...

You said this in a previous post:

>0.999... = 1 - 1/10^(n) with n integer starting at n = 1, then n upped limitlessly.

https://www.reddit.com/r/infinitenines/comments/1worjkl/what_is_1_1_0999/?utm_source=app_first_navigation&mweb_loid=t2_2nynhc1r1h

Did you mean that 0.999... is literally equal to the sequence [1 - 1/10^(1), 1 - 1/10^(2), ...]?

Just a yes or a no would be OK.

Let n be a positive integer. Consider the following sequences:

𝑎ₙ = 1, 1, 1, …

𝑏ₙ = 0.9, 0.99, 0.999, …

𝑐ₙ = 0.99, 0.9999, 0.999999, …

These sequences respectively have the following formulas:

𝑎ₙ = 1

𝑏ₙ = 1 − 1/10ⁿ

𝑐ₙ = 1 − 1/100ⁿ

We clearly have 𝑏ₙ < 𝑎ₙ for all n. We also have 𝑏ₙ < 𝑐ₙ for all n.

If we're using the notation 0.999... to represent the sequence 0.9, 0.99, 0.999, ..., that means we're using 0.999... to represent 𝑏ₙ. To the extent that it makes sense to write 𝑏ₙ = 0.999..., that would mean

𝑏ₙ < 𝑎ₙ implies 0.999... < 1.

But why couldn't we use the notation 0.999... to also represent the sequence 0.99, 0.9999, 0.999999, …? Then we would be using 0.999... to represent 𝑐ₙ. To the extent that it makes sense to write 𝑏ₙ = 0.999... and 𝑐ₙ = 0.999..., that would mean

𝑏ₙ < 𝑐ₙ implies 0.999... < 0.999...

The inequality 0.999... < 0.999... is ridiculous on its face.

The notation 0.999... is inadequate to describe the sequence given by 1 − 1/10ⁿ because it is ambiguous. It could just as well represent 1 − 1/100ⁿ, 1 − 1/1000ⁿ, or any other sequence that corresponds to 0.999... Using this notation in this way leads to nonsensical identities like 0.999... < 0.999...

I'm an anarchist, and I'm struggling to understand the concept of market anarchism. For example, I've been trying to read "Markets, Not Capitalism" by the C4SS:

https://www.minorcompositions.info/wp-content/uploads/2011/10/MarketsNotCapitalism-web.pdf

Among other things, the authors of this document critcize the "money monopoly," where the state gives certain individuals the privilege to issue currency and punishes counterfeiting. The authors then make claims about the various benefits that would arise "if the business of banking were made free for all."

What exactly does it mean to make the "business of banking" free for all? I don't understand how the modern-day system of central banking can be made free for all. Surely the whole point of a central bank is that it represents the monetary authority of a state (or monetary union). With no state and no monetary authority, what would be the purpose of a bank? If I wanted money and there were no counterfeit law, why would I bother interacting with a bank when I could simply create my own money? If money could be created by anyone at will and spent in markets without the risk of imprisonment, then why would sellers want money to begin with?

It seems oxymoronic to advocate for a market economy without some kind of monetary authority. In modern economies, states and central banks have the exclusive authority to create money; that's what makes money scarce. Sellers participate in markets in order to make money. If the creation of money were truly free for all, then what purpose would a market serve?

I have an unhealthy obsession with this subreddit. I want to understand the extent to which people insist on this idea that 0.9... cannot possibly be equal to 1.

What would you (anyone who believes in this theory) do if you were presented with math problems like the ones at this web page?

https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-2/e/geometric-series-of-constants

Or, for example, what would you do if you were asked to find this geometric series:

sum (-0.5)^n, n=0 to infinity

Would you be able to pass an exam with questions like these on it? Would you just reject these math problems as pure nonsense? Or would it be like a 12-dimensional chess situation where you'd know that the correct answer was 0.666... or whatever but you'd put down the answer of 2/3 because that's the answer that all the sheeple believe?

In summary, can you (would you) actually do math problems related to this subject?

In Analysis I by Terence Tao, Tao rigorously develops the real number system and proves basic results of real anlysis. In Appendix B, he formally defines the decimal system. Proposition B.2.3 in particular reads:

>Proposition B.2.3 (Failure of uniqueness of decimal representations). The number 1 has two different decimal representations: 1.000... and 0.999....

>Proof. The representation 1 = 1.000... is clear. Now let’s compute 0.999.... By definition, this is the limit of the Cauchy sequence

>0.9, 0.99, 0.999, 0.9999, ... .

>But this sequence has 1 as a formal limit by Proposition 5.2.8.

Now that this rookie mistake by Terence Tao has finally been exposed, should I throw this textbook in the trash?