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0.999...= 1

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Re: 0.999...= 1

#291

Post by thom »

ta1234567890 wrote: post #235
lukifer wrote: post #228

Earlier quoted context omitted.

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…

> As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.

Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.

Re: 0.999...= 1

#292

Post by ubercow13 »

dahart wrote: post #260
ubercow13 wrote: post #226

Earlier quoted context omitted.

>if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right >0.9bar7 is a completely nonsensical number For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. It's like you're trying to perform trans…

> For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s befor…

In your proof you say

>And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right

This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's.

This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.

Re: 0.999...= 1

#293

Post by CogitoCogito »

tom-thistime wrote: post #282
CogitoCogito wrote: post #269

Earlier quoted context omitted.

It's smart of you to bring up the surreal numbers: https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...

Strictly speaking I brought up the surreal numbers.

Then I guess you're the smart one. My apologies.

Re: 0.999...= 1

#294

Post by TuringTest »

lukifer wrote: post #228
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…

I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach[1] showed me that there's another, more static logical interpretation of infinity which also comes handy.

In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the generation process (i.e. it is higher than any item in the list, or smaller, or has happened after all of them)... This object doesn't really belong to the infinite generative sequence, it's an item outside all of them, and can't be reached by completing the sequence; it merely exist outside the process and happens to have the property of "dominating" all the items in it.

You can assume the existence in the same way you assume the existence of a number which is the square root of -1, or how you define triangles whose angles add up to more or less than 180 degrees. If you do that, this object "at the infinite" can be formally defined and treated axiomatically to find out what mathematical properties it possesses.

[1] https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

Re: 0.999...= 1

#295

Post by calibas »

__s wrote: post #287
calibas wrote: post #271

And this is why I prefer hyperreals. 0.999... = 1 - 1/∞ We talk about infinity all the time in mathematics, teachers use the concept to introduce calculus in a way that people can more easily understand, but using infinity directly is almost universally banned within classrooms. Nonstandard analysis is a much more intuitive way of understanding calculus, it's the whole "infinite number of infinitely small pieces" con…

I think what's important here is that if you're making that claim, 0.333... = 1/3 - 1/∞ Which implies 3/∞ = 1/∞

Apologies, I should have explained it differently.

0.999... implies a number infinitesimally smaller than 1. You wouldn't use 0.999... in a hyperreal system because you can represent it directly.

I shouldn't have mixed different systems and claimed they're mathematically equivalent, you've proven that doesn't work.

Re: 0.999...= 1

#296

Post by Bodell »

lmkg wrote: post #137
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.

"Players and painted stage took all my love, And not those things that they were emblems of."

The Circus Animal's Desertion, W. B. Yeats

Re: 0.999...= 1

#298

Post by da_chicken »

thinkharderdev wrote: post #288
traderjane wrote: post #259

Earlier quoted context omitted.

It's the smallest number bigger than 0.

That doesn't exist. An open interval doesn't have a smallest number.

Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal

If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist.

It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.

Re: 0.999...= 1

#299

Post by missedthecue »

parski wrote: post #214
ping_pong wrote: post #160

My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard…

I asked my math teacher this when I was a kid. He told me to accept that's the way it is so I did.

Kinda like why 5 rounds up instead of down

Re: 0.999...= 1

#300

Post by brutt »

2OEH8eoCRo0 wrote: post #221
gspr wrote: post #162

Earlier quoted context omitted.

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…

Ask for a number between .9 repeated and 1

0.00...1
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