Yet another discussion about 0.99...

[MNOWAX]

.999...=1
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MNOWAX

[Thok]

[bgg1996]

[Zag]

It's proven if Achilles ever actually catches up with the tortoise. Oh look! He's done it! There we go; let's call this one dead.

[Scurra]

Was this one actually tested experimentally though?
I mean there seems to be a lot of anecdotal reports, and accounts of thought experiments, but did Achilles actually try racing the tortoise? Or is it just one of those things that everyone says is obvious without proof?

(Note that I would be unwilling to accept e.g. Usain Bolt racing a tortoise, on the grounds that this would be an entirely different experiment and would prove nothing except that Usain Bolt can probably run faster than a tortoise. Also, Usain Bolt doesn't have that heel problem either.)
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[extro...*]

[MNOWAX]

. and this is why I brought it up.

It's like you guys can't stop talking about it when it is brought up on here.
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MNOWAX

[extro...*]

[bgg1996]

[Thok]

The real reason for having .999...=1 is translation invariance; we'd like the number line to look the same everywhere. If .999...=/=1, then some of the points on the number line have the property that there's an arbitrarily close second point to it, and others don't have that property.

A less mathy version of this is what is pi+1-.999...? It's either pi, in which case 1=.999..., or it's not pi, in which case I have to define lots of numbers like pi+17*(1-.999...), which is different from pi+16*(1-.999...), which is different from pi+15*(1-.999...), et cetera and none of those numbers have decimal representations. In neither case do you get the situation where each decimal representation represents a unique number and each number is represented by a unique decimal representation, which is what the typical person arguing for .999...=/=1 wants.

[Death Mage]

By that logic, arbitrarily large numbers are meaningless because it's too much hassle to differentiate between them when the differences are so small. A googolplex-pi would need to be defined, as would a googolplex-(pi*2), as would a googolplex-(pi*3).. and really, that's just too much work.

Yes, different numbers *ARE* different, and should not be treated the same way.
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[Thok]

[extropalopakettle]

[extropalopakettle]

X = 1 - 0.999...

To those saying X>0, I ask: Does X = 2*X? Does X = X/2?

[Zag]

[Jedo the Jedi]

My only question is if .99999...=1, then where does Samadhi live?
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[extropalopakettle]

Googolplex is tiny compared to Graham's number.

[Thok]

[referee]

Assume they're different

1 != .999...

multiply by 10

10 != 9.999...

If we subtract the first line from the second

10 - 1 != 9.999... - .999...
9 != 9

Contradiction! (Works because the third line is just the first line multiplied by nine, so if 9 equals 9, then 1 equals .999... qed)
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[Death Mage]

Infinity/infinite repeating does not work that way.
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[LordKinbote]

[Thok]

[extropalopakettle]

backfilling, a simple rational numbers like 1/3 can't be represented as finite decimal numbers in the same way 1/2, 1/5 and 1/8 can (0.5, 0.2 and 0.125 respectively). 1/3 can be represented as an infinite repeating decimal, 0.333..., where the ",,," denotes infinite repetition. However, this notation we use allows us to express some numbers in more than one way. While 1/3 can only be expressed as 0.333..., 1/2 can be expressed as 0.5 or as 0.4999...

Which rational numbers require "..." notation depends on the base of the number system being used. In base 3, "one third" is 0.1, while "one half" is 0.1111... (1/3 + 1/9 + 1/27 + 1/81 + ...).

[Death Mage]

Infinity - infinity is like dividing by zero. It's undefined. The same is true for infinitely repeating number lines. Simply put: You've changed the .999... after multiplying it by 10, therefore trying to subtract the two does NOT equal zero. The decimal numbering system we use simply doesn't allow you to note that change.
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[Zag]

[Dragon Phoenix]

1/3 = 0.3333333333333333..........
1/3 = 0.3333333333333333..........
1/3 = 0.3333333333333333..........
sum left and right:
1 = 0.9999999999999999.........
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[Death Mage]

Zag, your quote is right, but you're applying it to the wrong person. I'm not the one trying to apply infinity as a rational number.

All of the proofs for .999...=1 are flawed. There's one fundamental flaw that a majority of them use. It's that .333... or .111... or any other infinitely repeating string actually correctly represents the number. It doesn't. What it represents is the closest digitial representation of 1/3, or 1/9, etc. But the very fact that it infinitely repeats means it is not a fully accurate representation, but an estimation. And you're forgetting to round.

"But you can't round, you only round the last digit and there is no last digit with an infinite string!"

And that's why it's not an accurate representation, as as long as you continue to think it is you will never fully understand the real issue here.

But let me pose this to you:

What is the largest number for X that fits this statement:

X < 1

There is no "If you allow for infinitesimals." Trying to deny that a number can be infinitely small is the same as denying one can be infinitely big. It doesn't work.

[Thok]

[Death Mage]

The only times .999... = 1 is when you are exploiting a flaw in the decimal system. In other words: When you are writing it down wrong. There is already a way to represent 1 in decimal, and that's 1. That's the ONLY way that is completely accurate.

"There is no such number" is not a valid answer for "what is the largest value of X where X < 1".
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[extropalopakettle]

[Amb]

0.9999 stops at the planck length. After that, the digits are imaginary and cease to be useful.

[DM*]

What number is the average of 1 and infinity? What is infinity + 1? Are you going to argue that the concept of infinity can't exist beachwear you can't apply some basic mathematics to it? How on Earth can you justify simultaneously arguing for and against the concept of infinity? And what is 1 - (1/infinity)?

[extro...*]

[MNOWAX]

I jsut want to say something quick to you all.....


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HAHAHAHAHAHA
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MNOWAX

[DM*]

Beachwear? Really, Autocorrect? Really? "Because" translates to "beachwear" to you? [/i]

[DM*]

Beachwear? Really, Autocorrect? Really? "Because" translates to "beachwear" to you? [/i]

[Amb]

[Death Mage]

And it double posted it? Fuck posting from my cell phone. Grrrrrrrr.
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[Thok]