Current Events > Proof that 0.999...=1

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clearaflagrantj
06/01/17 8:17:17 AM
#51:


Look at all the math nerds in this topic.
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ChromaticAngel
06/02/17 2:25:28 PM
#52:


scar the 1 posted...
Nice to read about p-adic numbers. You don't see that very often. I guess one could say that hyperreals contain numbers that "kind of" lie in-between 1 and 0.999..., but hyperreals aren't so common either.


There is no number between 1 and 0.999... because 0.999... IS 1. You're fundamentally asking how many numbers exist between 1 and 1. The answer is 0. They are the same number.

I get that people want there to be a number that is infinitely close to 1 but isn't 1, and that number does exist but it is not 0.999...
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refmon
06/02/17 2:26:27 PM
#53:


why stop there?

0.998...=1
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ChromaticAngel
06/02/17 2:28:03 PM
#54:


refmon posted...
why stop there?

0.998...=1

You didn't stop anywhere because you never got to the end of 0.999...
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teepan95
06/02/17 2:31:51 PM
#55:


Good topic
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Cheese_Crackers
06/02/17 9:08:47 PM
#56:


teepan95 posted...
Good topic

Glad it's well-received. I saw a lot of topics about it but never really saw a meaty discussion about the complex math behind repeating decimals, which is really the best proof of the result.
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G_G
06/02/17 9:58:42 PM
#57:


i dont get it though >_______>

why do we arbitrarily decide .999... is 1 but not .999...8 ?
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Phantom_Nook
06/02/17 9:59:03 PM
#58:


refmon posted...
why stop there?

0.998...=1

x = 0.998...
100x = 99.8...
1000x = 998.8...

900x = 899
x = 899/900
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Phantom_Nook
06/02/17 10:00:21 PM
#59:


G_G posted...
i dont get it though >_______>

why do we arbitrarily decide .999... is 1 but not .999...8 ?

The periods mean that the 9 is repeating forever. If you have some number of 9's followed by an 8, it's not an infinite sequence.
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G_G
06/02/17 10:01:49 PM
#60:


thats not what i mean

im not asking why .999... is 1 and .999...8 isn't, im asking what decides that .999... is 1 instead of .999... being .999...8

why do we round up
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Steelers
06/02/17 10:05:26 PM
#61:


No whole numbers exist between one and two, are they the same number
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G_G
06/02/17 10:06:00 PM
#62:


i dont know <_<

im not very good at math, i just dont understand where the line comes from
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Cheese_Crackers
06/02/17 10:26:23 PM
#63:


G_G posted...
thats not what i mean

im not asking why .999... is 1 and .999...8 isn't, im asking what decides that .999... is 1 instead of .999... being .999...8

why do we round up

What do you mean by .999...8? An infinite sequence of 9s followed by an 8? That doesn't exist because then the sequence of 9s isn't infinite.

The fact that .999...=1 isn't because of rounding up. It's true that .99999999 would be rounded up to 1 in most circumstances, but .999... is a different creature. It's the difference between a finite string of 9s and an infinite one.

The only way to talk about an infinite beast like .999... equaling anything is to consider what happens with an extremely long but finite string of 9s, and try to notice a value that these numbers become closer to. In this case, we would consider the following list:

.9
.99
.999
.9999
.99999
.999999
.9999999

...and so on. Maybe you can see that these values, with each iteration, are becoming arbitrarily close to 1. Of course no entry in this list will be 1, but if you want to consider the "limit" of the list - that is, an infinite string of 9s - then you need to consider what value the list entries are becoming arbitrarily close to.
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Mr_Legendia
06/02/17 10:27:30 PM
#64:


I'm trying.
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G_G
06/02/17 10:31:32 PM
#65:


Cheese_Crackers posted...
G_G posted...
thats not what i mean

im not asking why .999... is 1 and .999...8 isn't, im asking what decides that .999... is 1 instead of .999... being .999...8

why do we round up

What do you mean by .999...8? An infinite sequence of 9s followed by an 8? That doesn't exist because then the sequence of 9s isn't infinite.

The fact that .999...=1 isn't because of rounding up. It's true that .99999999 would be rounded up to 1 in most circumstances, but .999... is a different creature. It's the difference between a finite string of 9s and an infinite one.

The only way to talk about an infinite beast like .999... equaling anything is to consider what happens with an extremely long but finite string of 9s, and try to notice a value that these numbers become closer to. In this case, we would consider the following list:

.9
.99
.999
.9999
.99999
.999999
.9999999

...and so on. Maybe you can see that these values, with each iteration, are becoming arbitrarily close to 1. Of course no entry in this list will be 1, but if you want to consider the "limit" of the list - that is, an infinite string of 9s - then you need to consider what value the list entries are becoming arbitrarily close to.

that makes more sense. thanks.

sorry for being stupid but is 1.999... 2?
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Cheese_Crackers
06/03/17 12:07:11 AM
#66:


G_G posted...
Cheese_Crackers posted...
G_G posted...
thats not what i mean

im not asking why .999... is 1 and .999...8 isn't, im asking what decides that .999... is 1 instead of .999... being .999...8

why do we round up

What do you mean by .999...8? An infinite sequence of 9s followed by an 8? That doesn't exist because then the sequence of 9s isn't infinite.

The fact that .999...=1 isn't because of rounding up. It's true that .99999999 would be rounded up to 1 in most circumstances, but .999... is a different creature. It's the difference between a finite string of 9s and an infinite one.

The only way to talk about an infinite beast like .999... equaling anything is to consider what happens with an extremely long but finite string of 9s, and try to notice a value that these numbers become closer to. In this case, we would consider the following list:

.9
.99
.999
.9999
.99999
.999999
.9999999

...and so on. Maybe you can see that these values, with each iteration, are becoming arbitrarily close to 1. Of course no entry in this list will be 1, but if you want to consider the "limit" of the list - that is, an infinite string of 9s - then you need to consider what value the list entries are becoming arbitrarily close to.

that makes more sense. thanks.

sorry for being stupid but is 1.999... 2?

You're not being stupid at all. And yeah, 1.999... = 2. It works for any number; 2.34999... = 2.35

I was rewatching Total Drama Island recently and there's an episode where two teams are doing a race or something; the final score is 10 vs 9.999..., and the team with 10 was declared the victor. The correct ruling is that the scores are the same because 9.999... = 10, so it would be a draw.
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FaultyGourry
06/03/17 12:10:16 AM
#67:


And yet if you take .001% away from my sandwich it's still going to be missing that .001% and not magically going to be a full sandwich. Chew on that, math!
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Albel-Nox
06/03/17 12:19:05 AM
#68:


Hardest game yet. Drained, evaded #65/66 and any ninja for now.


CONTINUE

YES NO



.......................
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G_G
06/03/17 12:21:20 AM
#69:


Cheese_Crackers posted...
You're not being stupid at all. And yeah, 1.999... = 2. It works for any number; 2.34999... = 2.35

wait it applies for decimals too???
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Cheese_Crackers
06/03/17 12:24:36 AM
#70:


G_G posted...
Cheese_Crackers posted...
You're not being stupid at all. And yeah, 1.999... = 2. It works for any number; 2.34999... = 2.35

wait it applies for decimals too???

Yeah, for the same reason.

2.349
2.3499
2.34999
2.349999
2.3499999

They're getting arbitrarily close to a single value, namely 2.35

Or you could view it as 2.34999... = 2.34 + .00999... = 2.34 + .01 = 2.35, where we applied the same principle as my original post to conclude that .00999... = .01

Basically any infinitely repeating string of nines can be replaced by a single 1
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Mr_Legendia
06/03/17 6:25:37 AM
#71:


Lmao!!

.999... is more than 1 due to infinitesimal. On!the number line the .999... would go on to to infinity while 1 stays still.

That would explain 2.999... being 3, because it can't be less than 3.

Or am I off base here?
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Mr_Legendia
06/03/17 7:09:06 AM
#72:


Yea, Total Drama was on some BS.

Thanks for the sandwich, really helped.
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Cheese_Crackers
06/03/17 10:12:19 AM
#73:


Mr_Legendia posted...
Lmao!!

.999... is more than 1 due to infinitesimal. On!the number line the .999... would go on to to infinity while 1 stays still.

That would explain 2.999... being 3, because it can't be less than 3.

Or am I off base here?

I'm not sure which side you're taking here.

.999... is exactly equal to 1; infinitesimals aren't part of standard analysis (analysis is the branch of math which deals with continuous processes and limits). If you want to go into non-standard analysis such as hyperreals, then be my guest, but that is definitely not, well, standard. No one will know you're thinking in that way unless you say so.

You're right that 2.999... = 3, because 2.999... = 2 + .999... = 2 + 1 = 3
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ChromaticAngel
06/03/17 11:17:21 AM
#74:


G_G posted...
im not asking why .999... is 1 and .999...8 isn't, im asking what decides that .999... is 1 instead of .999... being .999...8

.999...8 is not a number that exists.

Mr_Legendia posted...
.999... is more than 1 due to infinitesimal.

An infinitesimal is a hyperreal number defined such that 1 / infinitesimal = infinity (or -infinity in the case of a negative infinitesimal). Your theory is nice and all, except 0.999... is a real rational number, not a hyperreal. Hence, 1 - infinitesimal is a nice way of expressing a number that is infinitely close to one but not one, however, that number is not 0.999... as once you add an infinitesimal into the mix, you no longer end up with a real rational number.
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scar the 1
06/03/17 1:01:09 PM
#75:


ChromaticAngel posted...
scar the 1 posted...
Nice to read about p-adic numbers. You don't see that very often. I guess one could say that hyperreals contain numbers that "kind of" lie in-between 1 and 0.999..., but hyperreals aren't so common either.


There is no number between 1 and 0.999... because 0.999... IS 1. You're fundamentally asking how many numbers exist between 1 and 1. The answer is 0. They are the same number.

I get that people want there to be a number that is infinitely close to 1 but isn't 1, and that number does exist but it is not 0.999...

Yes yes, I was being a bit imprecise, hence the "kind of". Point is that hyperreals include the kind of itsy bitsy number that a lot of people are talking about. Kind of like 0.000...1. DISCLAIMER: I again used the phrase kind of.
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ChromaticAngel
06/03/17 2:14:58 PM
#76:


scar the 1 posted...
Yes yes, I was being a bit imprecise, hence the "kind of". Point is that hyperreals include the kind of itsy bitsy number that a lot of people are talking about. Kind of like 0.000...1. DISCLAIMER: I again used the phrase kind of.


Sure, but an infinitesimal is not something you can write with Arabic numerals.
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Mr_Legendia
06/03/17 5:30:50 PM
#77:


I don't think the postulate .999... is 1 is true. But It really is a monster. Absolute infinity, which seems correct, contradicts the postulate of Addictive Inverse.

Please bare with me for I cannot translate all the notions or operands for the sets in OP

If .999... is 1, and 1 still stays 1, then what is infinity? I ask this because negation. It brought me the idea that Absolute Infinity can only be terminated by another Absolute Infinity. But if .999... is 1, why if I Inverse either number, either way (left or right), I can't negate anything if I switch the results around.

Basically, if .999... and 1 real numbers that are the same, why can't -.999... negate 1 or vice versa.

Honestly still not sure.
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ChromaticAngel
06/03/17 5:34:31 PM
#78:


Mr_Legendia posted...
I don't think the postulate .999... is 1 is true. But It really is a monster. Absolute infinity, which seems correct, contradicts the postulate of Addictive Inverse.

Please bare with me for I cannot translate all the notions or operands for the sets in OP

If .999... is 1, and 1 still stays 1, then what is infinity? I ask this because negation. It brought me the idea that Absolute Infinity can only be terminated by another Absolute Infinity. But if .999... is 1, why if I Inverse either number, either way (left or right), I can't negate anything if I switch the results around.

Basically, if .999... and 1 real numbers that are the same, why can't -.999... negate 1 or vice versa.

Honestly still not sure.

-0.999... = -1
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Mr_Legendia
06/03/17 6:11:02 PM
#79:


I don't wanna go to hell.
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Mr_Legendia
06/03/17 6:21:04 PM
#80:


I've cried so long.

Everyone is evil at this point, huh?
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Albel-Nox
06/03/17 6:32:53 PM
#81:


WAIT!

I did finally figure it out. LMAO. Sooooooo simple.

It's NOT and never ever will be.

That was cool too, I guess.

And PS. Calculus SUCKS!
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Mr_Legendia
06/04/17 12:51:08 PM
#82:


So that 0.1 is what that 999... can never get, I get it.

Well if Jon is practically a 1 and logically a 2, I'm the inverse of both.

Life is truly a game, and its mastery only requires attention.
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Hexagon
06/04/17 1:33:31 PM
#83:


Cheese_Crackers posted...

It depends on how pedantic you're feeling. Writing down 0.333... = 1/3 isn't a given statement, it requires its own proof.


Is long division considered a proof now? It's like saying 2 times 5 equals 10 requires proof. Sounds silly to me.
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TheFuzz3451
06/04/17 11:15:17 PM
#84:


Are people itt trolling or just stupid
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MutantJohn
06/05/17 3:33:42 PM
#85:


TheFuzz3451 posted...
Are people itt trolling or just stupid

Exactly. 0.999.... cannot equal 1
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tiornys
06/05/17 3:45:53 PM
#86:


MutantJohn posted...
Exactly. 0.999.... cannot equal 1

Why not?
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MutantJohn
06/05/17 3:46:28 PM
#87:


tiornys posted...
MutantJohn posted...
Exactly. 0.999.... cannot equal 1

Why not?

Oh, I was trolling lol
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