1 and .9999999... the same number?

[IWntErinNess]

do the numbers 1 and .99999... [infinitely repeating] have the same value?

i would say yes. if the answer was no, then one could find the difference between these two numbers... which i don't think exists.

for instance, draw a number line... plot 1 on it. now plot say .9 on it... and .99. now try to put .9999... infinitely repeating on it. it doesn't fit anywhere.

[pokerbobo]

But our amps go to eleven....see its one louder.

[Silent A]

Ah, the endless debate. I've seen 30 page threads on this silly topic.

I've always been of the "same thing" camp because:

1 - 0.99999... = 0.000000..... = 0

or, better yet:

x = 0.99999999...
10x = 9.99999999...
10x - x = 9.9999999... - 0.9999999...
9x = 9.0
x = 1

0.9999999... = 1

[Neuge]

Yes, 1 and .99999... are the same number. A simple, but not a "true", proof is:

1/9 = .11111...
2/9 = .22222...
.
.
.
8/9 = .88888...
9/9 = .99999... = 1

A real proof would be:

lim x->inf ( sum{n=1->x} 9/10^n ) = lim x->inf ( .9(1-10^-(x+1))/(1-1/10) ) = lim x->inf ( .9(1-10^-(x+1))/(9/10) ) = .9/(9/10) = 1

[MidGe]

[ QUOTE ]
Ah, the endless debate. I've seen 30 page threads on this silly topic.

[/ QUOTE ]

Yes, that is true there has been many threads on this same topic. Unfortunately it is very hard to do a search for "1" and ".999...". So, I guess we are bound to see this topic come up again and again, every time that someone discover the meaning of .999... /images/graemlins/smile.gif

[vhawk01]

[ QUOTE ]
[ QUOTE ]
Ah, the endless debate. I've seen 30 page threads on this silly topic.

[/ QUOTE ]

Yes, that is true there has been many threads on this same topic. Unfortunately it is very hard to do a search for "1" and ".999...". So, I guess we are bound to see this topic come up again and again, every time that someone discover the meaning of .999... /images/graemlins/smile.gif

[/ QUOTE ]

Save us some time OP and just google it or something. Its a fun intuition problem, I made a big deal of it in like 7th grade math class or something, long story short, yes, they are the same. There are a few rigorous proofs and some less so but more helpful ones out there.

[MrMon]

Just think of the difference between the two as the smallest number possible without actually being zero.

The number is actually pretty useful, as it's essentially the basis of calculus. Remember all those tiny little boxes approaching an infinitely small width? That's this number.

[godBoy]

My first thought on this one was..

.99 [infinitely repeating] isn't a real number.
.99 is real, .999 is real - but infinitely repeating ?

I suppose all maths is purely conceptual - so nothing in maths is real - and if .999 repeating is proved useful then it's as real as 1.

I wouldn't call them the same number though.

[vhawk01]

[ QUOTE ]
My first thought on this one was..

.99 [infinitely repeating] isn't a real number.
.99 is real, .999 is real - but infinitely repeating ?

I suppose all maths is purely conceptual - so nothing in maths is real - and if .999 repeating is proved useful then it's as real as 1.

I wouldn't call them the same number though.

[/ QUOTE ]

Then you'd be wrong.

[AndysDaddy]

[ QUOTE ]
My first thought on this one was..

.99 [infinitely repeating] isn't a real number.

[/ QUOTE ]

Yes it is. For convenience we just use one of its other names instead:

1

[PairTheBoard]

[ QUOTE ]
do the numbers 1 and .99999... [infinitely repeating] have the same value?


[/ QUOTE ]

I suppose it depends on what you mean by the word "value". For example, if the Smiths believe in 1 and the Jones believe in .99999999... do they have the same family Value?

Or, if by "Value" you mean "Fruit", do 1 and .99999... taste the same?

If by "Value" you mean "Form" or "Representation" then they certainly look different.

If by "Value" you mean "Number" then what do you mean by "Number"? It looks like they might smoke differently. .999999... looks like a pretty fat doobie.

If you're talking about a mathematical "number" system, which Number System are you talking about? The Cardinal Number system? The Infitesimal Number System? The Real Number System constructed from the rationals by way of Dedekind Cuts?

Then of course you also have to decide on what the meaning of the word "is" is.

PairTheBoard

[CORed]

In mathematical terms, infinitely repeating decimals most certainly are real numbers, as are irrational (infinite non-repeating decimals like pi, e, the square root of two, etc.) Many common fractions, like 1/3, 1/9, 1/7, can only be represented by infinitely repeating decimals.

That 0.9999...... and 1 are equivalent is a proven mathematical fact. You can disagree with it, but you are just as wrong as if you decided that pi is equal to 3.

[thylacine]

[ QUOTE ]
do the numbers 1 and .99999... [infinitely repeating] have the same value?

i would say yes. if the answer was no, then one could find the difference between these two numbers... which i don't think exists.

for instance, draw a number line... plot 1 on it. now plot say .9 on it... and .99. now try to put .9999... infinitely repeating on it. it doesn't fit anywhere.

[/ QUOTE ]

Deja Vu

In the real numbers they are equal (i.e. exactly the same number).

In the surreal numbers they are not equal (i.e. they are two different numbers).

Okay. Discussion closed. The probability I am right on this is 1.

[jason1990]

[ QUOTE ]
Okay. Discussion closed. The probability I am right on this is 1.

[/ QUOTE ]
So you are only right "almost surely." Does this mean the discussion is still open?

[Matt R.]

oh [censored]

[thylacine]

[ QUOTE ]
[ QUOTE ]
Okay. Discussion closed. The probability I am right on this is 1.

[/ QUOTE ]
So you are only right "almost surely." Does this mean the discussion is still open?

[/ QUOTE ]

/images/graemlins/wink.gif

[CORed]

Could you tell me a little more about these surreal numbers of which you speak?

Are they used to estimate the cost of government projects, perhaps?

[thylacine]

[ QUOTE ]
Could you tell me a little more about these surreal numbers of which you speak?

Are they used to estimate the cost of government projects, perhaps?

[/ QUOTE ]

http://en.wikipedia.org/wiki/Surreal_number

http://en.wikipedia.org/wiki/Real_number

enjoy

[godBoy]

[ QUOTE ]
In mathematical terms, infinitely repeating decimals most certainly are real numbers, as are irrational (infinite non-repeating decimals like pi, e, the square root of two, etc.) Many common fractions, like 1/3, 1/9, 1/7, can only be represented by infinitely repeating decimals.

[/ QUOTE ]
I'm not a maths buff - i'm just saying what seems logical.

I would say that a fraction like 1/3 can not be represented perfectly by a decimal point number. But 1/3 is a perfect representation of the value.

[ QUOTE ]
That 0.9999...... and 1 are equivalent is a proven mathematical fact. You can disagree with it, but you are just as wrong as if you decided that pi is equal to 3.

[/ QUOTE ]
That's most likely.

[vhawk01]

[ QUOTE ]
[ QUOTE ]
In mathematical terms, infinitely repeating decimals most certainly are real numbers, as are irrational (infinite non-repeating decimals like pi, e, the square root of two, etc.) Many common fractions, like 1/3, 1/9, 1/7, can only be represented by infinitely repeating decimals.

[/ QUOTE ]
I'm not a maths buff - i'm just saying what seems logical.

I would say that a fraction like 1/3 can not be represented perfectly by a decimal point number. But 1/3 is a perfect representation of the value.

[ QUOTE ]
That 0.9999...... and 1 are equivalent is a proven mathematical fact. You can disagree with it, but you are just as wrong as if you decided that pi is equal to 3.

[/ QUOTE ]
That's most likely.

[/ QUOTE ]

Words and numbers are just symbols that try to represent different ideas. No reason that 0.9999... is any less of a perfect representation than 1/3...in fact, they mean the same thing! I suppose we could use HGM to represent this quantity as well, if we all agreed to, but we basically have 1/3, 2/6, 0.99999..., and a whole bunch of other symbols to represent it.

[doucy]

Here's one way I've tried to explain this in the past:

Given- If there is no number in between x and y, then that means x and y are the same number.

So let's say, hypothetically, that there was a rule in math that you are only allowed to go one place value past the decimal point (eg .1, .2, .3, etc). If that were true, then .9 and 1 would be the same number, since there is nothing between them.

Now let's say the rule is that you are only allowed to go 3 places past the decimal. If that were the case, then .999 and 1 would be the same number, since there would be nothing between them.

Now let's say you are only allowed to go infinity places past the decimal. If that is the case, then .9999-infinity would be the same as 1, since there is nothing between them. Oh wait, that is the rule. I guess they're equal then.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
Could you tell me a little more about these surreal numbers of which you speak?

Are they used to estimate the cost of government projects, perhaps?

[/ QUOTE ]

http://en.wikipedia.org/wiki/Surreal_number

http://en.wikipedia.org/wiki/Real_number

enjoy

[/ QUOTE ]

So .999999... and 1 are not equal as Surreal Numbers.

Using the notation of the Link,

.99999... = {.9,.99,.999,... | 1 }
and
1 = {0 | Nullset }

So .999... <= 1 because
1 is Not <= any element in {.9,.99,.999,...} and
no member of the Null set is <= to .999...

But it's not the case that 1<= .999... because although
.999... is Not <= 0
1 IS <= 1 which violates the condition required.

So as Surreal Numbers .999... is strictly less than 1. They differ by an infinitesimal.

However this may not be as satisfying to the Not-Equal People as they might think. .999... and 1 differ by an infinitesimal but not all infinitesimals are equal in the Surreals, if I understand correctly. The Binary Representation 0.111111 also differs from 1 by an infinitesimal but it may be a different infinitesimal than (1 - 0.999...) so that 0.999... and Binary 0.111... are also different Surreal numbers.

There are an awful lot of Surreal Numbers. There are so many of them they are not even a Set. They are a Class.

My take on Surreal Numbers?

Class Dismissed.

PairTheBoard

[fluorescenthippo]

.333333333... = 1/3

.33333333.. + .3333333333... + .3333333333.... = .9999999999 = 1

1/3 + 1/3 + 1/3 = 3/3 = 1

[vhawk01]

[ QUOTE ]
.333333333... = 1/3

.33333333.. + .3333333333... + .3333333333.... = .9999999999 = 1

1/3 + 1/3 + 1/3 = 3/3 = 1

[/ QUOTE ]

If we believed .333333.... = 1/3 we wouldn't have this thread to begin with!

[StepBangin]

[ QUOTE ]
Given- If there is no number in between x and y, then that means x and y are the same number...........

Now let's say the rule is that you are only allowed to go 3 places past the decimal. If that were the case, then .999 and 1 would be the same number, since there would be nothing between them.


[/ QUOTE ]

I have no math skills at all, but isnt this like saying

.991 = .992 ?

But then since .991 is the same as .992 and .992 is the same as .993 .991 should be the same as .993 but isnt since there is a number in between.

Please explain so I know why I am wrong /images/graemlins/smile.gif

[doucy]

[ QUOTE ]
[ QUOTE ]
Given- If there is no number in between x and y, then that means x and y are the same number...........

Now let's say the rule is that you are only allowed to go 3 places past the decimal. If that were the case, then .999 and 1 would be the same number, since there would be nothing between them.


[/ QUOTE ]

I have no math skills at all, but isnt this like saying

.991 = .992 ?

But then since .991 is the same as .992 and .992 is the same as .993 .991 should be the same as .993 but isnt since there is a number in between.

Please explain so I know why I am wrong /images/graemlins/smile.gif

[/ QUOTE ]

Yeah, you are right that using the logic of my post, every number is equal to every other number. I just made up that idiotic proof to try to convince a mathematically challenged person that .9repeating and 1 were equal. Luckily for me, that person was not as smart as you.

[StepBangin]

LOLISMART

[vhawk01]

[ QUOTE ]
[ QUOTE ]
Given- If there is no number in between x and y, then that means x and y are the same number...........

Now let's say the rule is that you are only allowed to go 3 places past the decimal. If that were the case, then .999 and 1 would be the same number, since there would be nothing between them.


[/ QUOTE ]

I have no math skills at all, but isnt this like saying

.991 = .992 ?

But then since .991 is the same as .992 and .992 is the same as .993 .991 should be the same as .993 but isnt since there is a number in between.

Please explain so I know why I am wrong /images/graemlins/smile.gif

[/ QUOTE ]

No, his example was really just talking about significant figures. In a world with only 2 significant figures, .991 and .992 really ARE the same number. .991 and 1.00 aren't. We live in a world with only infinite significant figures, so .999.... and 1 are the same.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
Given- If there is no number in between x and y, then that means x and y are the same number...........

Now let's say the rule is that you are only allowed to go 3 places past the decimal. If that were the case, then .999 and 1 would be the same number, since there would be nothing between them.


[/ QUOTE ]

I have no math skills at all, but isnt this like saying

.991 = .992 ?

But then since .991 is the same as .992 and .992 is the same as .993 .991 should be the same as .993 but isnt since there is a number in between.

Please explain so I know why I am wrong /images/graemlins/smile.gif

[/ QUOTE ]

You're right. He was ok until he started talking about only being able to go 3 places past the decimal. What he say there is like saying, As Integers 1 and 2 are the same because there is no Integer between them. His way of determining if two numbers are equal within a number system only works for number systems like the rationals or reals that are dense.

PairTheBoard

[Wyman]

Let me give this a whirl.

When I think of a number written in decimal, I think of it as the limit of a sequence. For instance,

0.33 = lim (x_n),
n-->infinity

where
x_1 = 0.3
x_2 = 0.33
x_3 = 0.330
x_4 = 0.3300
...

And the limit is clearly 33/100.

Similarly, 0.9999... is equal to

lim (x_n),
n-->infty

where
x_1 = 0.9
x_2 = 0.99
x_3 = 0.999
...

And this limit is clearly 1.

Remember that a "number" or a "value" is independent of the system/method that we choose to represent the number.

[Johnny Drama]

what if I have .999... airplanes on 1 conveyor belt? does it fly?

[godBoy]

LOL

That's fantastic!

[MidGe]

If you only take away (1 - .999...) part from it, it will fly, no problems!

[jogsxyz]

While playing fixed limit poker, .9 and 1 is the same number.

[Matt R.]

[ QUOTE ]
what if I have .999... airplanes on 1 conveyor belt? does it fly?

[/ QUOTE ]

awesome