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Author Topic:   0.99999~ = 1 ?
Huntard
Member (Idle past 1532 days)
Posts: 2870
From: Limburg, The Netherlands
Joined: 09-02-2008


Message 1 of 237 (543076)
01-15-2010 8:19 AM


I've seen this thing come along a couple of times on the internet, and was curious what our local math guys think about this.

It is my understanding that 0.9999~ does indeed equal 1.

I could be wrong though.

So, any thoughts on this?


I hunt for the truth

I am the one Orgasmatron, the outstretched grasping hand
My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
My name is called religion, sadistic, sacred whore.
-Lyrics by Lemmy Kilmister of Motorhead


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cavediver
Member (Idle past 2880 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 2 of 237 (543079)
01-15-2010 8:42 AM
Reply to: Message 1 by Huntard
01-15-2010 8:19 AM


It is my understanding that 0.9999~ does indeed equal 1.

Yes, indeed it does... unless you happen to be a constructivist, that is!


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Briterican
Member (Idle past 3186 days)
Posts: 340
Joined: 05-29-2008


Message 3 of 237 (543080)
01-15-2010 8:49 AM
Reply to: Message 2 by cavediver
01-15-2010 8:42 AM


Hmm, equal?
cavediver writes:

Huntard writes:

It is my understanding that 0.9999~ does indeed equal 1.

Yes, indeed it does... unless you happen to be a constructivist, that is!

Shouldn't there be some sort of conditional involved or something? I don't see how it could be considered accurate to say that 0.99999~ is precisely equivalent to 1.

I appreciate that, for all intents and purposes, they are equivalent. But mathematically they simply are not, right?

Maybe I'm missing something.


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Son Goku
Member
Posts: 1207
From: Ireland
Joined: 07-16-2005
Member Rating: 5.2


Message 4 of 237 (543082)
01-15-2010 9:00 AM
Reply to: Message 1 by Huntard
01-15-2010 8:19 AM


Totally right!
Yes, you are correct. 0.9999..... = 1. Equals exactly.
Although it looks confusing, it can be broken apart. It's easy to see what 0.99 means. It's 9/10 + 9/100, which is clearly not 1. However when you see an expression like 0.9999...., this is the infinite sum:
9/10 + 9/100 + 9/1000 + 9/10000 + .....................
So the real question is, what are infinite sums?

I can add more detail if people want, but under the conventional definition given to an infinite sum, then this infinite sum equals one.

Here is a proof:
1. One of the axioms of real numbers: Between any two distinct real numbers there is a third real number not equal to either. (The density property)

2. Adding two numbers and dividing by two, results in the number half way between them. By (1), if they are distinct, this should be a new number.

3. 0.999999..... + 1 = 1.999999999.....

4. (1.99999.....)/(2) = 0.99999....., you can check this with long division.

5. Hence, the number halfway between 1 and 0.9999.... is 0.99999....

6. By (2) this is in contradiction with (1), hence

7. 1 and 0.999.... are not distinct and are the same number.

QED

What this actually means is that there are two different ways to represent the real number 1.


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Huntard
Member (Idle past 1532 days)
Posts: 2870
From: Limburg, The Netherlands
Joined: 09-02-2008


Message 5 of 237 (543083)
01-15-2010 9:04 AM
Reply to: Message 3 by Briterican
01-15-2010 8:49 AM


Re: Hmm, equal?
Briterican writes:

Shouldn't there be some sort of conditional involved or something? I don't see how it could be considered accurate to say that 0.99999~ is precisely equivalent to 1.

I appreciate that, for all intents and purposes, they are equivalent. But mathematically they simply are not, right?

Maybe I'm missing something.


I guess you are then, because they are.

If you don't think they are equal, try answering this question: "What is the difference between 1 and 0.9999~." If they are not equal, there should be a difference, no?

Also try this explanation

X = 0.9999~
10X = 9.9999~
10X - X = 9
9X = 9
X = 1


I hunt for the truth

I am the one Orgasmatron, the outstretched grasping hand
My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
My name is called religion, sadistic, sacred whore.
-Lyrics by Lemmy Kilmister of Motorhead


This message is a reply to:
 Message 3 by Briterican, posted 01-15-2010 8:49 AM Briterican has seen this message

  
LucyTheApe
Inactive Member


Message 6 of 237 (543084)
01-15-2010 9:15 AM
Reply to: Message 1 by Huntard
01-15-2010 8:19 AM


Divisions
x = 0.99...
10x = 9.99..
9x=9
x=1


There no doubt exist natural laws, but once this fine reason of ours was corrupted, it corrupted everything.

blɛz paskal


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New Cat's Eye
Inactive Member


Message 7 of 237 (543087)
01-15-2010 10:01 AM


another simply way
1/3 = 0.333~
2/3 = 0.666~
3/3 = 0.999~

  
Dr Jack
Member (Idle past 1342 days)
Posts: 3507
From: Leicester, England
Joined: 07-14-2003


Message 8 of 237 (543094)
01-15-2010 10:25 AM
Reply to: Message 3 by Briterican
01-15-2010 8:49 AM


Re: Hmm, equal?
No, mathematically they're precisely equivalent - they are, in fact, different ways of writing the same thing. What's more every terminating decimal has two ways of being written, i.e.

0.5214 = 0.521399999~
7.2 = 7.199999~

And so on.


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Briterican
Member (Idle past 3186 days)
Posts: 340
Joined: 05-29-2008


Message 9 of 237 (543100)
01-15-2010 11:10 AM
Reply to: Message 8 by Dr Jack
01-15-2010 10:25 AM


Re: Hmm, equal?
Yeah I get it now.

My brain doesn't want to cooperate, but thanks to the specific examples given by you guys, I understand now.

Well... I am 99.99999~% sure that I understand anyway


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Shield
Member (Idle past 2099 days)
Posts: 482
Joined: 01-29-2008


Message 10 of 237 (543145)
01-15-2010 5:07 PM


The paradox
I am intoxicated right now by an psychoactive phenethylamine but i will attempt to respond. English is not my first language so bare with me.

0.9999~ will always equal 1 in calculations. Think of the infinity - hotel analogy.

I removed a lot of babbling from this post, seeing as i am too intoxicated to do a proper reply right now, ill will post more on these following days.


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New Cat's Eye
Inactive Member


Message 11 of 237 (543146)
01-15-2010 5:19 PM
Reply to: Message 10 by Shield
01-15-2010 5:07 PM


Re: The paradox
2C-I?

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RAZD
Member (Idle past 642 days)
Posts: 20714
From: the other end of the sidewalk
Joined: 03-14-2004


Message 12 of 237 (543149)
01-15-2010 5:37 PM
Reply to: Message 9 by Briterican
01-15-2010 11:10 AM


Re: Hmm, equal?
Hi Briterican,

I'm with Catholic Scientist on the simple approach, I just look at it a slightly different way:

(1/3)x3 == 1

1/3 = 0.3 + 0.1/3
= 0.3 + 0.03 + 0.001/3
= 0.3 + 0.03 + 0.003 + ... + 3x10^-n + 1x10^-n/3
= 0.33333333~ and there is always a remainder of 1 to divide by 3 wherever you stop.

Now multiply that by 3

(1/3)x3 = (0.33333333~)x3 = 0.99999999~ == 1

This is really a product of using base 10, not anything magic with the math.

If we used base 9 then 1/3 would be 0.3:

  0.3
3\1.0
1.0
0

However 1/2 would be 0.44444~

  0.444 ...
2\1.000
0.8
0.10
0.08
0.010
0.008
0.001
etc.

My personal favorite repeating decimal fraction is 7ths:

1/7 = 0.142857142857~ (where the underlined portion repeats)
2/7 = 0.285714285714~
3/7 = 0.428571428571~
4/7 = 0.571428571428~
5/7 = 0.714285714285~
6/7 = 0.857142857142~

Now 1/7 + 6/7 = 0.999999~ == 1
and 2/7 + 5/7 = 0.999999~ == 1
and 3/7 + 4/7 = 0.999999~ == 1

... now if we used base 7 ...

Enjoy.


we are limited in our ability to understand
by our ability to understand
Rebel American Zen Deist
... to learn ... to think ... to live ... to laugh ...
to share.


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PaulK
Member
Posts: 17167
Joined: 01-10-2003
Member Rating: 3.7


Message 13 of 237 (543151)
01-15-2010 5:51 PM
Reply to: Message 9 by Briterican
01-15-2010 11:10 AM


Re: Hmm, equal?
Son Goku's explanation is the one that deals best with an actual mathematical understanding of what is going on.

When I got on to my degree course, one of the first things we were taught was a new definition of equality. It's not the same as Son Goku's point, but it is very closely related.

That is, two numbers a and b are equal if there is no number, e such that
0 < e <|a - b|

(Which is the same as saying that there is no number between them, if you think about it).

In fact we needed this definition to deal with the sums of infinite sequences. And if we couldn't do that, integration wouldn't work (and we'd need to find a physical solution to Zeno's paradox, too !) So it is important to mathematics that 1 and 0.999... are the same number. If they were different we'd have real problems.

Edited by PaulK, : Corrected after prompting from Dr. Adequate.


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Briterican
Member (Idle past 3186 days)
Posts: 340
Joined: 05-29-2008


Message 14 of 237 (543155)
01-15-2010 6:14 PM
Reply to: Message 12 by RAZD
01-15-2010 5:37 PM


Equivalence for repeating decimals!
Hi RAZD

As always, nice presentation.

RAZD writes:

My personal favorite repeating decimal fraction is 7ths

These sorts of things are fascinating. Even the simple graphic representation you provided using a yellow font to show the repeating pattern is fascinating. (My favourite example of this kind of thing is the Mandlebrot Set, which has nothing to do with 1=0.999~ but is an example of visualised mathematics).

Even wiki has a large page regarding .999999~

I wonder what other sort of mathematics concepts I'm underinformed on hehe.


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Briterican
Member (Idle past 3186 days)
Posts: 340
Joined: 05-29-2008


Message 15 of 237 (543156)
01-15-2010 6:17 PM
Reply to: Message 13 by PaulK
01-15-2010 5:51 PM


Re: Hmm, equal?
PaulK writes:

That is, two numbers a and b are equal if there is no number, e such that
|a - b| < e

(Which is the same as saying that there is no number between them, if you think about it).

This illuminates the matter even further.

I thank you, and Huntard for the OP as well.


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