## Is 0.9 recurring equal to 1?

The place for chatting and discussing subjects unrelated to Wesnoth.

Moderators: Forum Moderators, Developers

CheeseLord
Posts: 147
Joined: February 19th, 2009, 11:07 am
Location: Knagla

### Is 0.9 recurring equal to 1?

What do you think?? This is a question I debated with my parents
Lord of Cheese.....especially Parmesan!!!
None shall stand against the Dwarfs in their homeland

Top 3 quotes by a developer:

1. Locked
2. Do it yourself
3. "No"
HomerJ
Posts: 812
Joined: April 25th, 2008, 1:22 pm
Location: Hannover, Germany

### Re: Is 0.9 recurring equal to 1?

That is a good question.
Also: Did JW leave his peanut butter in here?

Greetz
HomerJ
Six years without a signature!
Hulavuta
Posts: 1668
Joined: October 11th, 2008, 8:17 pm
Location: United States

### Re: Is 0.9 recurring equal to 1?

No.

At least in my opinion. (Which isn't possible because this is a fact question)

Because, lets say we had .99999 right? so plus .00001 would make it 1. right? So needing that .00001 makes it so that repeating it does not equal 1. Plus, if the answer is yes, and it was recurring forever, it would go more than one.

I got a C in Geometry in the 2009-2010 school year, so quite possibly I'm wrong. Gambit might know, he teaches Math.
F:tGJ, Saurian Campaign
The Southern Chains, a fanfic
“The difference between winners and champions is that champions are more consistent."
~Sierra
CheeseLord
Posts: 147
Joined: February 19th, 2009, 11:07 am
Location: Knagla

### Re: Is 0.9 recurring equal to 1?

The thing is, this question pits two mathematically accurate facts against each other, so it depends on which one is true

1/3 +2/3 = 1 ...
but 0.9 recurring does not equal 1 ...

and HomerJ: Am I missing a joke here ??
Lord of Cheese.....especially Parmesan!!!
None shall stand against the Dwarfs in their homeland

Top 3 quotes by a developer:

1. Locked
2. Do it yourself
3. "No"
Joram
Posts: 366
Joined: September 2nd, 2008, 5:36 am

### Re: Is 0.9 recurring equal to 1?

The answer is yes, and it can be proven.

Take a=0.9999999...
Take b=1

10*a = 9.999999...
9*b = 9
9*b + a = 9.9999999...

Therefore: 10*a = 9*b + a
Implies: 9*a = 9*b
Implies: a = b

That is just one of the proofs, I believe there are more.
The Fires of Pride 0.3, a heavily story based campaign.
On hold while I try and finish my book
PeterPorty
Translator
Posts: 310
Joined: January 12th, 2010, 2:25 am
Location: Chair, In-Front-Of-Computer

### Re: Is 0.9 recurring equal to 1?

0.9 recurring is in fact equal to 1. if you turn the decimal number to a fraction, you get 9/9, and 9/9=1/1 and 1/1=1. Therefore, yes, 9 recurring=1.

damn, ninja'd, but yeah....
"The real world is for people who can't imagine anything better."
CheeseLord
Posts: 147
Joined: February 19th, 2009, 11:07 am
Location: Knagla

### Re: Is 0.9 recurring equal to 1?

Those mathematical proofs are very nice ones . BUT, I don't see how it detracts from the fact that

0.9999....... DOESN'T equal 1 Like Hulavuta, You have to add 0.00...01 somewhere along the line for it to become 1

Like 0.49 recurring is not equal to 0.5

At least, this is what most lower school maths teachers tell us.
Lord of Cheese.....especially Parmesan!!!
None shall stand against the Dwarfs in their homeland

Top 3 quotes by a developer:

1. Locked
2. Do it yourself
3. "No"
Pentarctagon
Project Manager
Posts: 4494
Joined: March 22nd, 2009, 10:50 pm
Location: Earth (occasionally)

### Re: Is 0.9 recurring equal to 1?

1/3 = 0.3 repeating. 1/3 * 3 = 1 while 0.3 repeating * 3 =0.9 repeating. therefore 0.9 repeating = 1. on the other hand, that 0.00.....01 shouldn't be able to magically materialize out of nowhere. its probably just one of those things where it is mathematically possible but logically not possible (at least that's how I see it).
99 little bugs in the code, 99 little bugs
take one down, patch it around
-2,147,483,648 little bugs in the code
Hulavuta
Posts: 1668
Joined: October 11th, 2008, 8:17 pm
Location: United States

### Re: Is 0.9 recurring equal to 1?

PeterPorty: Fractions and Decimals both exist for a reason. 2/3 isn't exactly .6 repeating, because if 6 is two(2/3), then that means that 1 is 3, and 6+3 is 9, not 10.

So therefore .9 repeating might equal 9/9, but decimals can have tiny differences that fractions cover up. (Like the example above)
F:tGJ, Saurian Campaign
The Southern Chains, a fanfic
“The difference between winners and champions is that champions are more consistent."
~Sierra
Joram
Posts: 366
Joined: September 2nd, 2008, 5:36 am

### Re: Is 0.9 recurring equal to 1?

Those mathematical proofs are very nice ones . BUT, I don't see how it detracts from the fact that

0.9999....... DOESN'T equal 1 Like Hulavuta, You have to add 0.00...01 somewhere along the line for it to become 1
Where you are making your mistake is that the nines go on infinitely. There is no end to them, so if you add 0.00...01, you will end up with 1.00...00999...

Because of this, there is no number in existence that can be added to 0.999... that will result in 1 (other than 0)
Last edited by Joram on July 1st, 2010, 8:41 pm, edited 2 times in total.
The Fires of Pride 0.3, a heavily story based campaign.
On hold while I try and finish my book
Hulavuta
Posts: 1668
Joined: October 11th, 2008, 8:17 pm
Location: United States

### Re: Is 0.9 recurring equal to 1?

I'm assuming he meant that it would stop at one point, just far in. Does your proof only work with the only infinite 9's or does it also work with the nines stopping somewhere?
F:tGJ, Saurian Campaign
The Southern Chains, a fanfic
“The difference between winners and champions is that champions are more consistent."
~Sierra
CheeseLord
Posts: 147
Joined: February 19th, 2009, 11:07 am
Location: Knagla

### Re: Is 0.9 recurring equal to 1?

Joram wrote: Because of this, there is no number in existence that can be added to 0.999... that will result in 1.
but that doesn't mean that 0.999... is one, just like 0.49 recurring isn't 0.5

Hulavuta: I'm talking about infinite 9s BTW
Last edited by CheeseLord on July 1st, 2010, 8:41 pm, edited 1 time in total.
Lord of Cheese.....especially Parmesan!!!
None shall stand against the Dwarfs in their homeland

Top 3 quotes by a developer:

1. Locked
2. Do it yourself
3. "No"
PeterPorty
Translator
Posts: 310
Joined: January 12th, 2010, 2:25 am
Location: Chair, In-Front-Of-Computer

### Re: Is 0.9 recurring equal to 1?

Hula. Nope. read what Pent said. anyway, it is the same...
http://en.wikipedia.org/wiki/0.999...#Algebraic_proofs
"The real world is for people who can't imagine anything better."
Inactive Developer
Posts: 181
Joined: March 5th, 2007, 6:45 am
Location: Belgium

### Re: Is 0.9 recurring equal to 1?

Yes 0.99999... equals 1. The confusion only comes from a misunderstanding of number's representation. If you take a ruler and agree that any decimal number between 0 and 1 can be placed somewhere between the 0 and 1 marks. And, similarly, agree that any point between those marks has a decimal representation (possibly infinite like 0.9999...). Then 0.9999.. should be there too, but where? If we assume that it's not on 1, then there is a little space between 0.999.. and 1, and points between them can be represented by numbers. But how do you write a number bigger than 0.9999.. and smaller than 1? It's impossible. Thus, our initial assumption (0.999.. is not on 1) was false, and they are, in fact equals.

You must accept that one number (that is any point on the ruler) may have different numeric representations. Your intuition mislead you because when there is not an infinite number of decimals, things seems simpler. But you already accepted that 0.000... equals 0, why not accept that 0.999... equals 1. If there is a rule like "infinite sequence of 0 can be rounded to the lower integer", why not have a rule like "infinite sequence of 9 can be rounded to the upper integer"?
Joram
Posts: 366
Joined: September 2nd, 2008, 5:36 am

### Re: Is 0.9 recurring equal to 1?

To put what Alink was saying a little differently (as in, saying the same thing with different word choice), suppose you have two different positions on a ruler.

No matter how close you put them, there will always be some space in between (ignoring for the moment the physical theories that there is a "minimum distance"). You will be able to represent a point on that distance as a decimal or fraction. However, there are no points in between 0.999... and 1. Therefore, since there are no points between them, they must be the same number.

The reason I do this is because it is a very good point.
The Fires of Pride 0.3, a heavily story based campaign.
On hold while I try and finish my book