Is 0.9 recurring equal to 1?

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CheeseLord
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Is 0.9 recurring equal to 1?

Post by CheeseLord » July 1st, 2010, 7:52 pm

What do you think?? This is a question I debated with my parents :D
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Re: Is 0.9 recurring equal to 1?

Post by HomerJ » July 1st, 2010, 8:01 pm

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


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Re: Is 0.9 recurring equal to 1?

Post by Hulavuta » July 1st, 2010, 8:04 pm

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.
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Re: Is 0.9 recurring equal to 1?

Post by CheeseLord » July 1st, 2010, 8:07 pm

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

Post by Joram » July 1st, 2010, 8:12 pm

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.
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Re: Is 0.9 recurring equal to 1?

Post by PeterPorty » July 1st, 2010, 8:18 pm

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.

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Re: Is 0.9 recurring equal to 1?

Post by CheeseLord » July 1st, 2010, 8:23 pm

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.
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Re: Is 0.9 recurring equal to 1?

Post by Pentarctagon » July 1st, 2010, 8:28 pm

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).
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Re: Is 0.9 recurring equal to 1?

Post by Hulavuta » July 1st, 2010, 8:31 pm

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)
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Re: Is 0.9 recurring equal to 1?

Post by Joram » July 1st, 2010, 8:35 pm

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.
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Re: Is 0.9 recurring equal to 1?

Post by Hulavuta » July 1st, 2010, 8:37 pm

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?
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Re: Is 0.9 recurring equal to 1?

Post by CheeseLord » July 1st, 2010, 8:40 pm

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.
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Re: Is 0.9 recurring equal to 1?

Post by PeterPorty » July 1st, 2010, 8:40 pm

Hula. Nope. read what Pent said. anyway, it is the same...
http://en.wikipedia.org/wiki/0.999...#Algebraic_proofs
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Re: Is 0.9 recurring equal to 1?

Post by Alink » July 1st, 2010, 8:41 pm

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"?

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Re: Is 0.9 recurring equal to 1?

Post by Joram » July 1st, 2010, 8:45 pm

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. :)
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