## Is 0.9 recurring equal to 1?

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

Extract from GCSE maths paper:

The length of a pen is 14cm to the nearest cm
What is the maximum possible length of the pen?

A:15cm B:14.99 C:14.9 D:14.45 E:14.5

I went for E, but I wrote on the side (isn't it 14.49 recurring)
The teacher replied: Yes but they call it (virtually) 14.5

Isn't that not quite 14.5, just assumed for the sake of convenience??

If the majority of you guys are right, then doesn't that mean that every single textbook in the UK is wrong??
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Hulavuta
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### Re: Is 0.9 recurring equal to 1?

Oooooohhh...Now I see. Kind of.
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Joram
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### Re: Is 0.9 recurring equal to 1?

CheeseLord wrote:If the majority of you guys are right, then doesn't that mean that every single textbook in the UK is wrong??
If the majority of UK textbooks say that 14.49999... (on off to infinity) does not quite equal 14.5, then yes.
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Captain_Wrathbow
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### Re: Is 0.9 recurring equal to 1?

Hulavuta wrote:Gambit might know, he teaches Math.
He does? Aw, I wish he was my math teacher....
Sapient
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### Re: Is 0.9 recurring equal to 1?

I have a very simple proof that 0.9 recurring is not the same as 1

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

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

Captain_Wrathbow wrote:
Hulavuta wrote:Gambit might know, he teaches Math.
He does? Aw, I wish he was my math teacher....

He teaches online, so he could be.
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AI
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### Re: Is 0.9 recurring equal to 1?

There was a discussion on this only a few months ago in this very forum.

In short:
1=9/9=9 * 0.111111...=0.99999...

This is a fundamental property of the numbering system used, it's not a question of "belief".
Gambit
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### Re: Is 0.9 recurring equal to 1?

For the record I tutor math online. You have to have a college degree to teach.
And no 0.999... is not equal to 1. Whole numbers have an infinite number of significant digits. So it's 1.000... with infinite zeros (which isn't equal to 0.999...)
Last edited by Gambit on July 2nd, 2010, 12:58 am, edited 1 time in total.
Exasperation
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### Re: Is 0.9 recurring equal to 1?

0.999... does equal 1. The issue is that a recurring number is actually a limit of a convergent sequence.
0.999... is the limit as n -> infinity of (1-10^-n), which is 1-(the limit as n -> infinity of (10^-n)), which converges to 1-0=1.
Joram
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### Re: Is 0.9 recurring equal to 1?

And no 0.999... is not equal to 1. Whole numbers have an infinite number of significant digits. So it's 1.000... with infinite zeros (which isn't equal to 0.999...)
You have essentially answered the question: Does 1 = 0.999...?
By saying: 1 =/= 0.999...
While it isn't inherently wrong to answer a question like that, it isn't very rigorous either.

Exasperation is right in his post. We have thus proven that 1 = 0.999... by three different methods, using algebra, limits, and fraction/decimal equalities.
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Pentarctagon
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### Re: Is 0.9 recurring equal to 1?

but if they converge at infinity, doesn't that mean that they never actually converge (or at least that's what my math teacher said)?
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Midnight_Carnival
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### Re: Is 0.9 recurring equal to 1?

1 what?
1 apple? is 0.9999(rec,) or an apple the same as an apple, that is is an apple with an infinitely small bite taken out of it still and apple?

the value "1" is an abstraction. Is one part red 75000 parts blue still blue? it depends on how you define colours. Numbers and mathematics are made to be objective, this is becasue they are a tool used to measure things (mostly differences). You have to define "1" first. If you're dealing with countries, 0.999999(rec.) of a country could still be a country. Generally speaking, taken in isolation, numbers are (abstractions of) "pure" or "pepfect" values which don't exist in the real world. Don't try to apply "real-world logic" to numbers, it doesn't work. Mathematics is a codified system of conventions, you can work within the conventions or you can ignore them.

(guess which one I/we chose...)
...apparenly we can't go with it or something.
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### Re: Is 0.9 recurring equal to 1?

This is a common pattern, people arguing that 0.999.. is not 1 usually rely on intuition (which is misleading here, as often when dealing with infinity) or vague "proof" where others have easily indicated the error. In the other hand, proofs that 0.999.. = 1 are often rigorous like:
1 = 9/9 = 9*0.111.. = 0.999..
And nobody argues against that.

If you think that 0.999.. is not 1, you must then show the error in all these various proofs. Otherwise I don't see how you can still believe in something and, at the same time, have all these valid points contradicting your belief. And yes, if you can't point the error, you must accept them, or at least admit that you don't know since you have 2 contradicting "proofs" (only one is a real proof, since only one can be true, but you are unable to see which one).

edit:
Sorry, there is also people who accept the truth but are disturbed by its counter-intuitive aspects. To them, I suggest to check the underlying points used by their intuition (like "there is only one way to write a number")
pauxlo
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### Re: Is 0.9 recurring equal to 1?

The problem with all those arguments is, that one is not really sure what the used objects are.

If we look at the set of infinite chains of decimal digits with exactly one decimal point in it, then of course 0.999999999... and 1.000000... are not the same chain of digits.

But usually we calculate not with chains of digits, but with real numbers, who are represented by those "chains of digits". (If we only look at periodic chains, the rational numbers are sufficient.)

One (strict) way to define the real numbers [Wikipedia] is as the set of all equivalence classes of (infinite) cauchy sequences of rational numbers, where two sequences are considered equivalent when their difference approaches zero (where cauchy sequence (sequences where the later members do differ less and less) and approaches zero have also a formal definitions).

For a infinite decimal fraction, one can construct a representative of the real number by cutting the infinite chain, so 0.999... corresponds to the sequence [0, 0.9, 0.99, 0.999, ...] = [0, 9/10, 99/100, 999/1000, ...] and 1.000... corresponds to the sequence [1, 1, 1, 1, ...] = [1/1, 10/10, 100/100, 1000/1000, ...].
The difference sequence between those to are [1, 0.1, 0.01, 0.001, ...] or [1/1, 1/10, 1/100, 1/1000, ...]. Evidently, this series approaches zero, so the two series are equivalent, which means they represent the same real number.

So, in the real numbers, 0.9999... = 1 = 1.00000....

One can show that for every real number is exactly one infinite decimal fraction, with exception of those of the form x/10^y (with x,y integers), where there are exactly two decimal fractions (one with 99999... and one with 00000... after some point). (For other number systems there is a similar theorem.)

Paŭlo (a studied mathematician - but this is first semester stuff, partly already school stuff)