0.9 recurring=1 (2 Viewers)

[wilsondw]

I just recently found out about this and saw two proofs that actually show this


My friend believes it is a flaw...he tried to prove his point through a hypothetical situation where you can measure an object that is on an infinitely accurate scale

Meaning if you measure an object that weighs 0.9 recurring...it will say 0.9 recurring, not 1


Just wondering what are your thoughts on this

 

[nightweaver066]

http://en.wikipedia.org/wiki/0.999...

 

[SunnyScience]

I love this proof (but sure it's on the link) - our maths teacher taught us this in year 9.

Let x = 0.99999....

Therefore, 10x = 9.9999....
therefore, 9x = 10x - x
= 9.9999... - 0.9999....
= 9
So, 9x = 9
(dividing both sides by 9)
Therefore x = 1

But, x = 0.9999... (assumption)

Therefore,

0.9999.... = 1


LOVE this proof ( committed to memory)

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[jet]

They are the same number, there's no flaw here.

If you think about it more qualitatively:

0.9 is 0.1 away from 1.
0.99 is 0.01 away from 1.

0.99999999999 is 0.00000000001 away from 1.

As you gain more and more 9's you become closer and closer to 1. In fact, the distance between them becomes infinitely small which makes them indistinguishable.

 

[SunnyScience]

Philosophy states that if two numbers are different, you can fit a number between them e.g. their average. However, what number can fit between 0.999.. And 1?

The trend can also be seen when we consider 0.333... = 1/3. It's just that with 0.999... it's human nature to doubt it could equal such a perfect whole number like 1

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[mirakon]

They are the same number, there's no flaw here.

If you think about it more qualitatively:

0.9 is 0.1 away from 1.
0.99 is 0.01 away from 1.

0.99999999999 is 0.00000000001 away from 1.

As you gain more and more 9's you become closer and closer to 1. In fact, the distance between them becomes infinitely small which makes them indistinguishable.

Im pretty sure that approaching something doesnt mean its equal or indistinguishable

 

[D94]

They are the same number, there's no flaw here.

If you think about it more qualitatively:

0.9 is 0.1 away from 1.
0.99 is 0.01 away from 1.

0.99999999999 is 0.00000000001 away from 1.

As you gain more and more 9's you become closer and closer to 1. In fact, the distance between them becomes infinitely small which makes them indistinguishable.

But infinity is just a concept used by humans to explain unboundedness or no boundaries of an object (shown by Hilbert's paradox of the Grand Hotel); it doesn't exist in reality. By presupposing its existence, you immediately introduce a flaw in your argument. Conceptually correct, in reality, incorrect.

 

[lolcakes52]

http://www.youtube.com/watch?v=wsOXvQn3JuE&list=UUOGeU-1Fig3rrDjhm9Zs_wg&index=2&feature=plcp

 

[harrisony]

http://www.youtube.com/watch?v=wsOXvQn3JuE&list=UUOGeU-1Fig3rrDjhm9Zs_wg&index=2&feature=plcp

That was an april fools joke in response to her original video http://www.youtube.com/watch?v=TINfzxSnnIE

 

[wilsondw]

Well he tried to say its wrong by saying "What if" you measure an object that weighs 0.9 recurring grams on an infinitely accurate scale, it will say 0.9 recurring, not 1

But searched up definition of infinite: Limitless or endless in space, extent, or size; impossible to measure or calculate

 

[inJust]

A substitute taught us this in year 7. I can still remember the day where my mind got screwed.

 

[seanieg89]

This is just a formal peculiarity of the way we write down numbers: The decimal representation of a real number is not necessarily unique.

For similar reasons, 0.1111...=1 in binary.

And regarding the existence of infinity 'in reality' one may pose the question: "do real numbers exist?". Certainly we can not measure anything to arbitrary degrees of accuracy, we can never write down an arbitrary real number in its entirety. In what sense are real numbers more entitled to the right of existence than infinity? (Or for that matter complex numbers.)

My view is that these are just abstract objects we have constructed from the axioms of logic and set theory, the same axioms which we use to formally deduce their properties...independently of whether or not they 'exist in reality'...whatever that means.

 

[D94]

And regarding the existence of infinity 'in reality' one may pose the question: "do real numbers exist?". Certainly we can not measure anything to arbitrary degrees of accuracy, we can never write down an arbitrary real number in its entirety. In what sense are real numbers more entitled to the right of existence than infinity? (Or for that matter complex numbers.)

Paradoxically, infinity cannot exist in reality. Whether real numbers exist or not is a separate issue. The question presupposes truth and logic in real numbers and its operations, but questions the existence or reality of infinity (or 1/infinity).

 

[seanieg89]

Paradoxically, infinity cannot exist in reality..

Why?. (Not that I disagree with you. I have stated my philosophical view.)

 

[seremify007]

http://en.wikipedia.org/wiki/0.999...

Damn that is a long Wikipedia article for something which I assumed would be straightforward.


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[D94]

Why?. (Not that I disagree with you. I have stated my philosophical view.)

It can be shown through Hilbert's paradox of the Grand Hotel, which I guess thinking about it now, can also be scrutinised.

Edit: and this is presupposing all philosophical arguments about the world we live in being what we know as "reality".

 

[seanieg89]

Hilberts 'paradox' doesn't really demonstrate the nonexistence of infinity as a real object, it simply provides an example of a counterintuitive mathematical property that infinite sets (as a formal mathematical object) can have: namely that an infinite set can have proper subsets of the same cardinality.

More philosophical arguments are needed to discuss the notion of "truth". (Which is why I like to avoid the whole truth/existence quagmire when studying mathematics.)

 

[Demento1]

They are the same number, there's no flaw here.

If you think about it more qualitatively:

0.9 is 0.1 away from 1.
0.99 is 0.01 away from 1.

0.99999999999 is 0.00000000001 away from 1.

As you gain more and more 9's you become closer and closer to 1. In fact, the distance between them becomes infinitely small which makes them indistinguishable.

I'm not saying you're wrong of anything because you'll probably know more about mathematics then me but I was just thinking, by increasing the 9's you're gradually getting closer to 1 (as you mentioned) but then again, no matter how many 9's you place, it should never equal to 1 no matter what...

 

[Trebla]

I'm not saying you're wrong of anything because you'll probably know more about mathematics then me but I was just thinking, by increasing the 9's you're gradually getting closer to 1 (as you mentioned) but then again, no matter how many 9's you place, it should never equal to 1 no matter what...

By your argument 0.3333.... should never equal 1/3 for example.

If you always have a finite number of decimal places, 0.33333.... will never exactly equal 1/3. It is only exactly equal at the "limit" when you have infinitely many decimal places.

It's interesting to note that most people can see why 0.33333... = 1/3 but many cannot see why 0.99999... = 1/1 when it builds on exactly the same idea (it's just that 1 is a more elegant number than 1/3).

 

[D94]

Hilberts 'paradox' doesn't really demonstrate the nonexistence of infinity as a real object, it simply provides an example of a counterintuitive mathematical property that infinite sets (as a formal mathematical object) can have: namely that an infinite set can have proper subsets of the same cardinality.

More philosophical arguments are needed to discuss the notion of "truth". (Which is why I like to avoid the whole truth/existence quagmire when studying mathematics.)

I see your reasoning to this; but since Hilbert's Hotel shows the properties of "infinity" being different to the properties of a "finite" number or set of numbers, in this "real" world, we can conclude that "infinity" and a finite object cannot hold true. And then we come across the notion of "truth" again, so we come back to your point.