.9 recurring = 1

shamrock

Righto... why not start off the age old debate

.9 recurring is actually the same number as 1.

the reasoning goes like this:-
2 number are different if and only if the difference between them is greater than 0. (seems straight forward enough)

now if you take .999 from 1 you get .001

but .9 recurring is of infinite lenght (hence the recurring bit) and so the answer would be .0 recurring. so the difference between them is, technically speaking, 0.

So .9 recurring is the same as 1.

Anybody else?

Pablo

emm nope

stereo_steve

Ok, its past twelve nothings on tv and it would appear that only my modem wants to spent time with me so I'm gonna answer!

.9999 is not the same as 1. It gets close but never the same!

imagine on a larger scale. €999999999 is not the same as €1000000000 ! Theres a whole €1 in the difference and there will always be a euro in the difference! Same applies when going smaller always a difference of 0.something1!

hopefully thats the end of that!

But seriously there are more important questions like

1. If one synchronized swimmer drowns, do the rest have to drown too?

2.If you try to fail, and succeed, which have you done?

3.Why is it called tourist season if we can't shoot at them?

4. If the black box flight recorder is never damaged during a plane crash, why isn't the whole airplane made out of the same stuff? Come to think of it why not a tower?

5.If man evolved from monkeys and apes, why do we still have monkeys and apes?

6.If all those psychics know the winning lottery numbers, why are they all still working?

seamus

Yep, well in most mathematics anyway. Think about fractions...

1/1 = 1
1/2 = 0.5
1/10 = 0.1
1/1000 = 0.001

So the bigger the numerator gets, the smaller the value of the fraction. So, what is 1 divided by infinity? Well, it's generally accepted to be 0, even though it's more like

0.0000000000........(an infinite amount of more 0's)....0000000001

but it's just sooooo close to 0 (infinitely close in fact) that is is accepted to be 0. Well, actually that's a bit of a joke, because 1 divided by infinity can't be evaluated, but it just gives you the idea. In fact, any evaluatable number (eg 100, or 1,000,000,000) divided by infinity = 0, because infinity is an infinite number of times bigger than any number. (Infinity -10) == Infinity. That's just it's nature. Of course, there are plenty of people who can come on and contest, but this is the way infinity is accepted to behave in modern mathematics.

Now to get back on topic. Knowing the above, what is (1 - (1/infinity))? Well depending on what way you look at it, it is either equal to 1 or it is equal to .9 recurring. in fact it is equal to both, if we follow the rules of infinity. So therefore if

1 = (1 - (1/infinity)) = .999999.......

then .9 recurring = 1.
(that's if .9 recurring cannot be evaluated, e.g 3 * (1/3) can be
either 0.333333...... * 3, or, cancelling the threes, equal to 1 )

Can someone else explain this any better? please?

stereo_steve

Ok I was messing above, but seriously .9 recurring will always be .9 recurring.It doesn't change!
It will never ever equal 1. It will get so close that we could "call it 1" but it will never be equal to it!

Hopefully thats the end of it!

seamus

How do you know it will never be equal to one? That's one of the whole debates on infinity - what happens at the end? Does it ever end? To say .99 recurring will never be equal to 1 is incorrect because .99 recurring can never be evaluated, so quite simply maybe it is equal to 1, maybe it's not, we will never know.

I can see where you're coming from, and logic dictates that .9 recurring will always be that tiny bit smaller than 1, but due to the unknown and mysterious nature of infinity, all logic is out the door. It's just a matter of unlearning what you have learned, to quote Yoda.

stereo_steve

I realise that it it is very very close to one but it will never get to it. There will always be a small difference. By your logic, is 2 the same value as 3 when you have enough zeros in front of it? Every number has a unique value

stereo_steve

Ok, what started out as, what I thought was a stupid question, is really annoying me now! I went hunting around the internet and found a proof that shows .9999... is equal to 1.

x=0.9 rec.
10x=9.9 rec.
9x=10x-x=9
x=1

therefore 0.9 rec. = 1

i still find it very hard to accept! It should always be different!

seamus

Originally posted by stereo_steve
I realise that it it is very very close to one but it will never get to it. There will always be a small difference. By your logic, is 2 the same value as 3 when you have enough zeros in front of it? Every number has a unique value

Ah there'll just be no teaching ya! Wait til Occy gets here. 0 is a defined value, so your 2 = 3 is totally off the point.
{EDIT: got there b4 me - it's a noodle cooker alright!!:)}

1/3 is a number. It has 2 possible values. 0.33 recurring is (the theoretical) one, and 1/3 is (the real) one. 1/3 is more accurate in calculations, because no rounding is involved, but 0.33 recurring would be just as valid if we could calculate numbers with infinite recurrence. Therefore all numbers, by your logic cannot have a unique value, because, in your logic:

1/3 + 1/3 + 1/3 = 1, but

0.33* + 0.33* + 0.33* = 0.99* !=1

Therefore 1/3 does not have a unique value, rather 0.33* < 1/3, which is at odds to your explanation.

Guess we'll have to agree to disagree, at least until someone else brings in an opinion

stereo_steve

From the proof above, I have to accept it. Still though I find it hard to.

Bob the Unlucky Octopus

What steve is alluding to is a mathematical idea first disproved by DeMoivre (of DeMoivre's theorem regarding e^i*theta fame). Basically, seamus is right- as far as all numbers go, zero is a special case, it is neither an discrete integer, nor an approximation. It is an exact definition.

Now as far as the 3(0.33*) idea goes, 0.33* is just an approximation of the fraction 1/3. In fact, to some extent, all unitary prime decimal fractions are approximations (eg, 1/7). As such, using an approximation of value in an exact logical proof isn't going to wash.

And that's what this argument was about, approximations. If you wish to approximate 0.9999999999 to one...I suppose that would be reasonable in practical terms depending on the situation. After all, 0.9999999999 of a cookie can be considered one cookie for the purposes of consumption (unless it was a very big cookie you wouldn't notice :P). On the other hand, in theoretical terms of considering all ends...it cannot be considered one. In fact, even one divided by infinity isn't zero...just infintesimally small, and approaching zero. On the other hand, zero divided by anything is simply zero...since it is a defined conceptual number. Conceptual numbers are tricky because you cannot visualize them(I challenge you to show me 'zero' of anything) except in relation to defined integers (I had one cookie, and now I have zero).

Approximations are like conceptual numbers...in theory they are as good a guess as we have as to a value. 0.66666666666667 is very close to two thirds, but isn't quite the same.

It's vitally important to make the distinction between an approximated value and a defined value in a mathematical proof, otherwise the results of any such proof are immediately suspect. Ironically enough, DeMoivre later went on to construct a proof regarding imaginary numbers (e to the power i*theta = cos theta + i*sin theta)...which contains no less than three approximations, one constant (e) an undefined parameter (i) and an approximated angle measurement in radians (theta). Even the world of mathematics it would seem, is not without irony :P

Occy

leeroybrown

The limit of 1 - .9999 recurring as the number of recurring digits in the term apparoches infinity is zero.

There is still a difference but the more terms you take the smaller it becomes.

Pretty similar to the distance between a curve and its asymptote.

bonkey

Originally posted by stereo_steve
Ok, what started out as, what I thought was a stupid question, is really annoying me now! I went hunting around the internet and found a proof that shows .9999... is equal to 1.

x=0.9 rec.
10x=9.9 rec.
9x=10x-x=9
x=1

therefore 0.9 rec. = 1

i still find it very hard to accept! It should always be different!

This proof is subtly but fundamentally flawed.

Integer-based mathematics is not designed to cope with "recurring" numbers. Quite simply, while the concept of a number having its decimal places "going on to infinity" is one we can understand, we cannot actually work with infinities in a discrete numbering system.

For the proof above to work, it requires that 10x and x have the same number of figures after the decimal point. Now, while you can argue that this is true because there is an infinite number of places in each case, you are making the mistake of making two infinities equal.

To show that the proof above cannot be true, we simply need to take another example, and utilise the same mathematical boundaries. So, I can use infinities in mathematics (if you allow two recurring decimals to be considered to have the same number of decimal places for the purposes of mathematical expediency)

What the above proof is based on is the concept that while 10X has an infinity of decimal places, x has infinity + 1 decimal places. However, we are asserting that these cancel each other out (because there is no finite point where we can say "this is the last place) exactly, rather than approximately.

So, in effect, we are saying that :

infinity + 1 = infinity

Subtract 1 from each side :

infinity = infinity - 1

Now, by substituting infinity for infinity -1 on the RHS, I get :

infinity = infinity - 2

(I can do this because I have "proven" that infinity -1 and infinity are equal, and therefore interchangeable.

I can repeat this an infinite number of times, ultimately resulting in :

infinity = infinity - infinity

or

infinity = 0

Oops

jc

amen

stero_steve,

your equations/formula are incorrect

line 1 x=0.9rec correct
line 2 10x=9.9 rec correct
line 3 9x=10x-x = 9 incorrect 9x=10x-x =9x
line 4 x= 1 incorrect from line 3 9x=9x therefore x=x or 9 =9 not x=1

bonkey

Originally posted by amen
stero_steve,

your equations/formula are incorrect

line 1 x=0.9rec correct
line 2 10x=9.9 rec correct
line 3 9x=10x-x = 9 incorrect 9x=10x-x =9x

No - while its flawed, the progression is the same.

9x = 10x - x
= 9.9(rec) - 0.9(rec)
= 9

He just shortcutted it.

Its still wrong tho

jc

Kraken

Originally posted by stereo_steve
But seriously there are more important questions like

1. If one synchronized swimmer drowns, do the rest have to drown too?

2.If you try to fail, and succeed, which have you done?

3.Why is it called tourist season if we can't shoot at them?

4. If the black box flight recorder is never damaged during a plane crash, why isn't the whole airplane made out of the same stuff? Come to think of it why not a tower?

5.If man evolved from monkeys and apes, why do we still have monkeys and apes?

6.If all those psychics know the winning lottery numbers, why are they all still working?

1: Yes just to luck good.
2: you have succeded so there by failing i think.
3:You can shoot at them just dont get caught
4:point
5:Personaly i think that there is a large amount of confusion there i think that there may have been a mutation in cells in a certain type of monkey/ape and that is where we came from. there by confirming that we are related to monkeys/apes but also allowing a reason for there continual excistance
6:Science is blinding?

beaver

Kraken:
5:Personaly i think that there is a large amount of confusion there i think that there may have been a mutation in cells in a certain type of monkey/ape and that is where we came from. there by confirming that we are related to monkeys/apes but also allowing a reason for there continual excistance

Basically, we both had common ancestors but we boths branched off on our own seperate evolutionary paths.

stereo_steve

Ok, so the problem has finally been solved.... and everyone agrees.

0.999..... Will never equal one just get very close to it! I'm happy now!

As for the other questions, I don't think there are any answers!

Ruaidhri

we talking about discreet or continuous maths?because in one it's true,the other it's a hard one to prove....

shamrock

Well.. I still don't know wether it's true or not but tahnx for trying.

Better go to a lecture now.:o

common sence

stereo_steve wrote:

Ok, its past twelve nothings on tv and it would appear that only my modem wants to spent time with me so I'm gonna answer!

.9999 is not the same as 1. It gets close but never the same!

imagine on a larger scale. €999999999 is not the same as €1000000000 ! Theres a whole €1 in the difference and there will always be a euro in the difference! Same applies when going smaller always a difference of 0.something1!

hopefully thats the end of that!

But seriously there are more important questions like

1. If one synchronized swimmer drowns, do the rest have to drown too?

2.If you try to fail, and succeed, which have you done?

3.Why is it called tourist season if we can't shoot at them?

4. If the black box flight recorder is never damaged during a plane crash, why isn't the whole airplane made out of the same stuff? Come to think of it why not a tower?

5.If man evolved from monkeys and apes, why do we still have monkeys and apes?

6.If all those psychics know the winning lottery numbers, why are they all still working?

the common arguement that states .999... = 1
this is because when u times .999... by 10 it equals 9.999... minus .999... it equals 9
technically thus making .999... X 9 =9 and so .999... = 1

however this can be proven incorrect in numerousy as thirds do not exist in numerousy. The 1/3 principal can only be used in algebra as algebra constitues theory and numerousy is based around reality. to cross over would result in a perpetual motion machine(explained at the end)

first yes its completely correct that .999... does not = 1
proof on a calculator (try for your self) using algebraic maths to constitute n in the place of an amount we can assume that n amount of nines in .999...
n = infinity.
as n is a standard in algrbra the we can deduce that n can = any amount so lets test n

1.0n =1.000... = 1
1.1n =1.111...

so far makes sence ok folks on a calculator type this equasion for your self and see the trend occur.
.9 X 9 = 8.1
.99 X 9 = 8.91
.999 X 9 = 8.991 THIS EQUATES TO .999... X 9 = 8.999...1
.9999 X 9 = 8.9991 AND NOT 9
.99999 X 9 = 8.99991
.999999 X 9 = 8.999991
so having the trend as u can see the algrabr would be

.n x 9 = 8.m
n being set amount
m being .n8 less than n

So if n is infinate then in this equasion even algebra states .999... cant possibly = 1 unless u rig the results aka
.999 x 10 = 9.99 - .99 = 9 therefore .999 x 9 = 1 (not .999 { the number in bold is one less than the original number forcing the equasion to work aka human error)
the real equasion has to = 8.9n1 aka 8.999...1

result .999... < 1 or
0.999 is less than 1 !!!!
in answering to the rest of the questions...

1. if one drowns the rest will see her drown under water and as they all have life saving grades they will revive her
2. you have failed to succeed and succeded in failing both are the same.
= fail, whether u wanted to or not
3. a season for tourists where is the word "hunting"??? like rabbit hunting season
we will shoot at tourist when its tourist hunting season.
4. if the whole plane was made out of the flight recorder material, flights would cost around 30000 times more than they do now and planes would weigh more requiring more fuel and become a waste of money. lil box is all they need.
5. because they survived
6.becuase they are frauds LOL

please feel free to quote my mathematics on any calculator and see the trend urself.
perpetual motion theory
if .999... was to = 1 then you would be getting that .000...1 from nothing and to get something from nothing is... well... not only impossible but would create an infinately large amount rendering a limitlessly large number and "equating to power" an infinatly increasing never ending power source or the perpetual motion machine
why because the difference between 1 and 0 is greater than the bounds of infinity thus rendering this arguement ( .999... = 1 moot)

common sence

stereo_steve wrote:

Ok, its past twelve nothings on tv and it would appear that only my modem wants to spent time with me so I'm gonna answer!

.9999 is not the same as 1. It gets close but never the same!

imagine on a larger scale. €999999999 is not the same as €1000000000 ! Theres a whole €1 in the difference and there will always be a euro in the difference! Same applies when going smaller always a difference of 0.something1!

hopefully thats the end of that!

But seriously there are more important questions like

1. If one synchronized swimmer drowns, do the rest have to drown too?

2.If you try to fail, and succeed, which have you done?

3.Why is it called tourist season if we can't shoot at them?

4. If the black box flight recorder is never damaged during a plane crash, why isn't the whole airplane made out of the same stuff? Come to think of it why not a tower?

5.If man evolved from monkeys and apes, why do we still have monkeys and apes?

6.If all those psychics know the winning lottery numbers, why are they all still working?

the common arguement that states .999... = 1
this is because when u times .999... by 10 it equals 9.999... minus .999... it equals 9
technically thus making .999... X 9 =9 and so .999... = 1

however this can be proven incorrect in numerousy as thirds do not exist in numerousy. The 1/3 principal can only be used in algebra as algebra constitues theory and numerousy is based around reality. to cross over would result in a perpetual motion machine(explained at the end)

first yes its completely correct that .999... does not = 1
proof on a calculator (try for your self) using algebraic maths to constitute n in the place of an amount we can assume that n amount of nines in .999...
n = infinity.
as n is a standard in algrbra the we can deduce that n can = any amount so lets test n

1.0n =1.000... = 1
1.1n =1.111...

so far makes sence ok folks on a calculator type this equasion for your self and see the trend occur.
.9 X 9 = 8.1
.99 X 9 = 8.91
.999 X 9 = 8.991 THIS EQUATES TO .999... X 9 = 8.999...1
.9999 X 9 = 8.9991 AND NOT 9
.99999 X 9 = 8.99991
.999999 X 9 = 8.999991
so having the trend as u can see the algraba would be

.n x 9 = 8.m
n being set amount
m being .n8 less than n

So if n is infinate then in this equasion even algebra states .999... cant possibly = 1 unless u rig the results aka
.999 x 10 = 9.99 - .99 = 9 therefore .999 x 9 = 1 (not .999 { the number in bold is one less than the original number forcing the equasion to work aka human error)
the real equasion has to = 8.9n1 aka 8.999...1

result .999... < 1 or
0.999 is less than 1 !!!!
in answering to the rest of the questions...

1. if one drowns the rest will see her drown under water and as they all have life saving grades they will revive her
2. you have failed to succeed and succeded in failing both are the same.
= fail, whether u wanted to or not
3. a season for tourists where is the word "hunting"??? like rabbit hunting season
we will shoot at tourist when its tourist hunting season.
4. if the whole plane was made out of the flight recorder material, flights would cost around 30000 times more than they do now and planes would weigh more requiring more fuel and become a waste of money. lil box is all they need.
5. because they survived
6.becuase they are frauds LOL

please feel free to quote my mathematics on any calculator and see the trend urself.
perpetual motion theory
if .999... was to = 1 then you would be getting that .000...1 from nothing and to get something from nothing is... well... not only impossible but would create an infinately large amount rendering a limitlessly large number and "equating to power" an infinatly increasing never ending power source or the perpetual motion machine
why because the difference between 1 and 0 is greater than the bounds of infinity thus rendering this arguement ( .999... = 1 moot)

Tomlowe

not a proof but an intriguing point in favour of .9recurring = 1

1/9=.111111*
2/9=.222222*
3/9=.333333*
.
.
.
9/9=.999999*
but
9/9=1

Son Goku

0.9999999............ = 1 is definitely true. There are no points against it. Take a book like Munkres' Topology, Section 2 or any other decent topology book, there the axioms for the real numbers are listed. If follows from axioms 1-4, 5 and 7 that 0.9999....... = 1.

I know it isn't intuitive, but it's a fact. If it isn't true pi and e can't exist, because it is true by the same topological notion that gives pi and e their existence, the notion of completeness.

aequinoctium

.999999999999...... = 3x(.33333333......)

but .3333333333... = 1/3

=> .9999999... = 3x(1/3) = 1

Fremen

The real numbers form a dense set. Pick any two distinct real numbers, and you can find a third real number between the two of them.

Anyone feel like naming a real number between 1 and .99...?

Anyone?

A Primal Nut

stereo_steve wrote:

Ok, what started out as, what I thought was a stupid question, is really annoying me now! I went hunting around the internet and found a proof that shows .9999... is equal to 1.

x=0.9 rec.
10x=9.9 rec.
9x=10x-x=9
x=1

therefore 0.9 rec. = 1

i still find it very hard to accept! It should always be different!

wrong>>

10x = (9.9 rec)/1.0rec1 - remember when you multiply by 10 the final 9 gets removed the 1.0000----0001 removes the final 9 so 0.999*10=9.99. Thats the problem with that proof above >> when you multiply by ten its no longer recurring to infinity, its recurring to infinity -1.

A Primal Nut

stereo_steve wrote:

4. If the black box flight recorder is never damaged during a plane crash, why isn't the whole airplane made out of the same stuff? Come to think of it why not a tower?

Coz it would make the plane to heavy to take off? Just a guess considering aluminium is chosen for its light weight.

Fremen

remember when you multiply by 10 the final 9 gets removed

The point is that there is no final nine.What you're thinking of as "infinity minus one" is the same infinity (though arithmetic operations aren't defined on infinities).

If people can accept that 1/3 is .3 recurring, I don't see why they can't accept that 1 is .9 recurring.

Son Goku

A Primal Nut wrote:

wrong>>

10x = (9.9 rec)/1.0rec1 - remember when you multiply by 10 the final 9 gets removed the 1.0000----0001 removes the final 9 so 0.999*10=9.99. Thats the problem with that proof above >> when you multiply by ten its no longer recurring to infinity, its recurring to infinity -1.

You're applying a fact about the rationals to the Reals in general, which doesn't work.

Professor_Fink

A Primal Nut wrote:

wrong>>

10x = (9.9 rec)/1.0rec1 - remember when you multiply by 10 the final 9 gets removed the 1.0000----0001 removes the final 9 so 0.999*10=9.99. Thats the problem with that proof above >> when you multiply by ten its no longer recurring to infinity, its recurring to infinity -1.

I hate to point out the obvious, but (infinity - 1) is still infinity.

By the way, what you are arguing about is if lim_{x->infinity} (1 - 10^{-x})=1, and it is trivially true since lim_{x->infinity} (10^{-x})=0 and limits can be added.

sd123

A number divided by itself is always 1, right.

press the Pi button on a calculator. you should get 3.141592654. Now divide it by pi and you get 1. this is what we expect.

now press pi/ 3.141592654. the answer you get is .9999999999.

dunno how relivant this is but it is interesting nontheless. probably comes down to inaccuracies in the calculator itself

Professor_Fink

sd123 wrote:

now press pi/ 3.141592654. the answer you get is .9999999999.

dunno how relivant this is but it is interesting nontheless. probably comes down to inaccuracies in the calculator itself

Yes, it does. The number on the screen is not exactly the same as the internal representation of pi, so when you copy it down, and rekey it in, you introduce an inaccuracy.

sd123

yes but if you key it in ie put 3.141592654/pi, it'll come up 1. but if you key in pi/3.141592654 you'll get .99999999999

bonkey

And if you put in .99999999999 and hit 1/x, what do you get?

(If you're not using a scientific calculator, do 1 / .99999999999)

Seriously...calculators are *not* comparable to proper mathematics. And no...for the record...computers aren't either.

Son Goku

Just to be more specific 0.999999............. is a way of representing the infinite sequence:

9/10 + 9/100 + 9/1000 + 9/10000 + .....................

Let's set this equal to some number s.

9/10 + 9/100 + 9/1000 + 9/10000 + ..................... = s.

Pull out a factor of 1/10:

1/10 (9 + 9/10 + 9/100 + 9/1000 + ..........) = s

However s = 9/10 + 9/100 + 9/1000 + ..........

So this becomes:

1/10(9 + s) = s

Multiplying by 10

9 + s = 10s
9 = 9s
1 = s = 9/10 + 9/100 + 9/1000 + .......... = 0.999999......

I hope that proves it to people's satisfaction.

N.B.
Also remember infinity isn't a real number so expressions like (infinity - 1) are meaningless.

sd123

N.B.Also remember infinity isn't a real number so expressions like (infinity - 1) are meaningless.

That means that you just contradicted your own 'proof'

bleg

people,people wikipedia is your friend

and yes .99999... =1

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

bleg

people,people wikipedia is your friend

and yes .99999... =1

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

Pherekydes

sd123 wrote:

That means that you just contradicted your own 'proof'

eh? How?

Are you getting confused between infinite and infinity?

.9... is a real number. Infinity is not.

Son Goku

sd123 wrote:

That means that you just contradicted your own 'proof'

How so? I don't see where I did. Nowhere did I refer to a number called infinity and perform arithmetic on it.

Capt'n Midnight

since this thread is in science and not maths I can only assume you are refering to physical meanings

there are only about 10^90 atoms in the observable universe +/- a coulple of orders of magnitude
the number of different locations is when you divide a sphere of 15 billion light years diameter by the Planck Length
the number of time intervals is the 15 billion years divided by the Planck time
I don't know the smallest energy levels of photons, but electrons usually only jump between different band levels with specific energies, unless we are dealing with electron clouds

Multiply that all together and express it as powers of powers and you could write it all down. More importantly the number of digits in the number would represent the number of 9's you need such that for all physical meaning it would .9 recurring would be the same as 1

sd123

Son Goku wrote:

How so? I don't see where I did. Nowhere did I refer to a number called infinity and perform arithmetic on it.

The 's' you used above is a sum to invinity, therefore when you attempted to subtract numbers from it its a contradiction.

Son Goku

The 's' you used above is a sum to invinity, therefore when you attempted to subtract numbers from it its a contradiction.

When did I subtract from s?
Also s is an "infinite series". There is no numerical quantity called inifinity that I manipulated.