Is 0.999999.... (infinite) equal to 1?
Looks like we have lot of confusion over the subject here π .
So, why not dedicate a thread to it?
Is 0.999999.... (infinite) equal to 1? What do you have to say?
So, why not dedicate a thread to it?
Is 0.999999.... (infinite) equal to 1? What do you have to say?
Replies

mahulit can be proven mathematically, any1 willing can point out errors if any
let 0.9999..... = x
=> 100x = 99 + x (99.999999..... = 99 + .9999... = 99 + x)
=> 99x = 99
=> x = 1 (x <> 0)
any other number (other than 100) would also work
thus, 0.9999..... till infinite equals 1. 
raj87verma88The result which mahul derived and the solution that Biggie gave in another thread seem OK but still leave doubts in my mind.
In normal day to day life .60 = 1. And .999999.....infinity will always be 1. But if we take the real value of .9999...infinity then i don't think it will be = 1 ( till someone proves the result and hence proves me wrong).
If we can say .9999...infi. = 1 then we can also prove that the universe has limits. 
gohmYep, my line of thought was also the same as Patty's on this one from the earlier thread. Only one (or variables like 15/15) is equal to one in our current base 10 system. Excluding rounding for practical use:
If .9 repeating is = 1 then:
.9 to google plex= 1,
.99999999999999= 1,
.99999999999998= 1,
.9= 1,
.89= 1,
.75=1, etc.
Then all science theory breaks down into utter hopeless chaos... laugh. 
Kaustubh KatdareGosh, I love everyone here so much! π
Okay. Let's note something here : "Infinity". Shall we have a separate discussion on 'infinity'?
If we extend 0.999999... to infinity, probably we'll end up equaling 1. I do not quite agree with 0.999999 = 1 or 0.9999998 = 1. We all know that its NOT equal to one.
How do we prove that universe has limits? 
vissinWell... here's an old trick. I was rather intrigued when I first saw the equation 0.999... = 1 somewhere around 5 years back. And I tried to find out as many mathematical proofs of it as I can and see if I can find a flaw in it. At that time I was baffled but I'm not so any more. Here I'll first reproduce the simplest proof that I remember and then dare to propose my own penny's worth on it.
0.999... = 3 * 0.333... = 3 * 1/3 = 1
ie. 0.999... = 1
Now to address the question, "how is it even possible?"
Well Raj this does not prove that Universe has limits. In fact it proves nothing. It only goes on to display the limits of human perception.
Remember, Mathematics and all Science is nothing more than merely tools that we use. Allow me to quote Neils Bohr (as I have done many times ever since I first read it) "Physics tells us not what is but what we can say to each other regarding the world." Eventhough this was said about Physics, we can very well extend this to any Science.
Let's suppose I have a tray full of randomly cut pieces of apples. I want to use these pieces to make one full apple. In analogy I'm using fractions to create 'one'. Quite easily I make an assemble that looks very much like a whole apple but I notice that they are many gaps between the pieces. Instead of '1' I may have something like 0.999 (considering that I was born lucky). Now I cut even smaller pieces from the apple in the tray to fill these gaps. I want 1 and not an approximation to it. The pieces I cut keep getting smaller and smaller until I reach a single atom and then I go further into granualities of bosons, neutrons, protons, electrons and photons. What now. Do you think I've constructed one whole apple. I doubt it. But I'm sure to be very close. So close that I'm no longer able to see the difference even with the best technology that I may have. Now this is where 0.999... = 1. As also 1.000...1 = 1.
What is infinite. It's just a concept. Something that's beyond perception. The proof is 100% right. Mathematics is flawless. It agrees completely with the definitions that we ourselves have made for our own understanding. And it only gives us what is true for us (even if it may not be the absolute truth).
The above is a outcome of my petty mind and bears no external credentials (except Bohr) and is open to criticism.

raj87verma88Some new threads
1. On Infinity
2. Proving the limits of universe
3. Life outside earth
how about them biggie 
gohmExactly! So where do we draw the line to say one equals the other?
If we agree .999998=1 is false, then how is .99999repeating= 1.0000repeating? we have crossed over where acceptance is given to rounding thus creating equality. However the fact still remains there is a differnce, though microscopical in amount.
this is fun!
The_Big_KGosh, I love everyone here so much! π
Okay. Let's note something here : "Infinity". Shall we have a separate discussion on 'infinity'?
If we extend 0.999999... to infinity, probably we'll end up equaling 1. I do not quite agree with 0.999999 = 1 or 0.9999998 = 1. We all know that its NOT equal to one.
How do we prove that universe has limits? 
Ashraf HZVissin, great post! If I might add a totally useless comment: how do we get a complete "3" to multiply the 1/3? It could be 2.99999999... or 3.00000...1 π
1 in continous (note, not discrete) terms is so absolute, I'd doubt we can reach it in practice. Why don't we just appreciate that? We've seem to accept pi and e.. just bask in the beauty of natural numbers in life *grin* 
Kaustubh Katdare[another π useless post, maybe]
In my opinion, the moment you put an end to the series 0.999999......9... , it ceases to be equal to one.
When we say the series goes infinite, it equals one!
Now, in theory, we may say that since this series will never terminate, probably we'll never reach 1. In that case, 0.9999... continues to approach 1 but never becomes equal to 1.
What do you all have to say about it?
Anyone wants to discuss the concept of Infinity here? 
frodo.rokAbout the topic I am in complete agreement with Vissin.
About infinity:
Mathematicaly, Lim (1/X) =infinity
X>0
Physicaly, radius of the universe= infinity. 
suyashyes, according to abstract mathematical laws, it can be proved that 0.9999 (with infinite such 9's) will definitely be equal to 1 (as proved by raj87verma88)...
However, since the value of infinity can't be realized in real life, such an application is not possible for generating solutions to mathematical problems dealing with practical physics and other sciences π
I might sound confusing, but then "God doesn't exactly play dice" π, and there just might be some really great reason why we can't prove that 0.99999999 (till infinity) is not equal to 1 ... 
xheavenlyxI just could not hold myself here. Wow, amazing thread! π And a long reply, so read it, if interested π
I agree with Vissin 100.1111...%. Absolutely right. One of the best explainations!!!!
VISSIN:
What we prove is only to our abilities in science, we prove what we can perceive and measure. 1000 years ago we thought earth was flat till science proved otherwise. Anyway...
0.999... = 3 * 0.333... = 3 * 1/3 = 1
ie. 0.999... = 1
0.9999...inf. will equal to one depending on the perception of "0.99999...inf".
EXAMPLE: Radius of earth is 6,378.1 km or 6378.1 * 1000 * 100 * 10 mm= 6,378,100,000 mm so 6,378,100,000.99999... will also be 6378.1 km meaning we can accept it is equal to one when we are measuring a planet.
For some calculations we might need more accuracy and there we do not round it off till some numbers.
Depending on our use we can say 0.999999...inf is 1.
And biggie:
In my opinion, the moment you put an end to the series 0.999999......9... , it ceases to be equal to one.
I am hoping you dont mean this practically (theoretically maybe) because our cities and life is BUILT on these "inaccuracies" or approximations.
We dont say the car traveled 999.999998 meters but 1 km. We CAN say the ant traveled 999.999 meters OR 99999.90 cm because we MIGHT need more accuracy there.
I think Heisenberg's uncertainty principle explains it a bit too. May get confusing but keep reading it to understand.
...locating a particle (very accurately) (0.9999999...inf) in a small region of space makes the Momentum of the particle uncertain; and conversely, measuring the momentum of a particle precisely makes the position uncertain (1 maybe? *smiles*).
On a side note I think we confuse INFINITE with "never ending, forever", which does not exist except the universe. INFINITE in science/math (I think) is used to say that in a particular case the value may be too much for consideration. For example Canada is "infinitely big" for a snail, because it will never traverse through east to west in its lifetime. Same way "Limits" in mathematics is for calculations where "approaching 0" means we are happy with it even if it is not, does not make a difference. If we need a zero then we recalculate. 
MaRoI have a deep feeling telling me that there's no infinity, it's just a concept but there a finite number.
DON'T CONSIDER, just a feeling π 
Kaustubh KatdareThis is getting even more interesting now π
xHx, I'm talking about theory. I myself want to understand the concept ('concept') of Infinity in depth.
We dont say the car traveled 999.999998 meters but 1 km. We CAN say the ant traveled 999.999 meters OR 99999.90 cm because we MIGHT need more accuracy there.
That's right and explains my point.
For example Canada is "infinitely big" for a snail, because it will never traverse through east to west in its lifetime.
Good point. Extending on this logic, probably Universe is finite (okay, don't jump on me!) because its way too big for us to see what's at the end. I wonder if it has been mathematically proven that the Universe is infinite or its still a concept?
...and if universe is infinite, why do people say that its expanding? π [Out of context; sorry for that!] 
vissin
This and quite a few other posts here is speaking of the universe being finite or infinite. Here, I believe that we need to first understand what universe is. Another thread Big? I'll just give a few food for thought here as I don't want this post to be too long (and boring consequently)frodo.rokPhysicaly, radius of the universe= infinity.
While teaching lord Rama Ved Vashishtha spoke of many universes. Even today we speak of the many universes theory or the parallel universes. So in implication the whole expanse of the universe that we see as the universe is not all that's there.
If you believe in the big bang theory that inadvertently puts a limit to the size of the universe. Considering that the universe started expanding from one point and with the assertion that maximum speed attainable is that of light, if we know when the big bang occurred we can put an upper limit to the radius of the universe (d = s*t). We can find out when the big bang happened by studying the relative motion of galaxies and extrapolating it backwards to the center of the universe. I've heard of such figures as Universe being around 14 billion years old and the diameter of it being around 150 billion light years.
There is also a hypothesis that states that near big bang the matter coming out of the explosion probably traveled even faster than light.

frodo.rokBut is it feasible to take light speed constant at the time os Big Bang??
At Singularity(state of space time at the time of Big Bang where everything is concentrated in one point making space time INFINITELY curved) no physical law known to us holds.
So all this calculations are not trustworthy.
Isn't it?? 
xheavenlyx
Extending on this logic, probably Universe is finite (okay, don't jump on me!) because its way too big for us to see what's at the end. I wonder if it has been mathematically proven that the Universe is infinite or its still a concept?
Excellent point! basically we do not know for sure. depends on how far we can see in space and practical calculations on how galaxies are moving away from each other. Vissin also explains it well.
...and if universe is infinite, why do people say that its expanding? [Out of context; sorry for that!]
As for infinity, it should exist theoretically and it does not practically. I understand now what you were saying.
and frodo ur confusing urself and us, π. The breakdown of physis laws at singularity is just a theory. and moreover that point in time was only once, so we cannot base much logic for current argument. 
frodo.rokIf you believe in the big bang theory that inadvertently puts a limit to the size of the universe. Considering that the universe started expanding from one point and with the assertion that maximum speed attainable is that of light, if we know when the big bang occurred we can put an upper limit to the radius of the universe (d = s*t). We can find out when the big bang happened by studying the relative motion of galaxies and extrapolating it backwards to the center of the universe. I've heard of such figures as Universe being around 14 billion years old and the diameter of it being around 150 billion light years.Vissin
extrapolating it backwards to the center of the universeVissin
Xeahavlynx I am sorry if I have made you confuse but as per as my knowledge Big Bang is a theory yet to be proved.So I can help it if THERE LIES CONFUSSIONS.
Now on extrapolating backward You will not find the same laws working if you have any faith on GENERAL THEORY OF RELATIVITY.
Suppose the universe is now at a state where it has a radius of say 5 km and it is expanding.
Just imagine everything lies in that sphere with radius of 5 km.
What will happen at this moment??
General theory of relativity tells us due to almost infinitely high gravitational force space and time will be so curved that every laws of physics get broken down.
d = s*t....This simple formula will not work at that moment.
Also if you say singularity is a hypothesis then i am afraid theory of Big Bang is no exception.
THE EXPANSSION OF UNIVERSE IS JUST LIKE A BALLON WITH DOTS IN IT.
When you start blowing it the the distance between the dots increases.
Just in case of our universe distance between Galaxis is increasing.
But in comparission to the Gallactic distance the radius of Universe will always appear to be INFINITE. 
vissinThe discovery of the Cosmic Microwave Background (CMB) in the late 1960s gave tremendous boost to the Big bang theory and today it is the most widely accepted model of the Universe. Also General Theory of Relativity has only supported the Big Bang theory much to the discomfort of Einstein who believed in a static Universe.
The equation d = s*t has been proposed just to aid understanding and it's over simplification of the actual calculation. The Big bang has been divided into many stages called Epochs. The age and expanse of Universe has been calculated based on both Hubble expansion and CMB and has been found to be in agreement.
PS: The shape of the Universe is not very likely to be spherical. It might just be a lot flatter.

sataiutoNo. 0.99999....(infinite) is not equal to 1.
I can prove it.
so let's do the famous equation:
x=0.99999(infinite)
10*x=10*0.999999(infinite)
now let's do the calculations:
10x= 10*0.9999999<=>10x=9.999999999999999999999(infinite)1(which means that the last digit of the infinity is deleted...
now
10xx=9.999999999(infinite1)(infinite)
<=>
9x=8.9999999999(infinite)
<=>
9x/9 = 8.999999999(infinite)/9
if you do it with the calculator you'll indeed get 0.999999999 but it will stop being an infinite number!
you'll get 0.99999999(infinite1) which is finite.... the problem with this is that it's impossible to know how many digits this finite number has.
We can get one thing from this equation, it explains how infinite is nothing but finite. thus infinite=finite
Nothing is infinite! Infinite is just a loop of the same process again and again. 
Kaustubh KatdareI didn't get this 
10x= 10*0.9999999<=>10x=9.999999999999999999999(infinit e)1(which means that the last digit of the infinity is deleted...
Disagree with 
We can get one thing from this equation, it explains how infinite is nothing but finite. thus infinite=finite

gohmThis is a great brain teaser. it all depends on your view. If you talk to a cosmologist you will disover that infinity1=infinity, not finite. That's because by the nature of something being infinite it is impossible to take away enough to create a finite situation. ouch, my head is hurting now! =)

shalini_goel14
Good explanation given Mr mahul. Liked it. But can anyone tell me who all say 0.9999... (infinite) equal to 1? π I guess we are always taught to use approx.(~) in such cases. Correct me if wrong.mahulit can be proven mathematically, any1 willing can point out errors if any
let 0.9999..... = x
=> 100x = 99 + x (99.999999..... = 99 + .9999... = 99 + x)
=> 99x = 99
=> x = 1 (x <> 0)
any other number (other than 100) would also work
thus, 0.9999..... till infinite equals 1. 
sataiutoLet me explain it in a more clear way:
(maths are not like real life thus it's hard to understand this)
if you multiply 10 x 0.10 you'll get 1 but in this case (because it's infinite):
10 x 0.(infinite 9) equals the infinite number but with an end it means that the infinite number stopped being infinite. thus 9.9999999999(finite) the problem with this is that the decimals will be undefined; it will be impossible to know how many digits it has....
BTW I made a mistake on my last post, what I meant was that:
infinite=finite+1
thus it's impossible to find the infinite because every time we try to search for it it will indeed grow even higher and therefor not possible to ever reach the top.
I believe this supports the big bang theory. That while we are growing and reaching further and further to the answer(for what's beyond the universe or infinity?)that the universe is always 1 step ahead of us which means that the universe is always growing bigger x)...
Conclusion, infinity is just a number higher than another number and that that digit will always "run away" from us every time we try to look for it 
sataiutoshalini_goel14Good explanation given Mr mahul. Liked it. But can anyone tell me who all say 0.9999... (infinite) equal to 1? π I guess we are always taught to use approx.(~) in such cases. Correct me if wrong.
Actually mahul is pretty wrong lol
Let me explain why....
First of all, he doesn't even know how equations work... since when can you had 99 both sides and get x+99=100x rofl omg >.< what a noob,
only x+99x= 100x thus he's wrong...
if he had done it correctly you would've gotten this:
let 0.99999.... = x
=> x+99x = 99x+x
=> 100x = 99x+0.99999....
=> 100x = 100x
=> 100x = 99.9999999999999999(infinity9(1digit))
=> 100xx = 99.9999999999999999(infinity9(1digit))x
=> 99x = 98.99999999999999999(infinity9(1digit))
=> x= 0.9999999999999999(infinity9(1digit))
this means that x is both finite and infinite and that infinity=finite+1(digit)
1 is NOT EQUAL TO 0.9999999(infinite)....
the only conclusion we get from it is that infinite=finite+1
Good Luck, ;) 
sataiuto.............................

Kaustubh KatdareNot yet convinced with
"infinite=finite+1".
2 is 'finite'. So can I say 2+1 = 3 is infinite? Obviously not.
When we say 'infinite', we talk about something that is never ending and we definitely do not know what's at the end. However, through mathematics we can 'cancel' the infinite part (please correct me on this if I'm wrong)
That is, 0.99999...(infinite) minus 0.999999...(infinite) will equal to Zero. 
sataiuto
You got it wrong, what I meant was that a infinite number is a number that exists but it's impossible to know what it is at the same time, an infinite number is number that is always ahead of the finite number. If we try to find a final number it will be impossible. If we keep counting 999999999999999999999999999999999999999999999999 it will be impossible to find it because infinite is always 1 digit more than the number that we got when we stopped counting... when I said infinite=finite+1, the 1 doesn't mean a number but a digit or a "finger".The_Big_KNot yet convinced with
"infinite=finite+1".
2 is 'finite'. So can I say 2+1 = 3 is infinite? Obviously not.
When we say 'infinite', we talk about something that is never ending and we definitely do not know what's at the end. However, through mathematics we can 'cancel' the infinite part (please correct me on this if I'm wrong)
That is, 0.99999...(infinite) minus 0.999999...(infinite) will equal to Zero.
thus if someone says that infinite is 999999999999999999999999 he's wrong because it's actually 1 digit more than what he said but then if someone says that infinite is 999999999999999999999999+(9) he's wrong too because the final result will be always bigger than we can ever imagine, it's always growing and growing.....
let's say that in a classroom a teacher asks the students to define the number infinite and then one of them says: 999999999999999999999999999999999999999
and then another says "NO" it's 999999999999999999999999999999999999999999999999999999999999999999999999999999
and then another says "NO" it's 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999
they'll never find it because the infinite number is always "running away" from the number previously said....
Imagine that the number is afraid of us and that it keeps getting bigger so that we'll never find him. 
sheeko.99999999.......=1  ( 1 / ( 10 * infinity ) )
=1  ( 1 / infinity )
=1  0 =1 
sataiutosheeko.99999999.......=1  ( 1 / ( 10 * infinity ) )
=1  ( 1 / infinity )
=1  0 =1
hehe, you're so wrong...
10 times infinity is not infinity
"10*infinity= finite"
say: x=0.999999(infinite)
then
0.9999999(infinite)*10= 9.99999999...(infinityfinite digit)
actually....
it doesn't make any sense in reality or common sense but that's maths x)
if infinity is altered you'll get a concrete number even though it is not defined...
and 1/infinity is finite .... infinity are finite numbers following each other
thus
1  ( 1 / ( 10 * infinity )
= 1  1 / x (finite number)
= 1  x(finite number)
= 1  x
= infinity ;) 
skipperYes, that's more or less the key to it. Mathematicians regularly are upset by this question (I've noticed), since there is a succinct definition of a recurring fraction as a recurring series.
This illustrates that "1 does not exist", we can only ever approximate it. Despite what mathematicians have to say about it and the ratio of a circle's radius to its circumference. The mathematical descriptions/definitions still do not "generate" one of anything, nor complete the circle except on paper. As soon as you try to make "one bit" of storage you get an approximation, as soon as you try to quarter a circle or draw one accurately using the "perfect algorithm", accuracy and exactness vanish from the (physical) scene.
This appears to be a little detail that some mathwits step over continually (not the real bright ones, like 'meh'). Having tried this approximation a few times, I can personally vouch that it doesn't matter how many 9s you write down you can't get exactly one, ever. Never ever.
This is because, a circle is a mathematical idea, not a reality, ever. Nor is the value "one", except perhaps one electron, like when it hits a detector. But then, when or where does it do this?
Approximately "there" is where.
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