Is 0.999999.... (infinite) equal to 1?

[The_Big_K]

Looks like we have lot of confusion over the subject here .

So, why not dedicate a thread to it?

Is 0.999999.... (infinite) equal to 1? What do you have to say?

[mahul]

it 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.

[raj87verma88]

The 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.

[gohm]

Yep, 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.

[The_Big_K]

Gosh, I love everyone here so much!

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?

[vissin]

Well... 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.

[raj87verma88]

Some new threads
1. On Infinity
2. Proving the limits of universe
3. Life outside earth
how about them biggie

[gohm]

Exactly! 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!

Originally Posted by The_Big_K

Gosh, I love everyone here so much!

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?

[ash]

Vissin, 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

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*

[The_Big_K]

[another 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.rok]

About the topic I am in complete agreement with Vissin.

About infinity:
Mathematicaly, Lim (1/X) =infinity
X-->0

Physicaly, radius of the universe= infinity.

[suyash]

yes, 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

I might sound confusing, but then "God doesn't exactly play dice" , and there just might be some really great reason why we can't prove that 0.99999999 (till infinity) is not equal to 1 ...

[xheavenlyx]

I just could not hold myself here. Wow, amazing thread! And a long reply, so read it, if interested

I agree with Vissin 100.1111...%. Absolutely right. One of the best explainations!!!!

VISSIN:
0.999... = 3 * 0.333... = 3 * 1/3 = 1
ie. 0.999... = 1

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...

0.9999...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.

[MaRo]

I 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