Is 0.999... Really Equal to 1? Exploring the Mathematical Proofs

[wtfc]

I'm sure most of you already know this. The real point of the thread is finding different ways of approaching it. You see, I have a friend who refuses to accept what seems to me to be so obvious: .9 repeating (infinite 9s after the decimal) is exactly equal to the whole number 1.

Here is how I look at it.
You start of with .9999... and continue until you either run out of time or get tired. Now you are at a crossroads. At this stage you decide if it is still .9999... or 1. Conclusion is what one reaches when one is out of time. Just because you are tired of traveling through the infinite, .9999 does not equal 1 irrespective of all the mathematical equations.

Rounding off is only to solve equations. To further extend this, you either use whole numbers or fractions. You should not criss-cross domains like this.

 

[Hurkyl]

You start of with .9999... and continue until you either run out of time or get tired.

What is there to "continue"? .9999... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

 

[balkan]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

 

[Prometheus]

What is there to "continue"? .9999... is a decimal with an infinite number of nines; it is not a suggestion that you start with .9, then go to .99, then to .999, and continue in this fashion.

I sympathize with people who have trouble understanding infinity in mathematics.

Mathematical infinites exist by definition. It is not that there is any actual correlate to these infinities in the real world. For example, the series of numbers, 1, 2, 3, ..., is not infinite in the real world, where real people count them. Numbers are only infinite in the world of mathemeatics, where they can be defined as infinite, and thereby given attributes of this infinity.

When a person attempts to add actual instances of the number 9 to the end of this number, the world of mathematics is being substituted by the real world, and this is a mistake.

 

[Hurkyl]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities

(a) The real numbers are a mathematical "system" without "infinities".
(b) There are lots of other "system"s, each with their own pros and cons.
(c) You say it like the use of "infinities" is bad.
(d) What does science and quantization of matter have to do with anything?

 

[hello3719]

i can't wait till scientists figure out the smallest quantization of all matter and then use it to construct a new mathematical system without infinities
that'll rid the world of all this nonsense... or at least, i'll be a very happy man...

I'll take that as a joke.

 

[Prometheus]

I'll take that as a joke.

Are you laughing at his phraseology, or at the deeper meaning that he is trying to convey?

Or, do you understand the difference?

 

[loseyourname]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.

 

[HallsofIvy]

He's laughing because there is nothing nonsensical about infinity in mathematics. In fact, infinity, and particularly infinite sequences and series, can be quite useful.

and has nothing to do with physical discoveries.

 

[balkan]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

 

[hello3719]

well, to elaborate:
i'm perfectly fine with the concept of infinities within contemporary mathematics, it works fine for now and is quite usefull...
but i suspect that a full understanding of physics can only be made through a parallel evolvment of a new mathematical system... and i suspect this system not to involve infinities due to the simple fact that nothing in this universe is infinite, as far as we know... most likely not infinitely small either... only time will tell though...

the old mathematical system will still be appropriate for most tasks, just like classical physics compared to quantum mechanics, and it will have coherence, but it will not be usefull to describe the mechanisms of the universe to a sufficiently precise degree...

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

 

[Didd]

use your calculator to calculate 1/9.
It is 0.1111... and then 9/9=9*1/9=9*0.1111... but this means adding 0.1111...
itself 9 times. You will get 0.9999...
so, 9/9=9*1/9=0.9999...
But 9/9=1. So 1=0.999...

 

[Hurkyl]

And I'll reiterate; the theory of the real numbers has no infinities, and the only infinitessimal is 0.

 

[balkan]

hahaha, This would mean that our "old"(includes finite and infinite concepts) mathematical system would be more complete than the "new"(includes only finite concepts) one.

it would be more "complete", yes, but not more precise. I'm not just talking about a new array of functions, I'm talking about an entirely new system...
just like quantum mechanics are more precise dispite the fact that it deals with quantization and probability, where classical physics claims to predict precise locations and have fluent change of energy levels...
ironically, the "imperfect"/"unprecise" theory is more correct...

hurkyl:
i know, but that's not what I'm talking about... I'm talking about a new way of calculating...

 

[loseyourname]

Kind of like the way cellular automata have been used recently to model biological systems?

 

[balkan]

Kind of like the way cellular automata have been used recently to model biological systems?

are you asking me?
in that case, i'd say... erhm... something like it... but cellular automata uses regular math of course... on the other hand, cellular automata is time dependant and deals with uncertainties aswell, am i right? so i guess it could be paralleled...
it's hard to imagine another kind of math, since we're so used to think in the old mathematical patterns, but i think at some point it migth become usefull or even necessary to evolve it, just like classical physics...

 

[Hurkyl]

Speaking of quantum mechanics seems to be antithetical; mathematically, QM didn't replace anything, but it did push existing analysis to its limits (haha) and spurred developments in the subject. As far as I know, LQG and ST do the same.

 

[balkan]

Speaking of quantum mechanics seems to be antithetical; mathematically, QM didn't replace anything, but it did push existing analysis to its limits (haha) and spurred developments in the subject. As far as I know, LQG and ST do the same.

did i say it replaced anything? no... fine then... did i say the new math would replace the old? certainly not...
but it didn't just push existing analysis to its limits. classical physics were 100% unable to explain the phenomenoms that quantum mechanics did... so it did much more than that...

seriously, what is your problem with the idea of a new way of calculating? like i said, it should be coherent, so it wouldn't replace what you seem to hold so dear...

 

[Hurkyl]

classical physics were 100% unable to explain the phenomenoms that quantum mechanics did

Yes, but I was talking about the mathematics, remember?

seriously, what is your problem

I get the impression that you are calling for a change, not for any particular theoretical or practical reason, but because for whatever reason you think "infinities" are "bad".

This sort of affliction is far too common, and is a great contributor to the popular misunderstanding of mathematics and science, so I like to spend a good deal of effort trying to combat it. If I'm seeing it where it does not exist, I apologize.

 

[hello3719]

are you asking me?
in that case, i'd say... erhm... something like it... but cellular automata uses regular math of course... on the other hand, cellular automata is time dependant and deals with uncertainties aswell, am i right? so i guess it could be paralleled...
it's hard to imagine another kind of math, since we're so used to think in the old mathematical patterns, but i think at some point it migth become usefull or even necessary to evolve it, just like classical physics...

Mathematics is not at all like physics. In mathematics we establish the axioms, in physics, we do not even know if there is any or what are the axioms of the system.That's why we can develop ANY theory that is compatible with experiments, observations and the deductions. So even we change the mathematical system we would have to add axioms which won't make it a new kind of math.

 

[balkan]

Yes, but I was talking about the mathematics, remember?

I get the impression that you are calling for a change, not for any particular theoretical or practical reason, but because for whatever reason you think "infinities" are "bad". If I'm seeing it where it does not exist, I apologize.

you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...

Mathematics is not at all like physics. In mathematics we establish the axioms, in physics, we do not even know if there is any or what are the axioms of the system.That's why we can develop ANY theory that is compatible with experiments, observations and the deductions. So even we change the mathematical system we would have to add axioms which won't make it a new kind of math.

this makes no sense... since we don't know if there are axioms at all, we can develop any model and any theory using our mathematical system, which has inevitably got axioms involved?

anyway...
we wouldn't have to "add" axioms... the new system should simply just be backwards compatible with the new system (so that a proper adjustment of the new system would give the same results as working in the old one). But it should be more capable of explaining the physical phenomenons that seems to care **** about being in the dimension visible to humans...
and that's my entire point: there are phenomenons which a are very difficult to model and create theory for using our dimensional dependent math system...

look, I'm not expecting you to understand what i mean. it's obviously not that easy to swallow, but just like new concepts like the integral, infinity and zero have been involved, so can other new concepts... perhaps these concepts will be developed so that our present math system cannot copy it very well, while the new system could be used like the old one...

 

[arivero]

=1.0000000000...

0.999999999... is not equal to the whole number 1 but to the real number 1.000000000000...

Check the definition of real number; a pair of converging sequences is actually involved.

 

[selfAdjoint]

Actually you don't have to sum the series. The completeness property of the real numbers guarantees there is a real number, which you can prove to be unique, as the least upper bound of the set of numbers { x_i : x_i = 1.999...999 for i decimal places} . Proving that the number is neither greater nor less than 2 is then easy.

 

[hello3719]

you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...


this makes no sense... since we don't know if there are axioms at all, we can develop any model and any theory using our mathematical system, which has inevitably got axioms involved?

anyway...
we wouldn't have to "add" axioms... the new system should simply just be backwards compatible with the new system (so that a proper adjustment of the new system would give the same results as working in the old one). But it should be more capable of explaining the physical phenomenons that seems to care **** about being in the dimension visible to humans...
and that's my entire point: there are phenomenons which a are very difficult to model and create theory for using our dimensional dependent math system...

look, I'm not expecting you to understand what i mean. it's obviously not that easy to swallow, but just like new concepts like the integral, infinity and zero have been involved, so can other new concepts... perhaps these concepts will be developed so that our present math system cannot copy it very well, while the new system could be used like the old one...

Do you know what is an axiom ? I guess you can't swallow well what i said.

you are... infinities works splendidly... i would hate to do quantum mechanics without it... it is for a practical change however... to fully develop string theory e.g., many physicists believe a new mathematical system is neccessary...

Define "new mathematical system ", we are not on the same track.

 

[balkan]

Do you know what is an axiom ? I guess you can't swallow well what i said.

Define "new mathematical system ", we are not on the same track.

yes i do know what an axiom is... but your sentence made no sense...
"new mathematical system" like imaginary number theory was... just "bigger"... it can explain phenomenoms that the "old" theory can't, but is still coherent with the old... i wouldn't quite call it a new paradigm or anything, since it would still just be "math"...
they're trying it with string theory, but I'm having my doubts about it... but you got to start somewhere, right?

 

[MathematicalNewbie]

I was doing some research about .999~ = 1 tonight and I came across this page. I agree completely that .999~ is indeed equal to 1. However, I have question. Based on what I read, I concluded that .999~ can be expressed as an infinite geometric series such as 9/10 + 9/100 + 9/1000... etc. From what I've learned in math class, this should converge to 1, but not equal it. However, the proofs still show that .99~ is 1. I was wondering that since this series "approaches" 1, but still equals it, then can the sum of any infinite geometric series actually equal a sum instead of converging towards one? For example, if we are given a list of numbers in a series and I find that the sum of these numbers converges to say.. 30, then would there be a way I could prove that the sum is EQUAL to 30?

I may be confused about the whole concept of series in general, so please correct me if I'm wrong about anything here.

 

[chroot]

If a sequence converges to a number, then, when summed over an infinite number of terms, it would equal that number.

This is the meat of the entire debate that usually surrounds the 0.999... = 1 issue. 0.999... only equals one when you tack on an infinite number of nines, which makes some people uneasy. There is nothing mathematically wrong with summing an infinite number of terms, however, no matter how conceptually difficult it is for some people to grasp. The notation 0.999... specifically means an infinite number of nines, and is equal to one.

- Warren

 

[Hurkyl]

I find that the sum of these numbers converges to say.. 30, then would there be a way I could prove that the sum is EQUAL to 30?

Yes, it's true by definition. The term "sum of an infinite series" means "the value to which the partial sums converge".

So, if the partial sums converge to 30, then the sum is equal to 30.

 

[MathematicalNewbie]

Yes, it's true by definition. The term "sum of an infinite series" means "the value to which the partial sums converge".

So, if the partial sums converge to 30, then the sum is equal to 30.

Well, I guess that answers my question, but I still wonder why my math teacher specifically marks points off for writing "the sum of the series is 30" as opposed to "the sum of the series approaches 30 or converges to 30."

 

[unicorn]

I am new around here, so I don't know if anybody presented this already, but here goes:
0.999999...=x
10x=9.99999...=9-0.99999...=9-x
9x=9
x=1
And yes, sum of infinite geometrical series is one particular value when b(n+1)/b(n)<1. It's like in Zono's paradox with Achileus and a turtlle...

 

[musky_ox]

IMO its just rediculous to try to say that 0.9 repreating = 1. We KNOW that it is infinately close to, but will NEVER actually EQUAL 1. Maybe using mathematics we can say that, but it is not logically true, so there must be flaws in mathematics. You can give all the arguements you want to try and prove it, but its kind of a circular arguement. You have to assume that math is flawless, and i don't see why would would assume this when it contradicts itself.

Also, weird things happen with infinity. For example, what is infinity/intinity... what is infinity/infinity squared. How do you enter notation in this forum?

 

[chroot]

IMO its just rediculous to try to say that 0.9 repreating = 1. We KNOW that it is infinately close to, but will NEVER actually EQUAL 1.

It will be equal to one when there are an infinite number of nines, which is precisely what the notation 0.9... means.

Maybe using mathematics we can say that, but it is not logically true, so there must be flaws in mathematics.

Mathematics does not have flaws. It begins with a set of axioms, and derives a set of conclusions. You can start with different axioms if you like, and you'll get different conclusions. The axioms normally used in mathematics can be used to prove 0.9... = 1. There is no question about it, nor is there any room for debate.

You can give all the arguements you want to try and prove it, but its kind of a circular arguement. You have to assume that math is flawless, and i don't see why would would assume this when it contradicts itself.

It does not contradict itself. Once again, all you must do is select your axioms, and the results follow from them.

Also, weird things happen with infinity. For example, what is infinity/intinity...

Indeterminate.

what is infinity/infinity squared.

Indeterminate.

How do you enter notation in this forum?

Like this:

$$\frac{\infty}{\infty}$$

- Warren

 

[musky_ox]

It will be equal to one when there are an infinite number of nines, which is precisely what the notation 0.9... means.

1-0.999...=1/infinity

It is infinitly close to 0, but will NEVER reach it. It is like some immortal person writing down zeros forever and ever after the decimal point, with the idea that he is going to write a 1 as his last digit. We know that he will never reach his last digit, but his number will never equal zero, because it isn't complete... even though he gets infinitely close to it.

 

[chroot]

1-0.999...=1/infinity
It is infinitly close to 0, but will NEVER reach it. It is like some immortal person writing down zeros forever and ever after the decimal point, with the idea that he is going to write a 1 as his last digit. We know that he will never reach his last digit, but his number will never equal zero, because it isn't complete... even though he gets infinitely close to it.

This is a shaky argument, and I suspect you know it. No real person ever needs to be capable of writing down an infinite number of zeros to make an expression true. The operative word is if. If a person could write down an infinite number of nines, the resulting number would be equal to one. That's what the notation "0.99..." means, regardless of whether or not a human hand can write it out digit for digit.

- Warren

 

[HallsofIvy]

Not an argument at all! 1- 0.999... "is infinitely close to 0 but will never reach it" only means that the poster does not understand what 0.999... (or 1- 0.999...) means. Numbers are fixed, specific things- they are not "reaching" something and don't depend on any person, real or hypothetical, writing down digits. Digits are something we use for convenience- every number has an existence of its own completely independent of "digits". 0.999... IS 1.0 because the notation "0.999..." means "the limit of the sequence 0.9, 0.99, 0.999, etc." Notice that the definition is clear: an infinite decimal is the limit of such a sequence, not the sequence (a string of digits) itself.