.9999... = 1?

[Elemennop]

Hey guys,

so I have a friend in the grade above me taking calculus, and he told me that .99 repeating (think it means infinite 9's after the period) is the same as 1. HE said that's what his teacher told him, but it sounds ridiculous, and I think his teacher's ****in with him (he's done that before).

I tried googling it, but it seems like even mathematicians cant seem to agree on this. Is this like one of those unsolved problems in math?

 

[Gerbil]

It's true. I'm a Math major, and I can prove it to you. Just give me a second.

So, let's start with the following:

Would you agree that...

0.3333... (infinite 3s, we'll use "..." to define infinite)

0.333... = (1/3)

Would you agree that...

3 * 0.333... = 0.999...

Would you also agree that...

3 * (1/3) = 1

If you agreed to all of the above, then by substitution the following is true:

1 = 3 * (1/3) = 3 * 0.333... = 0.999...

Therefore, by the Transitive Property, 1 = 0.999...

Take a moment to review this. This is an entirely correct proof lol. (I know, My Mind was BLOWN when I saw this years ago LOL)

 

[Ch3s]

x=.9999999999

multiply by 10

10x=9.9999999999

subtract x

9x=9

divide by 9

x=1 there you go

or simpler is

1/3=.3333333

2/3=.6666666

1/3+2/3=1

and .666666666+.3333333=.9999999


Edit: no it isn't unsolved, and it's agreed upon. Without this fact, limits dont work. Another way to look at it is this:

rule: Any two real numbers MUST have one number in between them (this is a definite rule of math)

.9999999999.... and 1 have no number between them, so they HAVE to be the same thing.

 

[Gerbil]

Yeah, there are multiple ways to prove it. : )

Did you know...

There are no positive integers a, b, and n > 1 such that the following is true:

(a^n - b^n) | (a^n + b^n)

If you can prove that to be true, you've had some number theory courses lol

(hint: this is a problem I am having trouble with LMAO I was just curious if anyone on here can give me a good proof )

Calculus is AMAZING isn't it?

 

[Handorin]

x=.9999999999

multiply by 10

10x=9.9999999999

subtract x

9.1111111111111111111x = 9

divide by 9

.1111111111111 != 1 there you go

Fixed .

 

[o-Serin-o]

It's true. I'm a Math major, and I can prove it to you. Just give me a second.

So, let's start with the following:

Would you agree that...

0.3333... (infinite 3s, we'll use "..." to define infinite)

0.333... = (1/3)

Would you agree that...

3 * 0.333... = 0.999...

Would you also agree that...

3 * (1/3) = 1

If you agreed to all of the above, then by sunstitution the following is true:

1 = 3 * (1/3) = 3 * 0.333... = 0.999...

Therefore, by the Transitive Property, 1 = 0.999...

Take a moment to review this. This is an entirely correct proof lol. (I know, My Mind was BLOWN when I saw this years ago LOL)

Mind = Dumped

 

[Ch3s]

Fixed .

lol unfortunately 10x-x is 9x. sorry but you also did your math wrong, since if you did it the way you were doing it, it would be 9.00000000...1x. It may seem weird, but you can do it, and in fact must do it the way i did it.

 

[-ACE-]

Well, 0.9999... (repeating) isn't really a number, it's a concept (like infinity). It's essentially the closest thing there is to 1, without being 1. Just like 0.33333 (repeating) isn't a third, but it is often recognized as such because there is no closer way to accurately describe it in decimal (non-fraction) form.

 

[Gerbil]

lol unfortunately 10x-x is 9x. sorry but you also did your math wrong, since if you did it the way you were doing it, it would be 9.00000000...1x. It may seem weird, but you can do it, and in fact must do it the way i did it.

This.

Though, I see why Hando got confused. He subbed and actually subtracted the value for x.


What you should've said to make it more clear is to subtract 1x (since this is a forum, it's hard to explain in words a simple thing, 1x makes it easier for text imo)

 

[Handorin]

lol unfortunately 10x-x is 9x. sorry but you also did your math wrong, since if you did it the way you were doing it, it would be 9.00000000...1x. It may seem weird, but you can do it, and in fact must do it the way i did it.

I knew there was something wrong when I typed it, mainly that I forgot the X. lol

Theory is still bonkers though.

 

[Scidadle]

...Do my algebra for me? :D

 

[Ch3s]

Yeah, there are multiple ways to prove it. : )

Did you know...

There are no positive integers a, b, and n > 1 such that the following is true:

(a^n - b^n) | (a^n + b^n)

If you can prove that to be true, you've had some number theory courses lol

(hint: this is a problem I am having trouble with LMAO I was just curious if anyone on here can give me a good proof )

Calculus is AMAZING isn't it?

I did know that, but the last lecture i watched from terry tao he mentioned something about it being currently unproven. If thats the case, you are currently trying to find a proof for a 350 year old riddle. idk maybe im mistaken.

Edit: Handorin, i just found it hilarious that you are calling the theory that makes limits possible bonkers, and the thing under your name says "know your limits." Pretty hypocritical! lol jk but it is funny

 

[Gerbil]

The craziest PROVEN theory I know of involves quantum mechanics and physics...

The idea is that time is faster/slower in various places in our inverse. Proven many times.

The next one?

Time doesn't exist at all. Mind = blown. Go look for it lol

 

[Handorin]

The craziest PROVEN theory I know of involves quantum mechanics and physics...

The idea is that time is faster/slower in various places in our inverse. Proven many times.

The next one?

Time doesn't exist at all. Mind = blown. Go look for it lol

That's cause time is just a measurement we use to describe actions around us. >_>

Haven't they started two watches, and put one on a jet, and that one slowed down compared to the one on the ground?

 

[Gerbil]

Sorry for double post. I did know that, but the last lecture i watched from terry tao he mentioned something about it being currently unproven. If thats the case, you are currently trying to find a proof for a 350 year old riddle. idk maybe im mistaken.

I think this one is doable just because it's in the book and I can buy an answer key online for a few bucks LOL. But I know of a lot of similar problems to this that you may be thinking of.

The last problem I'll throw in this topic to derail it is the following:

I am doing a research project on a certain idea. The idea is this: is there some sort of Pyramidal Inequality set such that, given values for 4 areas of 4 distinct pyramids (yet not unique, i.e. they may share the same area), I can ALWAYS construct a pyramid given certain constrained limits?

The idea came from this (aside from my instructor LOL). We have what's known as the Triangle Inequality Theorem. It is as follows:

For a triangle to exist, the sum of any two sides of a triangle must exceed the length of the third side such as the following: Given sides a, b, and c...

a + b > c
a + c > b
b + c > a

We want to research on the idea of raising the geometry from 2D (triangles) to 3D (pyramid). When you go up a dimension, you go from Length to Areas.

Interesting yeah? I thought so too :D

 

[Gerbil]

That's cause time is just a measurement we use to describe actions around us. >_>

Haven't they started two watches, and put one on a jet, and that one slowed down compared to the one on the ground?

The concept of time in general is seeming not to exist according to their theory. It's very much beyond that idea lol.

As for the clocks, that's possible. I do know that they took atomic clocks that were synchronized and put them on satellites and left some on the ground, only to discover they were all different. Again, there's a TON of info on this. : )

 

[Ch3s]

That's cause time is just a measurement we use to describe actions around us. >_>

Haven't they started two watches, and put one on a jet, and that one slowed down compared to the one on the ground?

What your describing is Einsteins theory of relativity, where as you approach the speed of light, time slows, and all time stops in an inertial reference frame that is moving at the speed of light. What he is describing is that scientists proved that such a reference frame exists as something other than light. However if this were proven, all of Einsteins theory of relativity would be faulted. So as of know its a possible theory, with mathematical proofs that have variables in them. I think that is what he was talking about?

Edit: The clocks thing has happened, because they got it up to mach 5 (in that experiment) so the time slowed on the clock inside. They went for a while and the time was different when it got back. Unfortunately, other than the fact that it has to do with Einsteins theory of relativity, and that nothing can go faster than the speed of light, i can't tell you the reason why.

 

[Elemennop]

Yeah, there are multiple ways to prove it. : )

Did you know...

There are no positive integers a, b, and n > 1 such that the following is true:

(a^n - b^n) | (a^n + b^n)

If you can prove that to be true, you've had some number theory courses lol

(hint: this is a problem I am having trouble with LMAO I was just curious if anyone on here can give me a good proof )

Calculus is AMAZING isn't it?

(a^n + b^n) / (a^n - b^n) = 1 + 2* (b^n / (a^n - b^n) ) is an integer, so assuming some combo like that exists, we have

b^n / (a^n - b^n) = k/2 for k positive integer (assuming wlog that a > b, and gcd(a,b) = 1).

so rearranging, we have 1 / ((a/b)^n - 1) = k/2, so that k + 2 = k * (a/b)^n. LHS is positive integer, so RHS is too. hence, b^n | k (since a,b coprime), so k = l * b^n for some positive integer l. but then we have l*b^n + 2 = a^n * l, so that (a^n - b^n) * l = 2, where a, b, n, l are all positive integers, a,b,n>1.

From here it's pretty straightforward (just do casework for l = 1 and l = 2, and you just get that a^n - b^n is too big in any case).

Hence (a^n + b^n) / (a^n - b^n) is never an integer for a,b,n integers > 1.

Or something like that, I'm not that good at math.

 

[Ch3s]

where did you get that proof? i want to check out whoever did that's credentials to see if i should bother checking to make sure it works

 

[_KuyaSombreo_]

what exactly is infinty anyway?

 

[nevershootme]

simple...

















DO NOT DIVIDE BY ZERO!

 

[~Twitch~]

It's true. I'm a Math major, and I can prove it to you. Just give me a second.

So, let's start with the following:

Would you agree that...

0.3333... (infinite 3s, we'll use "..." to define infinite)

0.333... = (1/3)

Would you agree that...

3 * 0.333... = 0.999...

Would you also agree that...

3 * (1/3) = 1

If you agreed to all of the above, then by substitution the following is true:

1 = 3 * (1/3) = 3 * 0.333... = 0.999...

Therefore, by the Transitive Property, 1 = 0.999...

Take a moment to review this. This is an entirely correct proof lol. (I know, My Mind was BLOWN when I saw this years ago LOL)

*mind blown* this makes sense and i'm suprised it does because i just got out of middle school. this is really cool.

 

[Elemennop]

where did you get that proof? i want to check out whoever did that's credentials to see if i should bother checking to make sure it works

It's mine.

 

[dashdancedan]

There's a website with all of this information already on it.

 

[Ch3s]

It's mine.

I dont wanna be rude but i would be extremely surprised if it really was. Especially since you claim to suck at math, and .9 repeated =1 is important to number theory so you probably dont know that well. Regardless of who made the proof, it is very creative so if you made it don't take offense to this. the steps of the proof work, but as far as it proving the theory im not completely sure.

Infinity is an idea used to describe the largest number possible, but because there is no limit, the largest number possible can't actually exist, except for use in mathematical equations where it cancels out, otherwise it remains as a variable of sorts. (e.g. anything divided by infinity is so small it may as well be zero, so it is counted as such and the infinity term cancels. This is especially used in derivatives/calculus)

Dashdancedan: link please? Ive read most of it or watched lectures but sometimes it helps to have kids explaining it (this thread) because scientific terms can be hard to understand.

 

[Elemennop]

I'm an honors physics student at Purdue, I have some idea of what I'm talking about. I'm pretty confident that the proof is correct, regardless of my experience with the .999... = 1 debate.

 

[-ACE-]

Lol. I seriously can't believe this thread has gone this far.

what exactly is infinty anyway?

Infinity is a concept which essentially means "never-ending". Mathematically, infinity can be described as any number divided by zero (usually labeled as "undefined"). It is not a number, and it baffles me that some people still refer to it as so.

 

[j00t]

Someone get moogle in here

he has a master's in math and would love this thread

 

[dashdancedan]

www.wikipedia.org

Also, because there is no number between .9 repeating and 1...and other rigorous proofs blah blah math infinity blah limits blah

 

[Ch3s]

I'm an honors physics student at Purdue, I have some idea of what I'm talking about. I'm pretty confident that the proof is correct, regardless of my experience with the .999... = 1 debate.

my apologies, i guess its shallow but im more inclined to believe you now. But yes, i found no issues with your proof. and the thread went this far because math and physics are my favorite thing to talk about, my bad

 

[_Keno_]

I dont wanna be rude but i would be extremely surprised if it really was.

Actually he is a ****ing math genious, I cant tell if the topic itself is him trolling or not.

But .3333 repeating equaling 1/3 is literally the same argument as .9999 repeating, Gerbil.

.9999 repeating isn't one, it just might as well be.

 

[Gerbil]

@Cheap

The thing is, 1/3 is a rational number. (any digits that can be represented in the form of m / n is a rational number), therefore .333... exists. The proof is valid still from this perspective.

@OP

I think yuo misunderstood my presented problem (and this is probably my fault for how it's typed)

the " | " bar is not the divisor, it means "this part" divides "this part." It's not "this part" divided by "this part."

: )

 

[Kosine]

@ gerbil, his proof is correct, and i don't think he misinterpreted your notation

 

[Elemennop]

Gerbil,

If that's your criticism of my solution, you didn't even bother reading what I said.

 

[_Keno_]

@Cheap
The thing is, 1/3 is a rational number. (any digits that can be represented in the form of m / n is a rational number), therefore .333... exists. The proof is valid still from this perspective.

1/3 is not a rational number, in this case m/n = ~.333 repeating, not exactly .333 repeating. Saying that .333 repeating equals 1/3 is literally the exact same as saying that .999 repeating equals 1, which is what we were trying to prove. You can't use your answer to prove your answer. That is circular. I'm just saying that your first proof is wrong, as the first page is all that I read.

and wait, is elemennop seva?

 

[Gerbil]

Um.... 1/3 IS a rational number.

http://www.mathsisfun.com/rational-numbers.html

http://www.eduplace.com/math/mathsteps/7/a/index.html

http://science.jrank.org/pages/5743/Rational-Number.html

@OP

I hadn't read it quite all the way, I had glanced it late last night. I read it again today, and I personally can't find anything wrong with it. I'm gonna run it by my Prof and see what he says about it, and then when he gives me his solution I'll bring it back to compare. : )

EDIT:

I just realized this question is a huge debate for a lot of people, so I think I'm getting involved in something that I was just trying to be friendly about lol. I saw some other proofs for it, but some of them I don't really agree with (like the geometric series proof, I think it's only a show for convergence, but not an actual equivalence... but that's just me)

 

[Ch3s]

1/3 is not a rational number, in this case m/n = ~.333 repeating, not exactly .333 repeating. Saying that .333 repeating equals 1/3 is literally the exact same as saying that .999 repeating equals 1, which is what we were trying to prove. You can't use your answer to prove your answer. That is circular. I'm just saying that your first proof is wrong, as the first page is all that I read.

and wait, is elemennop seva?

read my first post for another more elegant proof. and 1/3 is by definition .333 repeating, although i thought what you thought at first. and in math, many proofs are using one form of an equation to prove that another form of that same equation, when altered without breaking any math rules, is equivalent, and therefore the equation works. Idk if that makes sense but basically, yes you can use your answer to prove your answer in a different form. and yes, it's circular, suck it up. lol

 

[blindghost]

Ok... I haven't read all the bull shnit on this thread. But the best way to approach this is to say for all intents and purposes in the physical world 10^-100 is a useless number (0.0000...90 more zeros...000001). Therefore 1-10^-100 is also a useless amount to subtract by (0.9999...90 more 9's...999999). The universe stops caring about such insignificance. Therefore the universe rounds it to 1.

The whole 1/3+1/3+1/3 is a semantic loophole and isn't math. And don't even try "what if I subtract a googleplex worth or 10^-100 but round in between each step?". It's just more semantics...

::EDIT:: I have now read all the bull shnit. I like this argument.

 

[_Keno_]

well, 1/3 is a rational number, but I do not agree that .333 repeating is. Nor do I agree that 1/3 = .333 repeating. In reality, they are never actually "exactly" the same number, though for mathematical purposes they might as well be the same number.

I'm arguing for "they aren't equal, but they might as well be" in both the OP's case and 1/3 case.

And to the guy above me, yes, even the universe rounds numbers to 1 in the case of .999 repeating.

 

[Elemennop]

well, 1/3 is a rational number, but I do not agree that .333 repeating is. Nor do I agree that 1/3 = .333 repeating. In reality, they are never actually "exactly" the same number, though for mathematical purposes they might as well be the same number..

why don't you think .33repeating is rational? Which decimal numbers do you think are rational?