Is there a rigorous proof that 1 equals 0.999...?

[micromass]

micromass submitted a new PF Insights post

Is There a Rigorous Proof Of 1 = 0.999...?

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[jedishrfu]

Great insight, thanks Micro.

 

[KSG4592]

Always find these fascinating, even if most of it goes over my head at this point.

 

[kreil]

Nice!

The informal proof I always share with people is that 1/9=0.111..., 2/9=0.222..., 3/9=0.333..., and so on until 9/9=0.999...=1

 

[Hornbein]

micromass submitted a new PF Insights post

Is There a Rigorous Proof Of 1 = 0.999...?

Continue reading the Original PF Insights Post.

I suggest 1 - 0.9999... = 0.0000...

 

[gill1109]

Here is a number system in which 1 is not equal to 0.9999... and it moreover is rather useful in game theory, and some people even imagine it might be useful sometime in the future, in physics: J H Conway's "surreal numbers". https://en.wikipedia.org/wiki/Surreal_number

 

[micromass]

Here is a number system in which 1 is not equal to 0.9999... and it moreover is rather useful in game theory, and some people even imagine it might be useful sometime in the future, in physics: J H Conway's "surreal numbers". https://en.wikipedia.org/wiki/Surreal_number

Whether $1=0.9999...$ in the surreals depends highly on the definitions for $0.9999...$. Some definitions make it equal, others don't.

 

[gill1109]

Whether $1=0.9999...$ in the surreals depends highly on the definitions for $0.9999...$. Some definitions make it equal, others don't.

Yes you are right. However the surreals do include numbers (lots and lots of them!) which are strictly between all of 0.99999...9 (any number of repetitions) and 1.

 

[micromass]

Yes you are right. However the surreals do include numbers (lots and lots of them!) which are strictly between all of 0.99999...9 (any number of repetitions) and 1.

That depends how many repetitions of $9$ you take in $0.999...$. If you take countably many, then sure. But if you index over all ordinals, then no.

 

[WWGD]

You can also use, though not as nice, the perspective of the Reals as a metric space, together with the Archimedean Principle: then d(x,y)=0 iff x=y. Let then $x=1 , y=0.9999...$Then $d(x,y)=|x-y|$ can be made indefinitely small ( by going farther along the string of 9's ), forcing $|x-y|=0$ , forcing $x=y$. More formally, for any $\epsilon >0$, we can make $|x-y|< \epsilon$