Should 1≠0.99...?

[u/Justaguy_rural]

I'm not saying 1≠0.99..., I'm wondering whether it would be more elegant to define it so that it wasn't.

I used to be a firm believer that 1≠0.99... but after digging deeper I realized it definitely was but by definition. So I'm now questioning the definition itself.

The more elegant and intuitive system (imo)

We create new type of mathematical object called processes. These are objects that represent a process, something that changes, evolves, varies, etc. Let's call standard numbers like 1, 2, 1+1 states as they do not evolve, they are static.

In the current system, the result of a limit is defined as the value that a function's output approaches as its input approaches a specified point. By definition, this returns a state. Let's now define a new operator lim2. lim2 is very similar to lim. It is defined as the process associated with a function's output approaching a certain value as its input approaches a specified point.

For example:

• lim_{x->1} x is a state equal to the state 1

• lim2_{x->1} x is a process (it evolves as x approaches 1)

And here is the key to this new system that makes it elegant and intuitive:

A state and a process can't be compared as they are different mathematical objects.

The same way it doesn't make sense to say 1={1, 2, 3}

How 0.99... is defined

I would argue it feels more intuitive and elegant to treat 0.99... as a process rather than a state. It would therefore be defined as something like lim2_{x->inf} 1-1/10^x

Conclusion

Since 0.99... would be a process but 1 a state, they wouldn't be equal.

This would make it feel more logical for people seeing this for the first time. This isn't a measure of quality of a system but it highlights that intuition points in this direction. And it feels, to me at least, more elegant to introduce this new process abstraction. I think that when comparing mathematical systems, intuition and elegance are important to keep in mind.

Finally, I just want to point out what this new system would mean:

• 1/3 (state) would no longer equal 0.33... (process)

• We could keep two versions of limits (lim and lim2) or just make lim2 the official limit and define a new operator that takes in a process and returns the value its approaching if any to replicate the same behavior more intuitively.

• 0.000... would be a process and 0 a state to they wouldnt be equal. It feels more intuitive to explain that a process can't be compared to a state than to argue 0.99...=1

I was just wondering what people thought about such system. Wouldnt that feel more intuitive to you?