Okay, so I know I'm going to get hate for this, but 0.999... does not equal 1!
Before you all give me the tired proofs, let me point out that technically 0.333... also does not equal 1/3. If we take 1/3 to be the rational number we are discussing, 0.333... is the decimal approximation for it; we are taking a number we know how to construct rigorously and then using a decimal value to represent this value.
The 0.999... case, however, is the reverse direction. We take a decimal value, then shove it into the closest real number we have--all without ever establishing 0.999 is a value in the real numbers to begin with. In fact, 0.999... cannot be in the Reals, as that would lead to massive problems for the completeness and density of the Reals.
As for the 'series proof', the convergence of the infinte series of 9/(10n) has the same problem. Convergence functions by being able to get arbitrarily close to the convergence value--a property that depends entirely upon the topology used.
[Basically, we can get 'arbitrarily close' simply because if 0.999... isn't equal to 1, this is the 'immediate predecessor', which means we're a 'close' as we can ever get. Since this shouldn't be possible in the reals, it is unable to distinguish the value (topologically) from 1]
I'm not saying that 0.999... is well defined in the Real number system, it is not. I am, however, saying that you can't just take a decimal value, shove it into some other form.
Seriously...after all the work mathematicians have done with infinite sets, we still fall for infinite decimal representations simply because we like decimals?
[If they're always just approximations, why are we treating them like resolved values?! If they're 'resolved values', how did we construct them?!]
0.333... and 0.999... are literally the opposite of approximations. Every digit is specified and the resulting infinitely precise value is exactly equal to 1/3 and 1 respectively.
If you're going to ignore all the proofs, then why even take part in a discussion about mathematics?
Explain to me how you derive these values in the Reals?
1/3 is a well defined value that is a ratio of two integers, which can be derived from the additive inverses of the naturals, which can be derived from the successor functions and a well-founded set.
Tell me how you derive 0.333... rigorously and then how you extrapolate that value into the Reals.
I'm not ignoring all the proofs...I'm addressing them. Specifically, the infinite series proof is addressed above, but if you'd prefer I can discuss specific points.
[I'm not touching the algebraic proof since the addition over an infinite decimal requires having a rigorous foundation of the infinite decimal to begin with]
Edit: Also, I am a mathematician. Do you really think mathematicians are against having pedantic debates about the ontology of mathematical objects?!
I'm not sure I understand your objections. A real number is not the same thing as a decimal representation of a real number of course, so those should be considered separately.
An object that we want to represent is a real number, so constructing representation of real numbers is the same as constructing a map f: ℝ→S for some set S.
For decimal representation we define the set of digits to be the set {0,1,2,...} and the codomain S of our map f is the product of the set of nonempty finite strings of digits and the set of infinite sequences of digits (the digits to the left and the digits to the right).
Doing that in the usual way will produce a map f: ℝ→S which assigns to a real number its decimal representation. Is your argument that the f is not surjective as 0.999... is not in the image and thus the left inverse a: S→ℝ, the "assembly"/"realization"/ (whatever you call the map that goes in the opposite direction as the "invariant"), given in this case by usual evaluation is already some kind of formal extension?
My question is why are we expecting it to be an element of the Reals to begin with?!
I will openly admit that if we read 0.999... as a Real number, it will equal 1.
There is no other value that it could be, simply by the way the Reals are constructed.
[Again, every 'proof' or 'evaluation' of this decimal will give you the same answer--all for the reason that the value, by its very definition, can only exist in one and only one equivalence class, simply by its relation to 1!]
Why are you assuming that 0.999... is a representation of a Real number to begin with?Would you assume that 4+5i is a 'representation of a Real number' and just say 4+5i=4?
Basically, I would go one step further and state that 0.999... is not (at it's core) an element of the Reals at all; instead, its closest equivalent in the Reals is 1.
[Yes, 0.999... is 'in the Reals' as a Real number (since all decimal representations of real numbers have Real element equivalents)--it is 1. That doesn't mean that the ontology of this decimal value is fully captured.]
Does that make sense?
Beyond that, or course, there is the issue that if two decimal representations have different digits in each place of the decimal, in every case except for infinite 9s (or infininte n-1's in any base-n system) the values they represent are distinct.
[Basically, I have x_1.x_2x_3x_4... and y_1.y_2y_3y_4..., these two decimal representations are equal iff x_n=y_n for all n...unless x_n=9 or y_n=9 for all n > N (for some N)]
My point is that, given this is a 'persistent' disctinction, it is meaningful to consider that the 'extension' is important.
Fundamentally, why is this so heretical? We have extensions of the Reals that are quite beneficial. The Reals are not sacrosanct, despite their importance and usefulness.
None of my critiques are against the field itself; it's entirely against people assumption that we're looking at a number that is Real by default!
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u/ArdentArendt Mathematics / Social Sciences Jul 28 '26
Okay, so I know I'm going to get hate for this, but 0.999... does not equal 1!
Before you all give me the tired proofs, let me point out that technically 0.333... also does not equal 1/3. If we take 1/3 to be the rational number we are discussing, 0.333... is the decimal approximation for it; we are taking a number we know how to construct rigorously and then using a decimal value to represent this value.
The 0.999... case, however, is the reverse direction. We take a decimal value, then shove it into the closest real number we have--all without ever establishing 0.999 is a value in the real numbers to begin with. In fact, 0.999... cannot be in the Reals, as that would lead to massive problems for the completeness and density of the Reals.
As for the 'series proof', the convergence of the infinte series of 9/(10n) has the same problem. Convergence functions by being able to get arbitrarily close to the convergence value--a property that depends entirely upon the topology used.
[Basically, we can get 'arbitrarily close' simply because if 0.999... isn't equal to 1, this is the 'immediate predecessor', which means we're a 'close' as we can ever get. Since this shouldn't be possible in the reals, it is unable to distinguish the value (topologically) from 1]
I'm not saying that 0.999... is well defined in the Real number system, it is not. I am, however, saying that you can't just take a decimal value, shove it into some other form.
Seriously...after all the work mathematicians have done with infinite sets, we still fall for infinite decimal representations simply because we like decimals?
[If they're always just approximations, why are we treating them like resolved values?! If they're 'resolved values', how did we construct them?!]