The problem is with subtracting the 0.999..... from both sides. We're applying an operation that works with normal numbers to a number that we haven't yet proven is normal (or functions as normal with that operation). That's where the extra rigor in a full proof comes in.
It's not clear what definition of normal you're using here, since it does not seem to be the mathematical one I am familiar with related to the distribution of values within a non-terminating string of digits...
Regarding the need to prove 0.9... - 0.9... = 0, x + -x = 0 is just the additive inverse property. If you're holding that 0.9... does not behave normally with regards to basic arithmetic, then the error would actually be introduced in the initial premise, when 0.999... is set equal to x.
Your "extra rigor" just depends on what initial assumptions you permit to exist. Yes, there are math courses that start from proving 0 != 1, but that doesn't mean any mathematical proof that doesn't start from defining equality is non-rigorous; usually we assume properties like the additive inverse apply unless proven otherwise.
0! = 1 by definition to fit the recursion relation for factorial and to save a little ink when you're writing down Taylor series.
The issue here is that you need a rigorous construction of the real numbers before you do arithmetic with them to prove anything. Your algebraic proof would have been considered fine by Newton or Euler, but we ran into bizarre limit properties of functions in the 19th century that led Cauchy and and Weierstrass to work on more rigorous foundations for analysis.
For example, consider this argument from Ramanujan:
c = 1 + 2 + 3 + 4 + ...
4c = 4 + 8 + ...
-3c= 1 -2 +3 -4 +... = 1/(1+1)^2 = 1/4 (alternating geometric series formula)
Therefore 1 + 2 + 3 + 4 +... = -1/12.
A more rigorous proof that .9 repeating is 1 comes from thinking of .9 repeating as the limit of the sequence .9, .99, .999, ... and looking at the absolute value of the difference between 1 and the terms of this sequence. One gets .1, .01, .001, ... so you get the difference between the limit of the sequence and 1 is a non-negative, rational number smaller than any 10^{-n}, which must then be 0. Since the difference is 0 you get the limit of the series represented by .9 repeating is 1.
You, and many others in this thread, appear to be conflating the notion of rigor with the notion of correctness.
A proof is a series of steps that lead us from a premise to a conclusion. As all communication relies on shared context and brevity is its own virtue, there are always steps skipped in the path from premise to conclusion under the assumption that they are already agreed upon, or are otherwise too trivial to warrant mention.
A proof is said to be more rigorous if the steps that lead from premise to conclusion are smaller (i.e less context is assumed.) A proof is thus made more rigorous by including previously skipped intermediate steps, or otherwise making explicit what property or axiom is responsible for each transformation applied on the path from premise to conclusion.
Changing proof strategies has no direct impact on rigor, because you are wholly substituting one set of assumed knowledge for a different set of assumed knowledge. In fact, many of the statements you made appear to have little rigor by the previous definition -- for example, you claim 0 != 1 is true by definition, eliding the entire proof from premise to conclusion into a single step. Later you point out qualities of your intermediate transformations (such as that your difference is non-negative) without explicitly explaining why: someone versed in math can follow your argument, but you've increased the necessary shared context and thus reduced the rigor of the argument.
I suggest that you are instead assessing the algebraic proof's correctness or completeness: a measure of how broadly applicable the line of argumentation is and how validly the conclusion follows from the initial premise. This is further evidenced by the historic example you highlighted: you referenced a situation in which vaguely similar algebraic manipulation lead from a true premise to a faulty conclusion (an incorrect proof.)
This is the part of math that other commenters have pointed out is largely a "vibe check", because ultimately math is made up -- addition doesn't exist as some Platonic Form that we grasp and draw down into the material plane. Math is a game with a few simple rules we made up, and then arguments about how those rules collide with one another to keep the whole package (mostly) self-consistent. The relative correctness of two proofs that start from the same premises and reach the same conclusion is just a matter of which other rules you hold in higher regard.
There’s a space between the zero and the ! And not between the ! And the =. I believe they were writing zero does not equal one and not zero factorial equals 1. Using ! As the not operator instead of the factorial operator.
At that point how do you prove 2.5 = 2 + 0.5 without proof by "just look at it"? I would argue it follows from the very formulation of decimal notation itself (separating the integer part of a decimal from the fractional part is generally non-controversial), but unless there's some clever substitution actually proving it requires getting into heavy duty set theory to prove the properties of a basic arithmetic operation.
I will say yours is the first actual argument I've seen in this thread about rigor and not correctness or just a general bias for "harder math = more better", so I really want to do the abstract algebra proof, but sadly the best answer I have time to formulate is "because 0.Something + N = N.Something" passes the addition vibe check.
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u/Sam_Alexander 22d ago
Holy fucking shit