Common counter-arguments and my responses
"0.9999... and 1 are obviously different numbers."
In mathematics, "obvious" means "a proof immediately springs to mind". If you don't have a proof in mind, then unfortunately no mathematical statement you make carries any weight.
"1 and 0.9999... are written differently, therefore they are different numbers."
There are many ways of writing any number. You could write 1/1, or 3-2, or 1.0, or 1.00, or 1.0000... or any number of other expressions, and all of them ultimately have the same meaning, "one".
"0.9999... is a concept, not a number."
All numbers are concepts.
"0.9999... can't exist in reality, but 1 can, therefore they are different."
Firstly, just because a number can't exist in reality doesn't mean it can't exist in mathematics.
Secondly, because 1 can exist in reality and 1 = 0.9999..., that means that 0.9999... can also exist in reality.
"There is a rounding error. 0.9999... and 1 are approximately equal."
Rounding errors only occur when we truncate a decimal expansion after a finite number of digits. All of the proofs above use the "..." notation at every step, which means that we always take into account all of the infinitely many decimal digits. There is no rounding, which means there is no error.
"0.9999... gets closer and closer to 1, but never reaches it."
0.9999... is a single number. It doesn't move, so it can't get closer and closer to anything. It is where it is.
"0.9999... is a decimal representation of infinity, not a number."
0.9999... is definitely less than 2, so it can't be infinitely large.
"Humans can't comprehend infinity, and not being able to comprehend infinity means you can't do mathematics with it."
Firstly, humans can comprehend infinity. Mathematicians do it all the time. It may impossible to literally "conceive of" infinite values, whatever that means, but that doesn't stop mathematicians from dealing with them without going nuts.
Secondly, infinity obeys rules. If something obeys rules in a consistent fashion, then you can do mathematics with it. Ordinal arithmetic is a good example.
In case the connection isn't clear, what is true of infinite values is equally true of infinite decimal expansions. There are rules and procedures and they work and give meaningful results. See "The Real Proof" above for a relatively tame glimpse of this, which is actually a vast region of mathematics known as "analysis", naturally based on rock-solid fundamental axioms.
"My mate/my dad/my mathematics teacher/Professor Stephen Hawking told me that 0.9999... and 1 were different numbers."
They were wrong.
"But they proved it, too!"
The proof was fallacious. Send it to me and I'll show you why.
"I still don't believe it and I'm entitled to my own opinion."
In regular science, we have theories. A theory is proposed in order to explain observations, and can be overturned in light of new, inexplicable observations. Multiple theories and opinions may compete with one another. There are fashions. There is room for debate.
In mathematics, we have theorems instead of theories. A theorem is the result of a mathematical proof. A theorem is a fact. A theorem cannot be overturned and is not a matter of opinion. Once proven, a theorem stands for eternity. Mathematics is not ideological.
Thanks to the many proofs above, "point nine recurring equals one" is just such a theorem. So, your opinion is wrong. And sorry, but no: you're not entitled to be wrong in mathematics. That's not how it works.
mathematician/an all knowing god.
