03-12-2009, 10:54 AM
If you really want to get into the history of math I can go there. But here is one of the best explainations as to why what you are thinking is wrong. From this website: http://qntm.org/?pointnine
Common counter-arguments and my responses
"0.9999... and 1 are obviously different numbers."
Not good enough. Intuition counts for nothing. In mathematics, proof is everything, and "obvious" means "a proof springs immediately to mind". Please PROVE that 0.9999... and 1 are not equal. Without proof, no hunch, feeling, or intuition is worth anything.
"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 2/2, or 9/9, or 2-1, 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. Some numbers, like 1, have stronger links to reality than others, but we are looking at mathematics here, not the real world. If you're going to throw away numbers which can't concretely exist, then you're throwing away pi, e, i, zero, and, frankly, almost all of mathematics.
"There is a rounding error. 0.9999... and 1 are approximately equal."
Do you see any rounding or approximation going on around here? That only happens when you stop counting after a certain number of decimal digits. But I have kept and counted every single one of the infinitely many decimal digits in my proofs. No rounding, no error.
"0.9999... gets closer and closer to 1, but never reaches it."
Closer and closer? How can it be getting closer and closer? It's one number! Try the limit argument, above.
"0.9999... is a decimal representation of infinity, not a number."
Well, how come it's DEFINITELY bigger than 0.5 and smaller than 2? Just because something has infinitely many pieces doesn't mean it's infinite. Zeno figured this out 2500 years ago.
"Humans can't comprehend infinity, and not being able to comprehend infinity means you can't do mathematics with it."
This statement is wrong on many levels. When you say humans cannot comprehend infinity, you are quite likely projecting your own inability to comprehend infinity onto everybody else in the world, among them many thousands of mathematicians who are perfectly capable of sitting down and dealing with infinite values in a sane and rational fashion. It may impossible to literally conceive of infinite values - whatever formal definition such conception could possibly have - but that does not and never will stop mathematicians from dealing with them. Mathematics is all about rules. Mathematicians have discovered that there are rules for dealing with infinity. These rules are perfectly consistent with our rules for dealing with other numbers, and given a little time you could probably learn to apply them yourself.
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. In science, credentials are as worthless as intuition (above). Proof is everything.
"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."
Mathematics is unlike regular science in that we can actually prove things, permanently, for real, instead of just finding increasing amounts of evidence supporting our hypotheses. That's why we have what we call "theorems" instead of theories. That point nine recurring equals one is just such a theorem (although it's so easy that it's barely worth the name). You aren't in a position to argue or debate about it. It's a fact. Your opinion is wrong.
You are entitled to be wrong, I suppose. However, if you are unable to admit that you are wrong when you have been proven wrong, then you have no business taking part in a mathematical discussion or pretending that you know anything about mathematics. Therefore, I must request that you please distance yourself from any future discussions that you may encounter on this topic, so that those who do want to learn, rather than just promote an ideology, can do so
Common counter-arguments and my responses
"0.9999... and 1 are obviously different numbers."
Not good enough. Intuition counts for nothing. In mathematics, proof is everything, and "obvious" means "a proof springs immediately to mind". Please PROVE that 0.9999... and 1 are not equal. Without proof, no hunch, feeling, or intuition is worth anything.
"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 2/2, or 9/9, or 2-1, 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. Some numbers, like 1, have stronger links to reality than others, but we are looking at mathematics here, not the real world. If you're going to throw away numbers which can't concretely exist, then you're throwing away pi, e, i, zero, and, frankly, almost all of mathematics.
"There is a rounding error. 0.9999... and 1 are approximately equal."
Do you see any rounding or approximation going on around here? That only happens when you stop counting after a certain number of decimal digits. But I have kept and counted every single one of the infinitely many decimal digits in my proofs. No rounding, no error.
"0.9999... gets closer and closer to 1, but never reaches it."
Closer and closer? How can it be getting closer and closer? It's one number! Try the limit argument, above.
"0.9999... is a decimal representation of infinity, not a number."
Well, how come it's DEFINITELY bigger than 0.5 and smaller than 2? Just because something has infinitely many pieces doesn't mean it's infinite. Zeno figured this out 2500 years ago.
"Humans can't comprehend infinity, and not being able to comprehend infinity means you can't do mathematics with it."
This statement is wrong on many levels. When you say humans cannot comprehend infinity, you are quite likely projecting your own inability to comprehend infinity onto everybody else in the world, among them many thousands of mathematicians who are perfectly capable of sitting down and dealing with infinite values in a sane and rational fashion. It may impossible to literally conceive of infinite values - whatever formal definition such conception could possibly have - but that does not and never will stop mathematicians from dealing with them. Mathematics is all about rules. Mathematicians have discovered that there are rules for dealing with infinity. These rules are perfectly consistent with our rules for dealing with other numbers, and given a little time you could probably learn to apply them yourself.
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. In science, credentials are as worthless as intuition (above). Proof is everything.
"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."
Mathematics is unlike regular science in that we can actually prove things, permanently, for real, instead of just finding increasing amounts of evidence supporting our hypotheses. That's why we have what we call "theorems" instead of theories. That point nine recurring equals one is just such a theorem (although it's so easy that it's barely worth the name). You aren't in a position to argue or debate about it. It's a fact. Your opinion is wrong.
You are entitled to be wrong, I suppose. However, if you are unable to admit that you are wrong when you have been proven wrong, then you have no business taking part in a mathematical discussion or pretending that you know anything about mathematics. Therefore, I must request that you please distance yourself from any future discussions that you may encounter on this topic, so that those who do want to learn, rather than just promote an ideology, can do so
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