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April 19th, 2017, 11:31 AM   #11
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Quote:
Originally Posted by Maschke View Post
No.
Then please give me the exact decimal representations of

$\sqrt{2}, \pi,\ \text { and } e.$
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April 19th, 2017, 11:47 AM   #12
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Quote:
Originally Posted by JeffM1 View Post
Then please give me the exact decimal representations of

$\sqrt{2}, \pi,\ \text { and } e.$
Computable numbers are easy. Each of those decimal expansions can be generated by a short computer program.

Are you making some sort of finitist argument? Saying that (for example) the set of natural numbers can't exist because we can't write them all down?
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April 19th, 2017, 12:10 PM   #13
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The question is whether a decimal expansion that is exact can be generated. Joppy already gave an exact answer to the original question. If we want (in the real world as opposed to the Platonic world of ideals) to generate an exact decimal expansion, there are many real numbers for which that cannot be done.

I agree that a very short computer program can generate successive digits in the decimal expansion of the square root of 2, but to generate the exact expansion in its entirety will involve a rather extended run time.

I am not a finitist in this sense: I am perfectly comfortable reasoning with the infinite and real numbers as ideal objects. I am a finitist in this sense: many things (including a physical exemplar of the set of natural numbers) do not exist in the physical universe.

I think I gave a correct and reasonable answer to the question that I believe was asked. There are many real numbers for which it is impossible in physical practice to provide an exact decimal expansion. We can give exact answers that are not decimal expansions, or we can give decimal expansions that are approximations.

So what are we arguing about again?
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April 19th, 2017, 12:15 PM   #14
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Originally Posted by JeffM1 View Post
So what are we arguing about again?
You denied that every real number has a decimal expansion. Since that claim contradicts standard math, I felt compelled to put in my 1.999... cents.

ps ... here's the original quote.

Quote:
Originally Posted by JeffM1 View Post
Do you understand that decimal notation cannot express exactly all real numbers?
I answered No to that question. I couldn't answer otherwise. Every real number has an exact decimal expression. Sometimes two of them.

Last edited by Maschke; April 19th, 2017 at 12:24 PM.
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April 19th, 2017, 12:35 PM   #15
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OK. You and I are going to disagree. In practice, you cannot give the OP an exact decimal expansion of the average of his surds. Even if he is willing to wait until long past the heat death of the universe, your computer program will still be grinding away.

Had we but world enough and time, your position would be true, but there is not time enough.
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