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February 24th, 2018, 03:04 PM   #41
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Quote:
 Originally Posted by zylo The count (a PROPERTY) of n-place decimals is finite as long as n is finite. As n $\displaystyle \rightarrow$ infinity, the count $\displaystyle \rightarrow$ {countable) infinity.
This is guesswork on your part. You have no evidence for it, and indeed it isn't true (quite trivially if you understand Cantor).

Quote:
 Originally Posted by zylo The first number in the list above is .0000....001, FOR ALL N, BY DEFINITION.
This is, of course, true for all $n$, but you must be careful to avoid the claim that such a number exists in the limiting case of infinite decimals: it clearly doesn't because infinite decimals do not terminate at all, let alone with the digit "1".

Quote:
 Originally Posted by zylo By definition of a real number, two real numbers are equal iff they have the same digits FOR ALL n. Period.
I don't know what definition you are using for the real numbers, but it is clearly wrong because $0.4999\ldots = \frac12 = 0.5$.
Quote:
 Originally Posted by zylo Two real numbers are the same if every digit is the same for all n, not if they approach the same \epsilon \delta limit.
This is absolutely false, not least because the real numbers are defined as the limits of sequences of rational numbers.

Quote:
 Originally Posted by zylo The real numbers are unique as I have defined them.
It's not up to you to define the real numbers. We already have several equivalent definitions of them. You can, by all means, come up with a different one but it must be equivalent to the definitions that already exist. Otherwise you are talking about something other than the set of real numbers.

 February 26th, 2018, 08:23 AM #42 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 For 4-place decimals the count should be .0000, .0001,...,.9999. That makes the count easier to visualize for all n. The count (a PROPERTY) of n-place decimals is finite as long as n is finite. As n $\displaystyle \rightarrow$ infinity, the count $\displaystyle \rightarrow$ (countable) infinity. I believe the term "countable infinity" comes from Cantor. I distinquish between limiting sum of an n-place decimal, which is not unique but rather a property of n-place decimals, and the typographical limit as n approaches infinity, which is unique. Two real numbers are the same if every digit is the same for all n, not if they approach the same $\displaystyle \epsilon, \delta$ limit. The real numbers are unique as I have defined them. Function: DEFINITION: Map from elements of one set to another. Continuity: PROPERTY of a function. Limit: PROPERTY of a function. $\displaystyle \epsilon, \delta$ for f(x) or $\displaystyle \epsilon$, M, n>M, for f(n) [QUOTE=zylo;588830]REAL NUMBERs are defined uniquely by "infinite" (unending) sequences of natural numbers. The sequence IS the real number. [QUOTE=zylo;589093] Specifically, the limiting case as n approaches infinity (for all n) of a finite sequence of digits. 0,1,2,3,...,9, as in .135396782......, n-places What is your definition of a real number?
February 26th, 2018, 08:43 AM   #43
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Quote:
 Originally Posted by zylo What is your definition of a real number?
A real number by definition is a subset $A$ of $\mathbb{Q}$ such that
- A is nonempty and $A\neq \mathbb{Q}$
- If $a\in A$, and $b>a$, then $b\in A$.
- If $a\in A$, then there is some $b<a$ that is also in $A$.

February 26th, 2018, 09:52 AM   #44
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Quote:
 Originally Posted by Micrm@ss A real number by definition is a subset $A$ of $\mathbb{Q}$ such that - A is nonempty and $A\neq \mathbb{Q}$ - If $a\in A$, and $b>a$, then $b\in A$. - If $a\in A$, then there is some $b You have defined a cut, which leads to the definition of a real number as a unique infinite decimal, and vice versa. If you feel comfortable dealing with analysis with a mental image of cuts, fine. You mentioned hyperreals in another thread. How do you think of hyperreals in terms of cuts? But I can see that an abstract mathematician might think of real numbers as defined things, names. EDIT What is the$\displaystyle \lim_{n \rightarrow \infty}$of f(n) in terms of cuts? In my opinion. cuts are a way to associate infinite decimals (real numbers, analysis) with points on a line (geometry). Last edited by zylo; February 26th, 2018 at 10:03 AM. February 26th, 2018, 11:46 AM #45 Math Team Joined: Dec 2013 From: Colombia Posts: 7,508 Thanks: 2513 Math Focus: Mainly analysis and algebra I've compared your post with the one before and there's almost nothing new. Quote:  Originally Posted by zylo for all n This adds nothing at all - unless for all n is back to including infinite decimals as it has in the past. But then your list becomes meaningless because you can't even construct the second element. Quote:  Originally Posted by zylo I believe the term "countable infinity" comes from Cantor. It does, but that doesn't mean that you can use it to describe anything that you guess is a countable infinity. It has a precise meaning that does not apply what you are saying. Quote:  Originally Posted by zylo I distinquish between limiting sum of an n-place decimal, which is not unique but rather a property of n-place decimals, and the typographical limit as n approaches infinity, which is unique. Two real numbers are the same if every digit is the same for all n, not if they approach the same$\displaystyle \epsilon, \delta$limit. This is rephrased, but it says the same as last time. You are back to just repeating wrong things as if that made them become true. Numbers do not have limits, but on the assumption that you are talking about the limit of the sequence$(0.49, 0.499, 0.4999, 0.49999, \ldots)$, for example you are still wrong. Real numbers are limits of sequences of rational numbers. That is one of the equivalent definitions that the mathematical world uses. For this reason$0.4999\ldots = 0.5000\ldots = \frac12\$ as you have admitted.

Quote:
 Originally Posted by zylo The real numbers are unique as I have defined them.
Again, your definition is wrong. Or perhaps you intend to talk about a different set to the one everyone thinks of when you talk about real numbers. But in that case, why choose deliberately confusing terminology.

Quote:
 Originally Posted by zylo REAL NUMBERs are defined uniquely by "infinite" (unending) sequences of natural numbers. The sequence IS the real number.
This also is false. It is the limit of a sequence of rational numbers that is the real number.

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