My Math Forum Real Numbers and Limits

 Real Analysis Real Analysis Math Forum

 February 22nd, 2018, 08:34 AM #31 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,854 Thanks: 1078 Math Focus: Elementary mathematics and beyond Please remain on topic. Thanks from topsquark
February 22nd, 2018, 09:51 AM   #32
Senior Member

Joined: Jun 2015
From: England

Posts: 853
Thanks: 258

Quote:
 Originally Posted by zylo I meant to elaborate on "Limit". Limit 1) Calculus. $\displaystyle \varepsilon, \delta$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = 1/2. property 2) Limit of n-place decimal as n $\displaystyle \rightarrow \infty$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = .4999999..... $\displaystyle \rightarrow$, definition
Limits were not part of 'calculus' when I studied it, although I will admit there is more than one calculus.

Last edited by skipjack; February 23rd, 2018 at 10:12 AM.

February 22nd, 2018, 11:45 AM   #33
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,374
Thanks: 2474

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by zylo I meant to elaborate on "Limit". Limit 1) Calculus. $\displaystyle \varepsilon, \delta$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = 1/2. property 2) Limit of n-place decimal as n $\displaystyle \rightarrow \infty$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = .4999999..... $\displaystyle \rightarrow$, definition
This highlights, I think, a confusion between the potential infinities of analysis and the actual infinities of sets theory.

In the first, the concept of limit is of great significance, because the expression has no value at infinity. We effectively define a value at infinity using the limit.

In the second, we don't. The infinite sequence actually exists (in the abstract). The limit is irrelevant to the value at infinity.

Last edited by skipjack; February 23rd, 2018 at 10:14 AM.

February 23rd, 2018, 07:08 AM   #34
Senior Member

Joined: Mar 2015
From: New Jersey

Posts: 1,440
Thanks: 106

Quote:
 Originally Posted by zylo I meant to elaborate on "Limit". Limit 1) Calculus. $\displaystyle \varepsilon, \delta$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = 1/2. property 2) Limit of n-place decimal as n $\displaystyle \rightarrow \infty$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = .4999999..... $\displaystyle \rightarrow$, definition
Another example of property is countabilility of the real numbers. All n-place decimals in [0,1) can be listed in numeric order from .000..00 to .999..99, n places. Therefore the n-place decimals in [0,1) are countable for all n, including the limit.

Last edited by skipjack; February 23rd, 2018 at 10:13 AM.

 February 23rd, 2018, 07:48 AM #35 Global Moderator   Joined: Dec 2006 Posts: 19,515 Thanks: 1745 Properties of finite cases needn't be properties that hold in the limiting case. For example, finiteness is a property of n in each of the individual cases, but not in the limiting case. Please address this point and also respond to my previous posts here and here. Thanks from topsquark
February 23rd, 2018, 09:25 AM   #36
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,374
Thanks: 2474

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by zylo Another example of property is countabilility of the real numbers. All n-place decimals in [0,1) can be listed in numeric order from .000..00 to .999..99, n places. Therefore the n-place decimals in [0,1) are countable for all n, including the limit.
This isn't new ground. Your errors have been pointed out before.

For example, by your logic, since every finite decimal of the form $0.49999\ldots$ is less than $\frac12$, you must have (using your notation) $$\lim_{n \to \infty}. 499999 = \frac12 \lt \frac12$$
This is obviously nonsense.

February 23rd, 2018, 09:28 AM   #37
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,374
Thanks: 2474

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by zylo I meant to elaborate on "Limit". Limit 1) Calculus. $\displaystyle \varepsilon, \delta$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = 1/2. property 2) Limit of n-place decimal as n $\displaystyle \rightarrow \infty$ $\displaystyle \lim_{\lim_{n\rightarrow \infty}}$ .499999 = .4999999..... $\displaystyle \rightarrow$, definition
While I have it in mind, you are completely wrong in this post in that the calculus limit is purely definition (not a property), while the infinite sequence of digits is a mathematical object, not a definition.

Last edited by skipjack; February 23rd, 2018 at 10:09 AM.

February 23rd, 2018, 02:45 PM   #38
Math Team

Joined: May 2013
From: The Astral plane

Posts: 1,887
Thanks: 765

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
 Originally Posted by zylo Another example of property is countabilility of the real numbers. All n-place decimals in [0,1) can be listed in numeric order from .000..00 to .999..99, n places. Therefore the n-place decimals in [0,1) are countable for all n, including the limit.
I almost agree with you. But your method only counts the rationals, which is indeed a countable set.

-Dan

February 24th, 2018, 12:09 AM   #39
Global Moderator

Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,854
Thanks: 1078

Math Focus: Elementary mathematics and beyond
Quote:
 Originally Posted by skipjack Properties of finite cases needn't be properties that hold in the limiting case. For example, finiteness is a property of n in each of the individual cases, but not in the limiting case. Please address this point and also respond to my previous posts here and here.

February 24th, 2018, 06:41 AM   #40
Senior Member

Joined: Mar 2015
From: New Jersey

Posts: 1,440
Thanks: 106

I neglected to mention a crucial number in my previous post: the second number in the abbreviated listing of n-place decimals:
For 4-place decimals the count should be .0000, .0001,...,.9999. That makes the count easier to visualize.

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.

The first number in the list above is .0000....001, FOR ALL N, BY DEFINITION. It has the PROPERTY that it approaches the functional limit zero as n $\displaystyle \rightarrow$ infinity. But that doesn't mean it IS 0. If it meant that it was zero, then the next number in the list would approach (be) zero and the next number would approach (be) 0 etc.

The confusion below comes from mixing up the definition of a real number with its properties. By definition of a real number, two real numbers are equal iff they have the same digits FOR ALL n. Period.

--------------------------------------------------------------------------------

Quote:
 Originally Posted by skipjack Properties of finite cases needn't be properties that hold in the limiting case. For example, finiteness is a property of n in each of the individual cases, but not in the limiting case.
Quote:
 Originally Posted by skipjack Would that point still apply if each "9" were replaced by "0" and the "4" were replaced by 5?
Yes. Reals include the rationals. .500000..... $\displaystyle \equiv$ 1/2
Two real numbers are equal iff they have the same digits for all n.

I distinguish 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 \epsilon \delta limit.

Quote:
 Originally Posted by skipjack Let's wait for zylo to explain what he meant, and why he used the word "uniquely". I'd also like to know what zylo meant by "limits are defined properties of real numbers", as one can certainly have limits of imaginary numbers.
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)

Of course one can have limits of imaginary numbers.

Quote:
 Originally Posted by greg1313 zylo, I expect that your next post will address the above.

Last edited by skipjack; February 26th, 2018 at 08:38 AM.

 Tags limits, numbers, real

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post zylo Topology 14 May 10th, 2017 01:57 AM Elektron Math 4 May 7th, 2017 11:47 AM Luiz Real Analysis 3 March 23rd, 2015 07:08 AM Luiz Real Analysis 1 March 18th, 2015 08:39 AM thehurtlooker Algebra 3 April 9th, 2013 12:58 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top