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January 17th, 2019, 06:42 AM   #21
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Quote:
Originally Posted by idontknow View Post
For each statement , a proof is needed.
Show a proof that $\displaystyle 0. \overline{9} =1$ .(or disprove it)
Yeah, so why don't you show a proof that 1 doesn't converge to 2. Or that 0.999... does converge to 1. Begin with defining what it means that a number converges to another.
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January 17th, 2019, 06:52 AM   #22
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Quote:
Originally Posted by zylo View Post

Defining the natural numbers by decimal digits is trivial: 1, 2, 3, ...
Application of a decimal point to all the natural numbers gives (defines) all the real numbers in [0,1). But that's been gone over many times in previous posts.

.9 and 1 are not the same thing. But considering them the same is part of the practical application of the decimal definition of the real numbers.

EDIT
One way to define all the real numbers in [0,1) is to list the natural numbers, put a period after them, and read them in reverse.
1. Is .1
2. Is .2
.
10. Is .01
.
That's a defintion of the real numbers using decimal digits. Period.

There is nothing about convergence or evaluation.

The only notion of limit is that some real numbers are an endless sequence (limit) of digits. For example Pi is the limit of a sequence of digits.

You can think of my definition as a system of symbols that allow you to talk about and practically work with real numbers.

You can use them and interpret them any way you like, just as you did in HS and Calculus. For example, you can associate a rational number, 1/3, with the real number consisting of the limit of an endless string of 3's, .333333333......, which you can interpret in the conventional sense as a sum of powers of 10 and then define a conventional {working) epsilon delta limit.
Note that the conventional Lim .3333....p, as the number of 3's becomes endless and p is ANY fixed number, is 1/3.
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January 17th, 2019, 07:06 AM   #23
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Quote:
Originally Posted by idontknow View Post
For each statement , a proof is needed.
Show a proof that $\displaystyle 0. \overline{9} =1$ .(or disprove it)
The question is more or less meaningless by definition: Construction of the real numbers
Quote:
Originally Posted by Wikipedia
By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a representation of x. This reflects the observation that one can often use different sequences to approximate the same real number.
The decimal representation is precisely a representation of such a Cauchy sequence. Specifically, $$a_0.a_1a_2a_3a_4\ldots = \sum_{n=0}^\infty 10^{-n}a_n$$
So you require a proof that
$$\sum_{n=1}^\infty \frac9{10^{n}} = 9 \sum_{n=1}^\infty \left(\frac1{10}\right)^{n} = 1$$ which is a triviality for anyone that has studied infinite series.
The fact that $$\frac1{1-x} = 1 + x + x^2 + x^3 + \ldots = 1 + \sum_{n=1}^\infty x^{n} \qquad (|x| < 1)$$ allows us to put $x = \frac1{10}$ and get
\begin{align}\sum_{n=1}^\infty \frac9{10^{n}} &= 9 \sum_{n=1}^\infty \left(\frac1{10}\right)^{n} \\ &= 9 \left(\frac1{1-\frac1{10}}-1\right) \\ &= 9\left( \frac1{\frac9{10}}-1\right) \\ &= 9 \left( \frac{10}9 - 1 \right) \\ &= 9 \left( \frac19 \right) \\ &= 1
\end{align}
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January 17th, 2019, 07:14 AM   #24
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Quote:
Originally Posted by zylo View Post
Defining the natural numbers by decimal digits is trivial: 1, 2, 3, ...
It most decidedly is not "trivial". You have just written a bunch of symbols. There's no definition there at all. Things just get worse when you hit the number 10. If you consider this to be a single symbol, then your argument via sequences simple disintegrates. Every natural number becomes a single (large in most cases) symbol, and your concept of dropping a decimal point in becomes meaningless.

Alternatively, the number 10 comprises two symbols (0 and 1) and there relative positioning has a meaning. This is a definition that you are relying on but refusing to acknowledge. The symbol ten means $0 + 1(10^1)$ and so any definition of a natural number via decimal digits requires that the corresponding sum can be evaluated.

This, again, is a concept that you refuse to admit - you have even taken to denying it on occasion.
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January 17th, 2019, 07:37 AM   #25
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Evaluation is strictly a matter of convention.

A computer programmer might decide to round off (evaluate) all numbers to ten decimal places, where the tenth place is the next higher or lower number (integer).

A mathematician might decide to evaluate .33333.... (endless string of 3's) as the epsilon delta sum (\Sigma).

An engineering lab might decide to evaluate all shock load data to the nearest integer. In this case ,9 = 1.
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January 17th, 2019, 08:49 AM   #26
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Quote:
Originally Posted by zylo View Post
Evaluation is strictly a matter of convention...
What a load of tosh! Approximation is approximation. Error correction is error correction. Nobody evaluates numbers differently.

To the extent that convention does apply, you should follow it or state that you are doing stuff that doesn't apply to the conventional system. You do neither.
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January 17th, 2019, 11:58 AM   #27
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Quote:
Originally Posted by v8archie View Post
What a load of tosh! Approximation is approximation. Error correction is error correction. Nobody evaluates numbers differently.

To the extent that convention does apply, you should follow it or state that you are doing stuff that doesn't apply to the conventional system. You do neither.
Archie

I have said all along that what zylo is doing is using a non-standard definition of real numbers and then saying that he has a proof that applies to the standard definition.

The exercise is beyond silly.
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January 17th, 2019, 02:13 PM   #28
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Quote:
Originally Posted by zylo View Post
1 is a real number by definition: it is a unique decimal sequence.
Why do you keep saying it's unique when we know of two different decimals that evaluate to 1?
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January 17th, 2019, 07:23 PM   #29
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How are the moderators still allowing this same nonsense to be spouted over dozens of threads, month after month after month? This is absurd. I understand not wanting to be as ruthlessly moderated as other sites such as stack exchange, but enough is enough. Kick this idiot off his podium already.
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January 18th, 2019, 04:50 AM   #30
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Quote:
Originally Posted by zylo View Post
All real numbers can be defined by unique decimal sequences . . .
That is how you started this topic, but you haven't explained what you meant by "unique". If the sequence 010 were used, why would it be unique?

Quote:
Originally Posted by zylo View Post
One way to define all the real numbers in [0,1) is to list the natural numbers, put a period after them, and read them in reverse.
How would the fraction 1/9, which is a real in [0, 1), be obtained in this "read them in reverse" way?
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