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January 22nd, 2019, 08:34 AM   #1
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Decimals and the Continuum

To define the real numbers in [0,1) list the natural numbers, put a period after them, and read them in reverse. Call them "decimals"

1. -> .1
2. -> .2
.
10. -> .01
.

The decimals can be associated uniquely with points on the line (the continuum) as follows:
Given a line segment, label its end points 0 and 1. Choose a point. Divide the line into ten closed intervals. If the point is, say, in third interval, S$\displaystyle _{1}$ =.2. Divide third interval into ten closed intervals. If point is in seventh interval, S$\displaystyle _{2}$=.26. Continue the process until the point is on an interval, which ends the sequence, or you have an infitite decimal (nested sequence} which uniquely defines the point.

To find the point corresponding to an infinite decimal. Divide the line into 10 closed intervals. If S$\displaystyle _{1}$=6, the point is in the fifth closed interval. Continue in this manner to define an infinite nest which uniquely defines a point on the line. If S$\displaystyle _{n}$ is finite, the point is at the end of an interval.

The above can be repeated for any radix to associate uniquely with every point on the line a radix fraction. If radix =2, Then any fraction in [0,1) is of the form .100110..., where each place is determined by a nest of intervals, divide the line in half. Then divide the interval in which the point resides in half again, and repeat till you get a finite or unending binary fraction.

EDIT:

The fact that decimals are a field follows from the fact that they are a field for n decimal places for all n.

Last edited by skipjack; January 22nd, 2019 at 01:10 PM.
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January 22nd, 2019, 08:45 AM   #2
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Why the need to make a new thread on it. Doesn't this fit perfectly in your other thread?
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January 22nd, 2019, 09:09 AM   #3
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Foundations of Analysis.

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Originally Posted by Micrm@ss View Post
Why the need to make a new thread on it. Doesn't this fit perfectly in your other thread?
I thought this forum was the appropriate place to give the foundations of analysis without getting buried.
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January 22nd, 2019, 09:23 AM   #4
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This has nothing to do with the foundations of analysis. It has nothing much to do with mathematics.
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January 22nd, 2019, 09:38 AM   #5
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Quote:
Originally Posted by zylo View Post
I thought this forum was the appropriate place to give the foundations of analysis without getting buried.
It is. But no need to make 10 threads with the same thing.
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January 22nd, 2019, 10:17 AM   #6
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At some point, you're going to have to admit the existence of natural numbers with an infinite amount of digits. Your ideas keep "getting buried" because no-one is agreeing with what you are saying.

Is this just a remake of your assertion that the reals form a countable set?

In the interest of keeping this a discussion (as opposed to a dissertation) this thread is closed. Please continue the discussion in your previous thread.
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Last edited by greg1313; January 22nd, 2019 at 10:45 AM.
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January 22nd, 2019, 01:57 PM   #7
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Quote:
Originally Posted by zylo View Post
Call them "decimals"
They don't include .11111111... (endless), which is why you haven't explained how that decimal would be obtained by your "start with the naturals" method.
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