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October 25th, 2012, 12:31 PM   #1
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Rational number

How I prove that beteen every some real numbers there is a rational number?
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October 25th, 2012, 12:37 PM   #2
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Re: Rational number

Look at the decimal expansions of the two real numbers and find the first decimal place where they differ.
Insert a finite decimal expansion in between.
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October 25th, 2012, 02:11 PM   #3
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Re: Rational number

What if they are irrational?
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October 25th, 2012, 08:39 PM   #4
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Re: Rational number

We can approach any real number as close as we like by a rational approximation. Suppose a, b are distinct, irrational, real numbers. There is a fixed, finite distance between them but we can make the distance between a and our rational smaller than any fixed distance thereby putting our rational in between a and b.

This is probably wrong but heres another idea...

suppose 2 distinct, real, irrational numbers a and b give interval [a... b]

divide both by a

now you have the interval [1... b/a] or [b/a...1] the ordering doesn't matter and b/a is most likely irrational but definitely fixed in value. You can approach 1 by a rational approximation as close as you like and since b/a is a fixed distance from 1 you can always insert a rational between them.

This should imply that there is a rational between any 2 distinct irrational real numbers, the size of the interval changes when dividing by 'a' but it is still a finite interval with fixed endpoints and 1 as an endpoint is very convenient to approach.

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October 25th, 2012, 09:04 PM   #5
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Re: Rational number

Quote:
Originally Posted by goodfeeling
What if they are irrational?
[color=#008000]mathman[/color]'s suggestion doesn't imply that the decimal expansions ever need to terminate or repeat, only that the expansions will inevitably differ at some point if the two real numbers are distinct.

Another approach would be to take two distinct real numbers and multiply both of them by an integer sufficiently large such that we may insert the integer between them:



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October 26th, 2012, 01:16 AM   #6
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Re: Rational number

Quote:
Originally Posted by MarkFL
Quote:
Originally Posted by goodfeeling
What if they are irrational?
[color=#008000]mathman[/color]'s suggestion doesn't imply that the decimal expansions ever need to terminate or repeat, only that the expansions will inevitably differ at some point if the two real numbers are distinct.

Another approach would be to take two distinct real numbers and multiply both of them by an integer sufficiently large such that we may insert the integer between them:



Oh wow! I definitely like this idea. I'm never going to forget this.

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October 26th, 2012, 03:10 AM   #7
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Re: Rational number

Another question: How I prove between two every irrational number there is irrational number?
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October 26th, 2012, 03:37 AM   #8
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Re: Rational number

Difficulty with [color=#40FF40]mathman[/color]'s approach comes when the difference of the two numbers is an integer. In the questions, There must be (at least) two different numbers.

For the latest question, multiply both numbers by a factor s.t. the denominators are equal. Between the numerators, there is an irrational number hence the quotient is irrational.
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October 26th, 2012, 04:50 AM   #9
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Re: Rational number

what is s.t.?
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October 26th, 2012, 06:29 AM   #10
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Re: Rational number

such that
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