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Topic: Can a fraction have none noneending and nonerepeating decimal representation?
Replies: 108   Last Post: Aug 16, 2013 5:22 PM

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 magidin@math.berkeley.edu Posts: 11,749 Registered: 12/4/04
Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted: Aug 1, 2013 7:59 PM

On Thursday, August 1, 2013 3:19:12 PM UTC-5, jonas.t...@gmail.com wrote:
> Den torsdagen den 1:e augusti 2013 kl. 21:55:51 UTC+2 skrev Arturo Magidin:
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> > On Thursday, August 1, 2013 12:45:54 PM UTC-5, jonas.t...@gmail.com wrote:
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> > [.snip.]
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> > > The thing is that i can express Pi as an exact fractional ratio of the perimeter of a polygon, without using trigonometry just continued fractions.
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> > Let's be clear:
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> > A rational number is one that can be expressed as a ratio of two INTEGERS. It is not enough to express something as "an exact fractional ratio", the two quantities must be INTEGERS.
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> > So the fact that you can express pi as "an exact fractional ratio" is irrelevant, because you are not expressing it as an exact ratio of INTEGERS.
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> > Also, the expression must be a single quotient, so continued fractions are likewise irrelevant to whether some real number is rational or not.
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> > > And that is both rational and a fact,
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> > It may be rational in the colloquial sense of the word, but that does not make it a Rational Number, that is it is not a quotient of two integers.
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> > > you can wave your hands shouting that is not a circle as much you want.
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> > I'm not waving my hands, I'm pointing out to you what the words mean and that they do not mean what you think they mean. As such, the one who is waving and shouting and dancing about is you, not me.
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> > Let me put it plainly: you don't know what you are talking about. You may continue to show off your ignorance and your inability to understand all you want, you'll still have no clue about this until you learn what it means for a number to be rational, and why everything you have said is completely and utterly irrelevant to the demonstrable fact that pi is not a rational number.
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> No

Yes.

>i think it is another way you may claim that 0.333... is a rational number because you know that is 1/3.

No. 0.3333 is *A DECIMAL REPRESENTATION* of the number 1/3, which is rational.

> And i may claim that Pi is a rational,

> because i know i can write it as a continued fraction.

Unfortunately for you, a number that can be expressed as a continued fraction is not guaranteed to be a rational, and in fact one can PROVE that there are continued fraction expansions whose value is NOT rational. As such, being able to write a number as a continued fraction does NOT establish its rationality.

> The real funny thing is that this was known to mathematicians 11 BC.

The real funny thing is that you have absolutely no idea what you are talking about, but you think you can school me.

Well, at least I can *spell* "sophisticated".

But I don't claim to be sophisticated. I just claim to be INFORMED, whereas you are not.

--
Arturo Magidin

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