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April 22nd, 2014, 06:14 AM   #1
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Fibonacci numbers

The number 0,112358...( the digits are the Fibonacci numbers) is rational/irational.How motivate? Thanks.
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April 22nd, 2014, 06:37 AM   #2
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Originally Posted by Roli View Post
The number 0,112358...( the digits are the Fibonacci numbers) is rational/irational.How motivate? Thanks.
I can't think of an easy proof (though I'm sure it's irrational).
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April 22nd, 2014, 03:20 PM   #3
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Hello, Roli!

Quote:
The number 0,112358...( the digits are the Fibonacci numbers) is rational/irrational?

If the decimal is constructed like this:

$\displaystyle \begin{array}{cccccccccccc}
0. & 0&1\\
& 0&0&1 \\
& 0&0&0&2 \\
& 0&0&0&0&3 \\
& 0&0&0&0&0&5 \\
& 0&0&0&0&0&0&8 \\
& 0&0&0&0&0&0&1&3 \\
& 0&0&0&0&0&0&0&2&1 \\
& 0&0&0&0&0&0&0&0&3&4 \\ \hline
0. & 0&1&1&2&3&5&9&5&4 & ... \end{array}$

its value is a rational number!


Let $\displaystyle F_0 = 0,\;F_1=1,\;F_2 = 1,\;F_3 = 3,\;F_4 = 5,\;\text{ . . .}$

$\displaystyle \begin{array}{cccccc}\text{Then we have:} & X &=& \frac{F_0}{10^1} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \frac{F_3}{10^4} + \frac{F_4}{10^5} + \cdots \\
\text{Multiply by }\tfrac{1}{10}\!: & \frac{1}{10}X &=& \qquad\;\;\;\frac{F_0}{10^2} + \frac{F_1}{10^3} + \frac{F_2}{10^4} + \frac{F_3}{10^5} + \cdots \end{array}$

$\displaystyle \text{Subtract: }\;\;\frac{9}{10}X \;=\; \frac{F_0}{10} + \frac{F_1-F_0}{10^2} + \frac{F_2-F_1}{10^3} + \frac{F_3-F_2}{10^4} + \frac{F_4-F_3}{10^5} + \cdots$

$\displaystyle \text{We have: }\;\; \frac{9}{10}X \;=\; 0 + \frac{1}{10^2} + \frac{F_0}{10^3} + \frac{F_1}{10^4} + \frac{F_2}{10^5} + \cdots $

$\displaystyle \qquad\qquad\;\;\; \frac{9}{10}X \;=\; \frac{1}{100} + \frac{1}{10^2}\underbrace{\left(\frac{F_0}{10^1} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \cdots \right)}_{\text{This is }X} $

$\displaystyle \qquad\qquad\;\;\;\frac{9}{10}X \;=\;\frac{1}{100} + \frac{1}{100}X$

$\displaystyle \;\;\; \frac{9}{10}X - \frac{1}{100}X \;=\;\frac{1}{100}$

$\displaystyle \qquad\qquad \frac{89}{100}X \;=\;\frac{1}{100}$

$\displaystyle \text{Therefore: }\:X \;=\;\frac{1}{89}$

Thanks from fysmat and v8archie
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April 22nd, 2014, 08:35 PM   #4
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Very nice!
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April 23rd, 2014, 01:42 AM   #5
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Yes , very nice but note that the number derived by my friend Soroban is not the same number as the OP.

I'm with CRG , the OP number is irrational. The addition method of Soroban would work if we increased the exponent of the denominator appropriately , whenever appropriate , for example ,

When you get to F(7) you increase the exponent in the denominator by 1 (in the Soroban derivation)

When you get to F(8 ) you increase the exponent in the denominator by 2 (in the Soroban derivation)


Last edited by agentredlum; April 23rd, 2014 at 01:44 AM. Reason: F(8) registered as smiley
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April 23rd, 2014, 01:54 PM   #6
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The proof the second poster gave seems correct to me.

Why is it wrong ?
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April 23rd, 2014, 01:56 PM   #7
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Originally Posted by William Labbett View Post
The proof the second poster gave seems correct to me.

Why is it wrong ?
I don't think anything is wrong with it, but I don't think that number is the one the OP intended.
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