April 22nd, 2014, 06:14 AM  #1 
Newbie Joined: Apr 2014 From: USA Posts: 24 Thanks: 1  Fibonacci numbers
The number 0,112358...( the digits are the Fibonacci numbers) is rational/irational.How motivate? Thanks.

April 22nd, 2014, 06:37 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  
April 22nd, 2014, 03:20 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Hello, Roli! Quote:
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 $\displaystyle F_0 = 0,\;F_1=1,\;F_2 = 1,\;F_3 = 3,\;F_4 = 5,\;\text{ . . .}$ $\displaystyle \begin{array}{cccccc}\text{Then we have:} & X &=& \frac{F_0}{10^1} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \frac{F_3}{10^4} + \frac{F_4}{10^5} + \cdots \\ \text{Multiply by }\tfrac{1}{10}\!: & \frac{1}{10}X &=& \qquad\;\;\;\frac{F_0}{10^2} + \frac{F_1}{10^3} + \frac{F_2}{10^4} + \frac{F_3}{10^5} + \cdots \end{array}$ $\displaystyle \text{Subtract: }\;\;\frac{9}{10}X \;=\; \frac{F_0}{10} + \frac{F_1F_0}{10^2} + \frac{F_2F_1}{10^3} + \frac{F_3F_2}{10^4} + \frac{F_4F_3}{10^5} + \cdots$ $\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} $ $\displaystyle \qquad\qquad\;\;\;\frac{9}{10}X \;=\;\frac{1}{100} + \frac{1}{100}X$ $\displaystyle \;\;\; \frac{9}{10}X  \frac{1}{100}X \;=\;\frac{1}{100}$ $\displaystyle \qquad\qquad \frac{89}{100}X \;=\;\frac{1}{100}$ $\displaystyle \text{Therefore: }\:X \;=\;\frac{1}{89}$  
April 22nd, 2014, 08:35 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,617 Thanks: 2608 Math Focus: Mainly analysis and algebra 
Very nice!

April 23rd, 2014, 01:42 AM  #5 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
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 
April 23rd, 2014, 01:54 PM  #6 
Member Joined: Apr 2014 From: norwich Posts: 84 Thanks: 9  
The proof the second poster gave seems correct to me. Why is it wrong ? 
April 23rd, 2014, 01:56 PM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  

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