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I know that $8/5$ comes close $8/5 - 5/8 = 39/40.$ Is there a fraction that comes closer?

amWhy
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rain
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  • Have you tried writing down an equation? – saulspatz Sep 22 '18 at 15:22
  • I did create an equation because in order for x (a number) to exceed 1/x, the equation 1-(1/x)=1 has to be true. I found that there were no rational solutions. However, I am not sure if there are any exceptions? – rain Sep 22 '18 at 15:24
  • That's correct. There are no rational solutions. When posting questions on this site, be sure to include the details of what you've done so far, so that people can give answers appropriate to your level of understanding of the problem. – saulspatz Sep 22 '18 at 15:27
  • Thank you for the advice! – rain Sep 22 '18 at 15:28
  • But there are better rational approximations for the solution(s), e.g. 809/500. – gammatester Sep 22 '18 at 15:32
  • Quotients of successive Fibonacci numbers get as close as you want to the Golden ratio, which is the actual solution. – Cheerful Parsnip Sep 22 '18 at 15:34
  • It is a basic characteristic of real numbers that if you have a real irrational number. You can find rational numbers that are as near the irrational number as you'd like. In fact if you just take the decimal expansion to any finite position that will be a rational that is close. Is $\frac 85$ is close but $\frac {161}{100}$ is closer And as $\frac {1 +\sqrt 5}{2} \approx 1.618033988749894848204586834365...$ then $\frac {809}{500}$ is closer yet and so on. You can do this forever. – fleablood Sep 22 '18 at 15:47

3 Answers3

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You want $x>0$ such that $$\frac{1}{x}+1=x$$ so try solving this for $x$.

Dave
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  • I found that there are no rational solutions and I am wondering if there are any exceptions that I am not aware of ? – rain Sep 22 '18 at 15:25
  • No, there are no other solutions. There is exactly one positive solution to this equation: the golden ratio. – Dave Sep 22 '18 at 15:28
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    Your question didn't specify it had to be rational. $\frac 1x + 1 = x \iff 1 + x = x^2 \iff x^2 - x - 1 \iff x = \frac {1 \pm \sqrt 5}2$. There are exactly two irrational values. – fleablood Sep 22 '18 at 15:29
  • A quadratic equation $ax^2 + bx + c =0$ will have at most two solutions ($\frac {-b +\sqrt {b^2 -4ac}}{2a}$ and $\frac {-b -\sqrt{b^2-4ac}}{2a}$ ). So $\frac 1x + 1 = x \iff x^2 - x -1 = 0$ will have at most two solutions. They are both irrational. So no, there are not any you missed. – fleablood Sep 22 '18 at 15:51
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" Is there a fraction (rational number) that comes closer?"

There is no rational number that is exact. But you can find a rational number as infinitely close as you like.

Using Daves answer there are two irrational solutions

$x = \frac {1 + \sqrt 5}2 $ and $x = \frac {1-\sqrt 5}2$

$\frac {1 + \sqrt 5} 2 \approx 1.6180339887498948482045868343656$ so although $\frac 89$ is close. $\frac {161}{100}$ is closer. And $\frac {161803}{100000}$ is even closer and $\frac {1618033988749894848204586834365}{10^{30}}$ is very very close.

There is no one rational number that is closest.

fleablood
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as in one of the later comments, take consecutive Fibonacci numbers

$$ \frac{8}{5} - \frac{5}{8} = \frac{39}{40} $$ $$ \frac{13}{8} - \frac{8}{13} = \frac{105}{104} $$ $$ \frac{21}{13} - \frac{13}{21} = \frac{272}{273} $$ $$ \frac{34}{21} - \frac{21}{34} = \frac{715}{714} $$ $$ \frac{55}{34} - \frac{34}{55} = \frac{1869}{1870} $$ $$ \frac{89}{55} - \frac{55}{89} = \frac{4896}{4895} $$

The results alternate, in every other line the numerator is larger (by $1$) than the denominator...

Will Jagy
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