Is it possible for a positive number to exceed its reciprocal by exactly one?

Question

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

Answer 1

" 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.

Answer 2

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

Answer 3

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...