Numbers whose self and reciprocal are finitely decimally expressable that are close to one? How would I go about finding numbers x such that x and 1/x are finitely decimally reciprocal and are also close to 1?
I'm not entirely certain how to phrase this question, but take for example 2. 2 and 1/2 can be represented with a finite number of decimal points.
The closest pair to 1 that I have found is 4/5 and 5/4. Are there closer pairs and how would I look for them?
 A: What you need is a power of $2$ and a power of $5$ which are close together. Your example has $4=2^2$ and $5=5^1$.
Since powers of $2$ are a multiple of $2$ apart, you can always get between $\frac 12$ and $2$.
To get the best results, though, we want $5^n=k2^m$ where $k$ is close to $1$.
Taking logs we have $n\log 5 = \log k + m\log 2$ and since $k$ is near to $1$ we have $\log k$ close to zero, and we see that $$\frac nm\approx\frac{\log 2}{\log 5}$$ and the best way of getting close is to use the continued fraction expansion of $\frac{\log 2}{\log 5}$.
A: The only rational numbers that have finite decimal representations are those whose denominator has only $2$ and $5$ as its prime factors. (because if $z$ has a finite decimal expansion, then $y=10^k z$ is an integer for some positive integer $k$, and thus $z = y/10^k$)
Thus, the pairs you are looking for are all of the form $2^m / 5^n$ and $5^n / 2^m$.
To see when they're close to 1, it's easier to turn it into an additive problem by taking logarithms: you want
$$ m \ln 2 - n \ln 5 \sim 0 $$
Rearranging, we want
$$ \frac{m}{n} \sim \frac{\ln 5}{\ln 2} $$
so the problem is to find very good rational approximations to $\ln 5 / \ln 2$.
The first few approximations given by continued fractions is:


*

*$m/n = 2$, and thus $x = 4/5 = 0.8$

*$m/n = 7/3$, and thus $x = 2^7 / 5^3 = 1.024$ and $1/x = 0.9765625$

*$m/n = 65/28$, and thus $x = 2^{65}/5^{28} = 0.99035\ldots$

