Can every number be represented as a sum of different reciprocal numbers? Can every rational number be represented as a finite sum of reciprocal numbers? You are only allowed to use each reciprocal number one time per expression (So for example 3/2 cannot be 1/2+1/2+1/2). 
You could then express these numbers in a kind of "binary-reciprocal" where a 1 in the nth place from the right denotes adding 1/n. E.g 17/10 could be 10011 as it is 1+1/2+1/5. 
Thanks in advance!

Every positive number is a subseries of the harmonic series (maybe infinite). You do the following: if $x$ is your number - add numbers till the next step is over $x$, Then don't add the following till adding again will give you something less or equal. Proceed in this manner.
For positive rationals there is a method to achive them in a finite manner - using the following replacement $1/k+1/k=1/k+1/(k+1)+1/k(k+1)$ - Botts, Truman (1967), "A chain reaction process in number theory", Mathematics Magazine, 40 (2): 55–65
