I made up this problem, and I have been thinking about it for a few days, but I do not know if it has a solution. Looking at other posts, this problem seems more mathematically oriented. Any help would be appreciated.
Suppose that we have an unlimited supply of resistors with resistances R1 Ω and R2 Ω. Is it always possible to make a circuit with equivalent resistance R1*R2 Ω using only a finite number of such resistors? (Note that R1 and R2 can be arbitrary real numbers, such as π=3.1415...)
The problem statement seems so simple, but the problem itself is actually quite elusive. I suspect that it is not possible, since, if R1=sqrt(3) and r2=sqrt(12), we need to make a circuit with equivalent resistance 6. A construction of such a circuit using only series and parallel combinations seems highly unlikely (I have tried in vain), and expressions for more complicated circuits get messy quickly.
If possible, could someone provide such a construction? If not possible, could someone provide a (rigorous) proof or counterexample explaining why it is not possible?
Thanks!
Suppose that we have an unlimited supply of resistors with resistances R1 Ω and R2 Ω. Is it always possible to make a circuit with equivalent resistance R1*R2 Ω using only a finite number of such resistors? (Note that R1 and R2 can be arbitrary real numbers, such as π=3.1415...)
The problem statement seems so simple, but the problem itself is actually quite elusive. I suspect that it is not possible, since, if R1=sqrt(3) and r2=sqrt(12), we need to make a circuit with equivalent resistance 6. A construction of such a circuit using only series and parallel combinations seems highly unlikely (I have tried in vain), and expressions for more complicated circuits get messy quickly.
If possible, could someone provide such a construction? If not possible, could someone provide a (rigorous) proof or counterexample explaining why it is not possible?
Thanks!