> One of the modern applications of Egyptian fractions is the request of a specific resistance value needed in the design of an electrical circuit, a problem called in the literature the 2- Ohm problem. College students know well from their physics class, that the equivalent resistance R of two parallel resistances and is given from a law very easy to deduce, based on equating the current passing through the fictitious equivalent resistance R with the two currents passing through both resistances while maintaining same potential. One direct application of this, suppose an engineer wishes to incorporate in one of his designs a resistor of so many ohms which the manufacturer does not produce; for it is impossible that the latter displays in the market all possible ohm-values for his resistors. First, the market cannot possibly sustain it, but more important, one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set. Rather, manufacturers display only in the market what they call an "E12 series", i.e. resistors in sets of 12 different values, namely

> 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 ......

> 100, 120, 150, 180, 220, 270, 330, 390, 470, 560, 680, 820......

> 1000, 1200.......... Ohms, etc.....

> Now suppose an engineer needs in one of his designs a resistor of 7 ohms, then he would resort to a parallel combination from the fraction 1/7=1/10+1/56+1/100+1/120+1/150 in which all the resistors belong to the E12 series, i.e. he will replace his 7 ohms resistor with 5 parallel resistors; this he reaches using a special software (computer programmes exist for such designs, yet the exact solution is by no means trivial). What I did myself instead, is to resort to Ahmes 2/n table and wrote 2/7=1/4+1/28 or that 1/7=1/8+1/56. Decomposing further 1/8 into 1/12+1/24=1/12+1/48+1/48, but then I shall have to use instead of the 48 ohms resistor a resistor of 47 ohms from the E12 table. My final fraction is 1/7=1/12+1/ 47+1/47+1/56, i.e. my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions). My solution is both minimal and optimal based on Ahmes table. The relative error of my design will not exceed 0.6 per cent, being negligible; especially that any manufactured resistor will itself be subject to some allowed tolerance of the same order.

https://web.archive.org/web/20130625181118/http://weekly.ahr...

So, a thousands of years ago? Ancient Egypt and before? :)

Also:

> The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.