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Comment by londons_explore | Hacker News Reader

When were the resistor series' first invented?

Is it possible that calculating the logarithmic scale numerically was quite a lot of effort, so instead a graphical approach, using a slide rule, was used? If so, small errors of a couple of percent could be expected in the results, especially if the rule wasn't precisely made?

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We had good logarithm tables in the 1620’s. http://www2.cfcc.edu/faculty/cmoore/LogarithmInfo.htm: “Napier died in 1617. Briggs published a table of logarithms to 14 places of numbers from 1 to 20,000 and from 90,000 to 100,000 in 1624. Adriaan Vlacq published a 10-place table for values from 1 to 100,000 in 1628, adding the 70,000 values“

These will have had errors, but I doubt they had them in the first 4 digits and if they had them, they would be easily spotted. (Edit: https://adsabs.harvard.edu/full/1872MNRAS..32..255G says there were errors)

(https://en.wikipedia.org/wiki/Adriaan_Vlacq)

> 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.

> then he would resort to a parallel combination from the fraction

Easier to find series combinations and then maybe clean them up with one parallel resistor. e.g. 4.7 + 2.2 ohms = 6.9 ohms; if you want better, 3.9+3.9 =7.8, and a 68 ohm resistor in parallel yields 6.997 ohms, an error of .04%.

Often that parallel resistor will be a trimpot or other adjustable means.

Or the greedy approach: 6.8+.22 in parallel with 2700 gives you 7.0018 with .026% error.

>my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions).

If we had just 1,2,5 series that would just be 2 resistors tho ? 3 resistors to get 9,8; 2 to get 6,4,3

The whole thing seems to be not that practical for electronics where you don't aim your amplifier to have amplification of "golden ratio" but in most cases some integer like x5 or x100

> 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

The rational numbers are an example of a countable dense set.

Unfortunately, countably-infinite product catalogs are still a bit unwieldy.

1/7 ≈ 1/8.2 + 1/47 gives about half the error (<.3%) with half the resistor count.

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