The numbers are not chosen arbitrarily at all. There is some (not a lot) of maths behind it.
The numbers are chosen so that they follow a geometric progression with a certain number of values per decade.
For E3 values (the earliest and now unused series, there were 3 values per decade. That is 1 a, b, 10, c, d, 100, e, f, 1000.
A geometric progression is one where you multiply the value of the previous value with a constant to get the next value.
In the case of E3, if the constant is k, then 1 * k * k * k = 10, so k is the cube root of 10. Thus k is 2.15.
so in the sequence a, b, 10, c, d, 100, e, f, 1000 given above, we can fill in the missing values:
1, 2.15, 4.64, 10, 21.5, 46.4, 100, 215, 464, 1000.
E3 was used in the era of 50% tolerance. Resistors were so inaccurate that if you tried to make a particular value, it could be anywhere between of that value. The sequence 1, 2.2, 4.7 (or 1, 2, 5) was used to denote nominal values.
Numerically, this series works because
1 * 150/100 = 1.5 and 2.2 * 100/150 = 1.47 (almost the same)
2.2 * 150/100 = 3.3 and 4.7 * 100/150 = 3.13 (almost the same)
4.7 * 150/100 = 7.05 and 10 * 100/150 = 6.7 (almost the same)
If you're worried by the "almost the same", repeat this with 47% and you'll find the values are much closer (46.78% is even closer)
However, when you're talking about such large tolerances, it doesn't make sense to worry about a few percent either way.
The E3 series lives on today in test equipment which has 1, 2, 5 series on the ranges. An oscilloscope might, for example, on the vertical gain settings have 10mV, 20mV, 50mV, 100mV, 200mV, 5000mV, 1V, 2V, 5V, 10V, ... as options for the "volts per division". This series often exists on even the most modern digital oscilloscopes.
As resistors (and capacitors, etc) began to be able to be manufactured in tighter and tighter tolerances, it made sense to have values between the E3 values.
The next step was E6. These were nominally 20% resistors, and for some time I had some of these which were hand painted with their values! (using the dot rather than stripe scheme).
E6 goes 1, 1.5, 2.2, 3.3, 4.7, 6.8, 10. It is no accident that the values inserted between the E3 values to make E6 as the midpoints seen when calculated above.
E6 is technically a geometric series with the multiplier being the 6th root of 10 -- 10^(1/6) = 1.47, and numerically speaking the tolerance would be the square root of this (21.15%).
Note the relationship between the mathematical tolerance and the multiplier for the next range.
After this E12 was used for 10% resistors, and E24 for 5%, and so on. For a long time after the introduction of these new-fangled 5% resistors, they were commonly only stocked in the E12 series, and even today many designs will use E12 series values exclusively.
For the higher tolerance ranges (E48, E96, and E192) the mathematical fudges which were employed to get values like 4.7 (instead of the closer 4.6 -- remember it was 4.64?) have significant effects. In these series, rather than the intermediate values slipping between the previous values, there are areas where the old values change. If you look at this page, you'll see that E6, E12, and E24 have a 4.7, but it is replaced with (surprise!) 4.64 in the E48, E96, and E192 series. See
here.
As a little challenge, can you find someone selling a 470ohm 1% resistor? Why? (
hint)
There's lots more information on the web.
here and
here would be places to start.
