Why are the E-series numbers different from the powers of 10?

The E-series numbers are the common values used in resistors. For example, the E6 values are:

• 1.0
• 1.5
• 2.2
• 3.3
• 4.7
• 6.8

As you can see, each is about $10^\frac16$ apart. But I wonder why they aren't the powers of $10^\frac16$ rounded to 2 significant figures.

• $10^\frac16 \approx 1.4678$
• $10^\frac26 \approx 2.1544$
• $10^\frac36 \approx 3.1623$
• $10^\frac46 \approx 4.6416$
• $10^\frac56 \approx 6.8129$

3.1623 should't round to 3.3 no matter rounding upwards or downwards. And by rounding to the closest number, 4.6416 rounds to 4.6.

The same happens in other E-series values. For example, the powers of $10^\frac{1}{12}$ rounded to 2 significant figures are:

• $10^\frac{0}{12} \approx 1.0$
• $10^\frac{1}{12} \approx 1.2$
• $10^\frac{2}{12} \approx 1.5$
• $10^\frac{3}{12} \approx 1.8$
• $10^\frac{4}{12} \approx 2.2$
• $10^\frac{5}{12} \approx 2.6$
• $10^\frac{6}{12} \approx 3.2$
• $10^\frac{7}{12} \approx 3.8$
• $10^\frac{8}{12} \approx 4.6$
• $10^\frac{9}{12} \approx 5.6$
• $10^\frac{10}{12} \approx 6.8$
• $10^\frac{11}{12} \approx 8.3$

While the E12 values are:

• 1.0
• 1.2
• 1.5
• 1.8
• 2.2
• 2.7
• 3.3
• 3.9
• 4.7
• 5.6
• 6.8
• 8.2

The numbers 2.7, 3.3, 3.9, 4.7, and 8.2 from E12 are different from their corresponding ones computed above.

So why are the E-series of preferred numbers different from the powers of 10 rounded to the closest number?

• It's odd, isn't it? However, 'why did history turn out the way it did' rarely gets good answers. Generally, if the difference between actual practice and ideal theory is unimportant, and practice has been going on long enough, practice rarely gets changed. Perhaps the 'original engineer' had a bent slide rule? Jun 24 '18 at 17:10
• The values are as you describe: resistorguide.com/resistor-values however there is no rounding. Jun 24 '18 at 17:12
• The primary purpose of the E numbers is to ensure that some E number is within ±20%/±10%/±5%/etc (depending on whether you use E3 or E6 or E12 or...) of any value you might need. Since the current numbers do that, there's not really too much incentive to change that. That said, I couldn't tell you why they were originally like that. Jun 24 '18 at 17:20
• Perhaps the aesthetics of the color code figured into it. ;-) 4.7 is quite attractive. Or maybe they preferred to grab some values from the E3 series. Jun 24 '18 at 17:34
• Yes, the middle of the span has been "fudged". @Andy_aka did a nice graph showing the deviation in this item: electronics.stackexchange.com/questions/67975/… Jun 24 '18 at 18:01

I've really enjoyed your question and definitely upped it. Your question made me think about and do some additional reading on the topic. And I really appreciate what I've learned from the process and that you stimulated that process for me. Thanks!

Historical Context

I'm not going to go back to the Babylonian days here. (Probably, the whole concept does go back that far, and further.) But I'll start about a century ago.

Charles Renard proposed a few specific ways of arranging numbers to divide (decimal) intervals. He focused on dividing a decade range in 5, 10, 20, and 40 steps, where the logarithm of each step value would form an arithmetic series. And these became known as R5, R10, R20, and R40. Of course, there are many other choices one could make. But those were his, at the time.

Obviously, a decade range can be divided up in many ways (and besides, you don't have to focus on a decade range, either.) One extension idea I saw used Renard numbering systems of R10/3, R20/3, and R40/3. These were interpreted to mean that you would rely upon the R10, R20, and R40 decade series approach but would step the values, three at a time. So for example, R20/3 means to develop numbers based upon R20, but to select only every 3rd term: $$\10\cdot 10^\frac{0}{20}\approx 10\$$, $$\10\cdot 10^\frac{3}{20}\approx 14\$$, $$\10\cdot 10^\frac{6}{20}\approx 20\$$, $$\10\cdot 10^\frac{9}{20}\approx 28\$$, $$\10\cdot 10^\frac{12}{20}\approx 40\$$, $$\10\cdot 10^\frac{15}{20}\approx 56\$$, and $$\10\cdot 10^\frac{18}{20}\approx 79\$$. They also suggested that if you were looking for nice steps only between $$\10\$$ and $$\40\$$ then you could use just the first few of that set: 10, 14, 20, 28, and 40.

If you want to read further, the above and a lot more can be found in a publication called NBS Technical Note 990 (1978). (The National Bureau of Standards [NBS] is now NIST.)

Meanwhile, after WW II, there was a strong push towards standardizing manufactured parts. So various groups, at various times, worked pretty hard on "rationalizing" standard values to aid manufacturing, instrumentation, the numbers of teeth on gears, and ... well, most everything.

Skim the E Series of Preferred Numbers and take note of the associated documents and their history. However, the documents referred to in that Wikipedia page don't cover how those preferred numbers were chosen. For that, there is "ISO 497:1973, Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers." and also "ISO 17:1973, Guide to the use of preferred numbers and of series of preferred numbers." I don't have access to those documents, so I wasn't able to read them despite the fact that in particular ISO 497:1973 seemed like a good place to go.

E-Series (Geometric)

I haven't yet found any specifics about the precise algorithm applied some decades ago for the question you asked. The idea of "rationalizing numbers" isn't a hard idea, but the exact process that was applied is far beyond my ability to be certain of reverse-engineering now. And I was not able to uncover an historical document that disclosed it. Some of the elements can only be brought to light by possessing the full documents related to their final choices. And I haven't found those documents, yet. But I'm confident I was able to work out what must have been their process for the resistor question.

One of the things mentioned in NBS Pub. 990, is the fact that differences and sums of preferred numbers should not, themselves, be preferred numbers. This is in an attempt to provide coverage for other values in the decade range when explicit values fail to meet a need (by using two values in a sum or difference arrangement.)

Keep in mind that this coverage question is more important for the series such as E3 and E6 and is almost not at all important for E24, for example, which directly contains many intervening values. With that, in mind, the following is my thinking about their thinking. Perhaps it won't stray too far from the actual reasoning for their process of "rationalizing" values and making a final decision about the preferred values they ultimately chose to use.

My Reasoning

There is a very nice, simple sheet to look that which summaries the E-series values for resistors: Vishay E-Series.

Here is my image of the two-digit E-series values which includes the calculated values, as well: Here is my process, given the above, which I believe may be at least similar to the reasoning used many years ago:

1. The idea of coverage is most crucial for E3 and least crucial for E24. A quick glance at E3 suggests a problem with the rounded values of 10, 22, and 46. They are all even numbers and there is no possible way of composing odd numbers using only even numbers. So one of these numbers must change. They cannot change 10. And for changing one, the only remaining two possibilities are: (1) 10, 22, 47; or (2) 10, 23, 46. But option (2) has a problem: the difference between 46 and 23 is 23, which is itself a number in the sequence. And that is enough of a reason to eliminate option (2). This leaves only option (1) 10, 22, and . So this determines E3. (I'll use [] to surround modified sequence values and <> to surround values that must be preserved from the prior sequence.)
2. For E6, it must preserve the value choices of E3, inserting its own values in between. Nominally, E6 is then <10>, 15, <22>, 32, , and 68. However, the difference between 32 and 22 is 10 and this is one of the values already in the sequence. Also, 47 minus 32 is 15. Again, 32 is involved in a problem situation. Neither 22, nor 47 can be changed (they are inherited.) So the obvious (and only) choice is to adjust the E6 sequence to <10>, 15, <22>, , , and 68. The difference and sum values now provide coverage, too.
3. For E12, it must preserve the value choices of E6, inserting its own values. Nominally, E12 is then <10>, 12, <15>, 18, <22>, 26, , 38, , 56, <68>, and 83. The number 83 already has a problem, since 83 minus 68 is 15 and that is already in the sequence. 82 is the closest alternative. Also, the span between 22 and 26 is 4, while the span between 26 and 33 is 7. The spans should, roughly speaking, be monotonically increasing. This situation is serious and the only option is to adjust 26 to the next nearest choice, 27. The sequence is now <10>, 12, <15>, 18, <22>, , , 38, , 56, <68>, and . But we have again a problem with 38, with a preceding span of 5 and a following span of 9. Again, the only fix for this is to adjust 38 to its next nearest choice, 39. So E12 is adjusted to a final: <10>, 12, <15>, 18, <22>, , , , , 56, <68>, and .
4. E24 goes through a similar process. It starts out, nominally, as: <10>, 11, <12>, 13, <15>, 16, <18>, 20, <22>, 24, , 29, , 35, , 42, , 51, <56>, 62, <68>, 75, , and 91. I think by now, you can apply the logic I've applied earlier and get the final sequence of (not dropping the <> but leaving the [] indicator): 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, , , , , , , , 51, 56, 62, 68, 75, , and 91.

I think you'll agree this process is rational and leads directly to what we see, today.

(I didn't go through the logic applied to all the 3-digit E-series values: E48, E96, and E192. But I think there is enough above already and I believe it will pan out similarly. If you find anything differently, I'll be glad to look it over, too.)

The final rationalization process, towards preferred numbers, then looks something like this: Above, you can see the steps involved and where the changes are made and how they are then carried forward (reading right to left, of course.)

Relevant Notes

• The sum or difference of preferred numbers tend to avoid being a preferred number, where possible. This is required in order to provide as much coverage as possible.
• The product, or quotient, or any integral positive or negative power of preferred numbers will be a preferred number.
• Squaring a preferred number in the E12 series produces a value in the E6 series. Similarly, squaring a preferred number in the E24 series produces a value in the E12 series. Etc.
• Taking the square root of a preferred number in the E12 series produces an intermediate value in the E24 series that isn't present in the E12 series. Similarly, taking the square root of a preferred number in the E6 series produces an intermediate value in the E12 series that isn't present in the E6 series. Etc.

The above is exactly true when using the theoretical values rather than the preferred values. (The preferred values have been adjusted, so there will be some deviation due to that fact, using preferred values instead of the exact values.)

Interesting question that caused me to dig in and learn some of the history of the problems and the reasoning behind preferred numbers which I hadn't as fully apprehended before.

So, thanks!

Footnote: This post is related to another I've added here.

• E48 to E192 are simply powers of 10 rounded to 3 significant digits with only 1 exception. Somehow $10^{\frac{185}{192}} \approx 9.19479$ becomes 9.20. Apr 18 '21 at 2:57