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Mental maths, nice numbers, and Fibbonaci

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Dr. Slack

Fri Nov 17 2017, 07:30AM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

For a decade now, I've been (more or less unsuccessfully) been trying to teach myself mental logs, mainly by 'times table' like repetition, using approximations to 10^0.1x. You'll know the alternate terms from standard capacitor voltage ratings, which go up in ratios of around 1.6.

log10(x)    x
0.0        1.0
0.1        1.25
0.2        1.6
0.3        2.0
0.4        2.5
0.5        3.2
0.6        4.0
0.7        5.0
0.8        6.3
0.9        8.0
1.0       10.0

As the log is base 10, you'll also recognise the dB scale in there, but for a factor of 10 or 20. With a little thought, I can recall the table, but it doesn't yet spring to mind as automatically as (say) <chant> eight 7s are 56 </chant>.

I was reading about Fibonacci numbers recently, and an interesting use in software estimation via 'Planning Poker'. In order to make everybody think, and eliminate anchor bias, the whole team are given cards, and they each place one face down, and then they're all revealed. The cards are numbered in the Fibonacci sequence, for several reasons, in addition to their being cool. One, it has constant relative precision, big tasks are estimated relatively as accurately as small tasks. Second, it allows a wide dynamic range with a small number of cards. And third, each number is the sum of the previous, so if a task is split, the new tasks get estimated quickly as 'is this part more or less than half of the original?' and the appropriate one chosen.

Now I already knew that the ratio of Fibonacci terms tended to 1.618..., which struck me as very close to the 1.6 ratio of successive capacitor voltages. Why hadn't I seen that before? So they are almost the same series?

I took 5 steps of capacitor voltages, 10, 16, 25, 40, 63, 100, and 5 steps of Fibonacci 5, 8, 13, 21, 34, 55. The former (obviously) increases by a factor of 10. The Fibonacci series increases by a factor of 11, and in the limit of big numbers by about 11.090.

The line 'so if she floats, she's a duck, and so she's made of wood!' came to mind. Are the capacitor voltages approximately sums of the previous two? It turns out, very close, and always slightly above. 10+16 is about 25. 16+25 is close to 40. 25+40 is not far off 63. And 40+63 is more or less 100.

So will this help me remember my log table? Probably not. Does it have any other useful fallout? Mmmmm, thinking, probably not. But I feel it's still mildly interesting.

Uspring

Fri Nov 17 2017, 11:00AM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

klugesmith

Sun Nov 26 2017, 10:59PM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716

Interesting stuff.

That table of values spaced by 1/10 of a decade was taught to me decades ago, for hand-drawing log scales on regular graph paper. (e.g. for Bode plots).

If a decade (sometimes 20 dB) is 10 divisions, then we can draw 2, 4, and 8 at divisions 3, 6, and 9. Those same factors, measured back from 10, give us 5, 2.5, and 1.25 at divisions 7, 4, and 1.
The power-of-two value after 8 is 16, giving us 1.6 and 6.25 for divisions 2 and 8.
Division 5 is, of course, sqrt(10). We have to remember it's 3.16, not far above pi.

Before Dr Slack's post, I had never noticed the regularity of standard capacitor voltage ratings.
5 per decade is handy, eh?

6 per decade, as found in e6 standard values of capacitance and resistance, is just a different set of numbers most of us have learned. Who can enlighten us about the unnecessarily irregular ratios found in the e6, e12, e24 series?

I think the Golden ratio / Fibonacci similarity is just a coincidence. log10(Ï†) is about 0.209. Not nearly as close to a round value as log10(2) being 0.301, handy for charting as noted above. And making log10(2^10) about 3.01. So close to 3, that a generation has grown up thinking that Kilo means 1024 of anything related to data storage.

Uspring

Sat Dec 16 2017, 06:54PM

Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711

Who can enlighten us about the unnecessarily irregular ratios found in the e6, e12, e24 series?

From the point of view of a circuit designer, who calculates a required resistance value and then selects the closest one out of a series, he would prefer a series in which the max ratio between consecutive list values is minimal, because this implies the least errors when selecting a list value. If the list values are of 2 digit accuracy, then the the best set of list values don't need to be equal to the rounded ideal numbers.

Ash Small

Wed Jan 31 2018, 01:25PM

Registered Member #3414 Joined: Sun Nov 14 2010, 05:05PM
Location: UK
Posts: 4245

I was under the impression that the capacitor/resistor values were more to do with the 20% tolerance thing.

How this fits in with the above, I'm not sure.

EDIT: 33, 39, 47 are all 20% apart, for example. (to the nearest whole number)

(It's more about ten percent tolerance, ie, if it's within 10% of 33, it's 33, the same for 39 and 47, etc.)

Sulaiman

Wed Jan 31 2018, 02:01PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3141

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