what Prefered Numbers are

Illustration with an example for programmers: you wish to show result list entries and provide a drop-down that selects the number of displayed items per page. What values do you put there?

The number system in common use is base 10, so we likely want to have 10, 100, 1000 in there. That's not fine-grained enough for practical use in most cases. Let's make it evenly spaced in between each magnitude and explore for seven and three steps, each.

› $a = 10
10
› sprintf '%.f', $a += (100-10)/7
23
› sprintf '%.f', $a += (100-10)/7
36
› sprintf '%.f', $a += (100-10)/7
49
› sprintf '%.f', $a += (100-10)/7
61
› sprintf '%.f', $a += (100-10)/7
74
› sprintf '%.f', $a += (100-10)/7
87
› sprintf '%.f', $a += (100-10)/7
100

› $a = 10
10
› sprintf '%.f', $a += (100-10)/3
40
› sprintf '%.f', $a += (100-10)/3
70
› sprintf '%.f', $a += (100-10)/3
100
[download]

That does not feel good for the seven scale. There is no meaningful difference between 74 and 87, when confronted with that choice, a user might pick either one by random chance. That's because humans instinctively operate on a scale that is lopsided with increasingly larger steps towards the "heavy" end: the geometric progression. It's still evenly spaced, just not additively, but multiplicatively. From observing nature, we also know that's much more common. Trying it out:

› $a = 10
10
› sprintf '%.f', $a *= 10**(1/7)
14
› sprintf '%.f', $a *= 10**(1/7)
19
› sprintf '%.f', $a *= 10**(1/7)
27
› sprintf '%.f', $a *= 10**(1/7)
37
› sprintf '%.f', $a *= 10**(1/7)
52
› sprintf '%.f', $a *= 10**(1/7)
72
› sprintf '%.f', $a *= 10**(1/7)
100

› $a = 10
10
› sprintf '%.f', $a *= 10**(1/3)
22
› sprintf '%.f', $a *= 10**(1/3)
46
› sprintf '%.f', $a *= 10**(1/3)
100
[download]

That feels better. On the seven scale, the initial values are too tight, that's because we have so many steps, let's discard this scale. For the remaining scale, rouding the numbers (to most significant digit), we get preferred numbers, expressed as the series: 10, 20, 50, 100, 200, 500, 1000, …. Let's put those in the drop-down.

# playground
my ($start, $end) = (10, 100);  # try: (35, 7400)
for my $step (1..20) {
    printf 'step %2d:', $step;
    my $i = $start;
    for (1..$step+1) {
        printf ' %.f', $i;
        $i *= ($end/$start)**(1/$step);
    }
    print "\n";
}
__END__
step  1: 10 100
step  2: 10 32 100
step  3: 10 22 46 100
step  4: 10 18 32 56 100
step  5: 10 16 25 40 63 100
step  6: 10 15 22 32 46 68 100
step  7: 10 14 19 27 37 52 72 100
step  8: 10 13 18 24 32 42 56 75 100
step  9: 10 13 17 22 28 36 46 60 77 100
step 10: 10 13 16 20 25 32 40 50 63 79 100
step 11: 10 12 15 19 23 28 35 43 53 66 81 100
step 12: 10 12 15 18 22 26 32 38 46 56 68 83 100
step 13: 10 12 14 17 20 24 29 35 41 49 59 70 84 100
step 14: 10 12 14 16 19 23 27 32 37 44 52 61 72 85 100
step 15: 10 12 14 16 18 22 25 29 34 40 46 54 63 74 86 100
step 16: 10 12 13 15 18 21 24 27 32 37 42 49 56 65 75 87 100
step 17: 10 11 13 15 17 20 23 26 30 34 39 44 51 58 67 76 87 100
step 18: 10 11 13 15 17 19 22 24 28 32 36 41 46 53 60 68 77 88 100
step 19: 10 11 13 14 16 18 21 23 26 30 34 38 43 48 55 62 70 78 89 100
step 20: 10 11 13 14 16 18 20 22 25 28 32 35 40 45 50 56 63 71 79 89 1
+00
[download]

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Also see – division of European money: 0.01, 0.02, 0.05, 0.10, 0.20, 0.50, 1, 2, 5, 10, 20, 50, 100… Pity the fools who have to deal with 0.25 units with no corresponding 2.5 and 25.