Colin Wright pointed out a pattern in my previous post that I hadn’t seen before. He wrote about it here a couple years ago.

Start with the powers of 2 from the top of the post:

2^{1} = 2

2^{2} = 4

2^{3} = 8

2^{4} = 16

2^{5} = 32

2^{6} = 64

2^{7} = 128

2^{8} = 256

2^{9} = 512

and sort the numbers on the right hand side in *lexical* order, i.e. sort them as you’d sort words. This is almost never what you want to do, but here it is [1].

128

16

2

256

32

4

512

64

8

Next, add a decimal point after the first digit in each number.

1.28

1.6

2

2.56

3.2

4

5.12

6.4

8

Now compare the decibel values listed at the bottom of the previous post.

1.25

1.6

2

2.5

3.2

4

5

6.3

8

Five of the numbers are the same and the remaining 4 are close. And where the numbers differ, the exact decibel value is between the the two approximate values.

Here’s a table to compare the exact value rounded to 3 decimals, the approximation given before, and the value obtained by sorting powers of 2 and moving the decimal.

|---+-----------+--------+-------| | n | 10^(n/10) | approx | power | |---+-----------+--------+-------| | 1 | 1.259 | 1.25 | 1.28 | | 2 | 1.585 | 1.60 | 1.60 | | 3 | 1.995 | 2.00 | 2.00 | | 4 | 2.512 | 2.50 | 2.56 | | 5 | 3.162 | 3.20 | 3.20 | | 6 | 3.981 | 4.00 | 4.00 | | 7 | 5.012 | 5.00 | 5.12 | | 8 | 6.310 | 6.30 | 6.40 | | 9 | 7.943 | 8.00 | 8.00 | |---+-----------+--------+-------|

So the numbers at the top and bottom of my list are practically the same, but in a different order.

**Related post**: 2s, 5s, and decibels.

[1] One reason I use ISO dates (YYYY-MM-DD) in my personal work is that that lexical order equals chronological order. Otherwise you can get weird things like December coming before March because D comes before M or because 1 (as in 12) comes before 3. Using year-month-day and padding days and months with zeros as needed eliminates this problem.

Traditional shutter speeds in photography are 1/n seconds, with n = 1, 2, 4, 8, 15, 30, 60, 125, 250, 500. These are basically powers of two but pushed to round numbers. If you do the same operation on them you get

1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8

which is an even coarser approximation (but with fewer digits!) to the powers of 10^(1/10).

It’s not hard to explain why this works.

Let G be the multiplicative group generated by 10^(1/10), and let H be the subgroup generated by 10. Then G/H is cyclic of order 10. Shifting the decimal point simply results in a different representative of an equivalence class in G/H.

Since 3 is relatively prime to 10, G/H is also generated by 10^(3/10). As everyone who has argued about the size of a kilobyte knows, 2^10 ≈ 10^3, so 10^(3/10) ≈ 2.