[prnqdlyx] approximations among decimal powers of 10

combine the numerical coincidences 10^.3 ~= 2 and 10^.7 ~= 5 (equivalently 10^.1 ~= 2^(1/3) ~= 5^(1/7) ) in various ways, shifting by powers of 10 as necessary, to yield the approximations below.  multiple routes can result in disagreeing approximations.  for example, 10^-0.9 = 10^(0.7*3 - 3) = 5^3/1000 = 0.125 or 10^-0.9 = 10^(0.3*7 - 3) = 2^7/1000 = 0.128 .  the exact values (to 3 significant figures) are also included in bold.  the exact values can also be distinguished because they usually are not digit patterns from powers of two: 625 125 25 5 1 2 4 8 16 32 64 128 256 512.

inspiration was to determine whether 10^.8 ~= sqrt 40 is related to the above concidences.  it is, because we can derive (10^.8)^2 = 10^1.6 = 10 * 10^.6 ~= 10 * 4 = 40 from the table below.

10^-0.9 = 0.125
10^-0.9 = 0.126
10^-0.9 = 0.128
10^-0.8 = 0.158
10^-0.8 = 0.16
10^-0.7 = 0.2
10^-0.6 = 0.25
10^-0.6 = 0.251
10^-0.6 = 0.256
10^-0.5 = 0.316
10^-0.5 = 0.32
10^-0.4 = 0.398
10^-0.4 = 0.4
10^-0.3 = 0.5
10^-0.3 = 0.501
10^-0.3 = 0.512
10^-0.2 = 0.625
10^-0.2 = 0.631
10^-0.2 = 0.64
10^-0.1 = 0.794
10^-0.1 = 0.8
10^0.0 = 1
10^0.1 = 1.25
10^0.1 = 1.26
10^0.1 = 1.28
10^0.2 = 1.58
10^0.2 = 1.6
10^0.3 = 2
10^0.4 = 2.5
10^0.4 = 2.51
10^0.4 = 2.56
10^0.5 = 3.16
10^0.5 = 3.2
10^0.6 = 3.98
10^0.6 = 4
10^0.7 = 5
10^0.7 = 5.01
10^0.7 = 5.12
10^0.8 = 6.25
10^0.8 = 6.31
10^0.8 = 6.4
10^0.9 = 7.94
10^0.9 = 8

perl -we 'for ( $i = -4 ; $i <= 9 ; ++$i ) { for ( $j = -4 ; $j <= 3 ; ++$j ) { $x = 10 ** $j * 2 ** $i ; $l = log ( $x ) / log ( 10 ) ; next if ( $l < -1.999 ) or ( 1.999 < $l ) ; printf ( "%.1f %g\n" , $l , $x ) } } for ( $i = -19 ; $i <= 19 ; $i++) { $x = $i / 10. ; $y = sprintf ( "%.2e" , 10 ** $x ) ; printf ( "%.1f %g\n" , $x , $y ) }' | sort -g | uniq

next, we consider errors of a specific choice of approximation:

	approx	exact	relative error	min. range	max. range
10^0.1	1.25	1.259	-0.71%	1.244	1.274
10^0.2	1.6	1.5849	0.95%	1.566	1.604
10^0.3	2	1.9953	0.24%	1.971	2.019
10^0.4	2.5	2.512	-0.47%	2.482	2.542
10^0.5	3.2	3.162	1.19%	3.1246	3.2
10^0.6	4	3.981	0.48%	3.934	4.029
10^0.7	5	5.012	-0.24%	4.952	5.072
10^0.8	6.3	6.310	-0.15%	6.234	6.3848
10^0.9	8	7.943	0.71%	7.849	8.038
10^1.0	10	10	0%	9.881	10.119

Pari/GP:

a=[1.25,1.6,2,2.5,3.2,4,5,6.3,8,10]
maxerror=(3.2-sqrt(10))/sqrt(10) ; for(i=1,10,x=a[i] ; y=10^(i/10) ; z=(x-y)/y*100 ; printf("<tr> <td>10^%.1f</td> <td>%5.2f</td> <td>%6.3f</td> <td>%5.2f%%</td> <td>%6.3f</td> <td>%6.3f</td> </tr>\n",i/10,x,y,z,y-maxerror*y,y+maxerror*y))

the "approx" column is the aesthetically chosen approximation.  "exact" is to 3 (ish) decimal digits.  the largest relative error, 1.19%, happens at 10^0.5 = sqrt(10).  the min and max range columns gives the bounds for 1.19% error around all the exact values.

assuming we're not willing choose a more accurate approximation for sqrt 10 than 3.2, the aesthetic problem is to choose the "simplest" number in each range as the approximation.  should 10^.1 be 1.26 or 1.25 ?  should 10^.9 be 7.9 or 8 ?

previously, common logarithms, tenth root of ten, etc.