The way in which some of our physical units where designed means that some numbers
have no reason whatsoever to be "round numbers".
For example,
the fact that the speed of light (Einstein's constant)
is very nearly 300 000 km/s is a pure coincidence
(it's now equated de jure to
299792458 m/s to define the SI meter in term
of the atomic second of time).
Another pure coincidence is the fact that the diameter
of the Earth is very nearly half a billion inches:
The polar diameter is 500531678 inches, the equatorial diameter is
502215511 inches.
Rydberg's constant expressed as a
frequency
is very nearly
p^{2}/3 10^{15} Hz.
The accuracy
of this pure coincidence is better than 8 ppm:
3.289841960364(17) 10^{15} Hz = c R_{¥}
[ CODATA 2010 ]
3.28986813369645287294483...
= p^{2}/3
On the other hand, some "coincidences" were engineered or
enhanced by metrologists...
Consider some "nearly correct" conversion factors:
 A typographer's point is 0.013837".
That's about 72.2700007227 points to the inch._{ }
 An inch is 0.0254 m.
That's about 39.37007874015748... inches to the meter.
In both cases, the accuracy of the rounded inverse conversion factor
keeps alive a competing unit based on that. The relevant numerical relations are:
254 . 3937 = 999998
13837 . 7227 = 99999999
Pairs of decimal numbers which have roughly the same number of digits and are
very nearly reciprocals of each other can be designed
around the factorizations into primes of numbers close to
powers of ten. For example:
The second entry is from the Iranian
gold trade :
100 g = 217 Ms
Number  Factorization  Pairs of Factors 

999998 2 ppm  2 . 31 . 127^{2} 
2 . 499999 31 . 32258  62 . 16129 127 . 7874 
254 . 3937 
10000011 1.1 ppm  3 . 7 . 31 . 15361 
3 . 3333337 7 . 1428573 
21 . 476191 31 . 322581 93 . 107527 
217 . 46083 651 . 15361 
999999 1 ppm  3^{3} . 7 . 11 . 13 . 37 
3 . 333333 7 . 142857 9 . 111111 11 . 90909 13 . 76923
21 . 47619 27 . 37037 33 . 30303 37 . 27027 39 . 25641 63 . 15873 
77 . 12987 91 . 10989 99 . 10101 111 . 9009 117 . 8547
143 . 6993 189 . 5291 231 . 4329 259 . 3861 273 . 3663 297 . 3367 
333 . 3003 351 . 2849 407 . 2457 429 . 2331
481 . 2079 693 . 1443 777 . 1287 819 . 1221 999 . 1001

1000001  101 . 9901  101 . 9901  
1000002  2 . 3 . 166667 
2 . 500001  3 . 333334  6 . 166667 
9999999 0.1 ppm  3^{2} . 239 . 4649 
3 . 3333333 9 . 1111111  239 . 41841 717 . 13947 
2151 . 4649 
99999999 0.01 ppm  3^{2} . 11 . 73 .101 .137 
3 . 33333333 9 . 11111111 11 . 9090909 33 . 3030303 73 . 1369863
99 . 1010101 101 . 990099 137 . 729927 
219 . 456621 303 . 330033 411 . 243309 657 . 152207 803 . 124533
909 . 110011 1111 . 90009 1233 . 81103 
1507 . 66357 2409 . 41511 3333 . 30003 4521 . 22119
7227 . 13837 7373 . 13563 9999 . 10001 
Not all such pairs of factors are interesting, because...
At a time when authors of arithmetic textbook wanted to reward
their students with interesting results to elementary problems, they would often build
some of their exercises on nontrivial factorizations of
10^{n}1
(Cunningham numbers).
The most valued prizes were
products of two factors with the same number of digits: