I would like to know if and how the powers of 10 are related to the printing of scientific notation in the console. I've searched R docs and haven't found anything relevant, or that I really understand.
First off, my
# scipen digits
# 0 7
#  10 100 1000 10000
#  1e+01 1e+02 1e+03 1e+04 1e+05
#  11 121 1331 14641 161051
#  100 10000
#  1e+02 1e+04 1e+06
Well, the answer is actually there in the definition of
?options, although it's pretty hard to understand what it means without playing around with some examples:
‘scipen’: integer. A penalty to be applied when deciding to print numeric values in fixed or exponential notation. Positive values bias towards fixed and negative towards scientific notation: fixed notation will be preferred unless it is more than ‘scipen’ digits wider.
To see what that means, examine the following three pairs of exactly identical numbers. In the first two cases, the width in characters of the fixed notation that is less than or equal to the width of the scientific, so fixed notation is preferred.
In the third case, though, the fixed notation is wider (i.e. "more than 0 digits wider"), because the 5 zeros amount to more characters than the 4 characters used to represent the same value using
e+nn. As a result, in that case scientific notation is preferred.
1e+03 1000 #  1000 1e+04 10000 #  10000 1e+05 100000 ## <- wider #  1e+05
Next, examine some numbers that also end with lots of zeros, but whose representation in scientific notation will require use of a
.. For these numbers, scientific notation will be used once you have 6 or more zeros (i.e. more than the 5 characters taken up by one
. and the characters
1.1e+06 1100000 #  1100000 1.1e+07 11000000 ## <- wider #  1.1e+07
Reasoning about the tradeoff gets a bit trickier for most other numbers, for which the values of both
options("digits") come into play, but the general idea is exactly the same.
To see some of the slightly surprising complications that come into play, you might want to paste the following into your console (perhaps after first trying to predict where within each series the switch to scientific notation will occur).
100001 1000001 10000001 100000001 1000000001 10000000001 100000000001 1000000000001 111111 1111111 11111111 111111111 1111111111 11111111111 111111111111 1111111111111