Rounding to an order of magnitude 

One way to simplify a number is to
round it to the nearest "power of ten".
This is an extreme form of rounding,
and usually uses exponential notation.
These orders of magnitude are very easy to work with.
So 256 is order 100, 365 is order 1000.
0.3 to 3 is "on the order of magnitude of 1".
3 to 30 is "on the order of magnitude of 10".
30 to 300 is "on the order of magnitude of 100".
Or "mumble mumble has an order magnitude of".
It quickly becomes more convenient to instead say
"... order of magnitude of ten to the zero" (10^{0} = 1)
"... order of magnitude of ten to the one" (10^{1} = 10)
"... order of magnitude of ten to the two" (10^{2} = 100)
250,000,000 (250 million, US population) is "on the order of magnitude of ten to the eight" (10^{8}).
The maximum error/uncertainty is a factor of 3.
10^{a} × 10^{b} = 10^{ a + b } 10^{a} / 10^{b} = 10^{ a  b } 10^{bigger} + 10^{smaller} ~= 10^{bigger} 10^{bigger}  10^{smaller} ~= 10^{bigger} 10^{same} + 10^{same} ~= 10^{same} 10^{same}  10^{same} ~= don't do it (10^{a})^{c} = 10^{ a × c }
Actually, rounding up from 3 isn't quite right.
The best boundries are not 3 and 30 but rather 3.1623 to 31.623 .
However, this only increases the maximum error 5%, from 3 to 3.162277
(the square root of ten). It does however introduce a systematic
underestimate.
For almost all calculations, the systematic bias doesn't matter even
a little bit. And if you are worried about the tiny extra error, you
are better off rounding less agressively
by leaving a significant digit.
It would be nice to break out oom math into its own page, with examples.
Some discussion of uncertainty needs to get written and linked in.
Needs an overhaul.
Comments encouraged.  Mitchell N Charity <mcharity@lcs.mit.edu> 
History: 1999.Oct.15 Corrected error re math rule (10^a)^c = 10^(ac) (never 10^(a/c)). My thanks to a reader. 1998.Jun.17 Corrected error re magnitude boundaries (it's 3.162, not 3.333). long ago previous change