Binary number double the number to be converted, record if the result is at least 1, and then throw away the integer part. For example, formula_10, in binary, is: Thus the repeating decimal fraction 0... is equivalent to the repeating binary fraction 0... . Or for example, 0.1, in binary, is: This is also a repeating binary fraction 0.0... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic.
Combinatorial number system "C"<nowiki>'</nowiki>, since they have the same largest part 9, but the next largest part 6 of "C" is less than the next largest part 7 of "C"<nowiki>'</nowiki>; the sequences compared lexicographically are (9,6,4,3,0) and (9,7,3,1,0). Another way to describe this ordering is view combinations as describing the "k" raised bits in the binary representation of a number, so that "C" = {"c"...,"c"} describes the number (this associates distinct numbers to "all" finite sets of natural numbers); then comparison of "k"-combinations can be done by comparing the associated binary numbers. In the example "C" and "C"<nowiki>'</nowiki> correspond to numbers 1001011001 =