A Preliminary Derivation
Given a fraction a/b, one may multiply or divide its top and bottom ("numerator and denominator") by the same number c:
(a/b) = (ac)/(bc)
where (remember?) the two letters ac stand for "a
times c" and similarly for bc.
That is so because (c/c) = 1, no matter what the value of c is (except of course zero: "Thou Shalt not Divide by Zero") and multiplying anything by 1 does not change its value. In multiplying fractions, the rule is to multiply top with top, bottom with bottom, so we get
(a/b) (c/c) = (ac)/(bc)
As for dividing top and bottom by the same number d
(a/b) = [a/d]/ [b/d]
it follows at once from the preceding, if we choose the number c to equal 1/d.
Working with Small Quantities
Some equations, identities or formulas contain small quantities, and these can be made much simpler and easier to use by sacrificing a little accuracy. In fact, some equations which have no simple solution at all (like Kepler's equation in section (12a)) can yield in this way an approximate solution, often good enough for most uses, or else open to further improvement.
Many such calculations make use of the following observation. When we derive squares, 3rd powers, 4th powers etc. of numbers larger than 1, the results are always bigger, while for numbers smaller than 1, the results are always smaller. For example: