complex fraction

A complex fraction is a fraction stacked inside a fraction — its numerator, its denominator, or both contain fractions of their own. Something like (1 + 1/x) / (1 - 1/x) is a complex fraction: a little fraction lives on each floor of the big one. (Here “complex” means complicated-looking, and has nothing to do with complex numbers.)

There are two standard ways to simplify it. The first: combine the top into a single fraction and the bottom into a single fraction, then divide — which means flip the bottom and multiply. The second, often faster: multiply the entire complex fraction, top and bottom, by the LCD of all the little inner denominators, which clears every small fraction in one stroke.

Either route ends at an ordinary rational expression. Throughout, watch the restrictions: every inner denominator, and the overall denominator, must be nonzero, so a complex fraction can carry several excluded values at once.