reciprocal of an expression

The reciprocal of an expression is what you get by turning it upside down — swapping its numerator and denominator. The reciprocal of 3/4 is 4/3, and the reciprocal of (x + 1)/(x - 2) is (x - 2)/(x + 1). A whole expression like x is secretly x/1, so its reciprocal is 1/x.

What makes the reciprocal special is multiplication: an expression times its reciprocal equals 1. This is why reciprocals are the engine of division — “invert and multiply” works because dividing by something is the same as multiplying by its reciprocal. In the language of structure, the reciprocal is the multiplicative inverse.

There is a value that has no reciprocal: zero. Since flipping 0/1 would give 1/0, and dividing by zero is undefined, only nonzero quantities have reciprocals. For an expression, this means its reciprocal is defined only where the original expression is itself nonzero.