Zero is the one number that has no reciprocal
Flip almost any number and you get its reciprocal: five becomes one fifth, a quarter becomes four. Multiply the pair and the answer is always one. Zero is the lone exception, because anything times zero is zero, never one. That gap is exactly why dividing by zero has no meaning.
A reciprocal, or multiplicative inverse, of a number is whatever you multiply it by to get 1. You find it by dividing 1 by the number, so the reciprocal of 0.25 is 4. That gives a handy shortcut: multiplying by a number does the same job as dividing by its reciprocal, so multiplying by 0.8 matches dividing by 1.25. The word reciprocal was already used this way in the third edition of the Encyclopaedia Britannica in 1797, and a 1570 English Euclid used it for quantities in inverse proportion.
The idea sorts number systems into families. Among real, rational and complex numbers, everything except zero has a reciprocal of the same kind, and that property is part of what mathematicians call a field. Whole numbers fail the test: only 1 and minus 1 have whole-number reciprocals, so the integers are not a field. Clock arithmetic has its own version. Working modulo 11, the inverse of 3 is 4, because 3 times 4 is 12, which leaves remainder 1. Such an inverse exists only when the number shares no factor with the modulus.
The notation causes famous confusion. Writing sin to the power minus one can mean one over the sine, which is the cosecant, or it can mean the inverse sine function, arcsin, which is something else entirely. Authors disagree on which word should mean which; in French, for instance, an inverse function is called a bijection réciproque. The trigonometric functions come in reciprocal pairs anyway: cotangent with tangent, secant with cosine, cosecant with sine.
Complex numbers add a curiosity. To flip one, you take the reciprocal of its size and reverse its angle. The imaginary unit i is special: its reciprocal and its negative are the same number, minus i, and i and minus i are the only complex numbers with that property.
Source: Multiplicative inverse
