Reciprocal
The reciprocal of a number is 1 divided by that number; for a fraction a/b, the reciprocal is b/a.
Formula
\text{reciprocal of } \frac{a}{b} = \frac{b}{a}
Definition
The reciprocal of a fraction is what you get when you flip it upside down, swapping the numerator and denominator: the reciprocal of $3/4$ is $4/3$, and any number multiplied by its reciprocal always equals $1$. More generally, the reciprocal of a nonzero number $x$ is $1/x$, also called the multiplicative inverse; for a fraction $a/b$ (with $a \neq 0$), the reciprocal is $b/a$. Formally, the reciprocal of a nonzero element $x$ in a field $F$ is its multiplicative inverse $x^{-1}$, the unique element satisfying $x \cdot x^{-1} = 1$; in a ring that is not a field, not every nonzero element has a multiplicative inverse (for example, $2$ in $\mathbb{Z}$ has no reciprocal in $\mathbb{Z}$).
Example
The reciprocal of $2/5$ is $5/2$: check, $2/5 \times 5/2 = 10/10 = 1$; the reciprocal of the whole number $4$ is $1/4$, since $4 \times 1/4 = 1$. Dividing by $2/3$ is the same as multiplying by its reciprocal $3/2$: $5/(2/3) = 5 \times (3/2) = 15/2 = 7.5$, the "keep-change-flip" method for fraction division. In $\mathbb{Z}_7$ (integers mod $7$), the reciprocal of $3$ is $5$, since $3 \times 5 = 15 \equiv 1 \pmod{7}$; finding such modular inverses uses the extended Euclidean algorithm and is central to RSA encryption.
Key Insight
Reciprocals are "undo" numbers for multiplication: multiplying by $3/4$ shrinks something, and multiplying by its reciprocal $4/3$ undoes that shrinking, bringing you back to the start. This is the deeper reason division works at all, division by $x$ is defined as multiplication by $x$'s multiplicative inverse, which must exist for division to be valid. The field axioms require every nonzero element to have a multiplicative inverse, and this existence of reciprocals is exactly what distinguishes a field from a ring: it is why $\mathbb{Q}$ supports division while $\mathbb{Z}$ does not, integer division leaves remainders precisely when the reciprocal does not exist in $\mathbb{Z}$.