Reciprocal
Fractions & DecimalsThe 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 - swap the numerator and denominator. The reciprocal of $3/4$ is $4/3$. Any number multiplied by its reciprocal always equals $1$.
Example
Reciprocal of $2/5$ is $5/2$. Check: $2/5 \times 5/2 = 10/10 = 1$. Reciprocal of the whole number $4$ is $1/4$. Check: $4 \times 1/4 = 1$.
Key Insight
Reciprocals are "undo" numbers for multiplication. Multiplying by $3/4$ shrinks something; multiplying by its reciprocal $4/3$ undoes that shrinking and brings you back to the start.
Definition
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 $b/a$. The product of any nonzero number and its reciprocal always equals $1$: $x \times (1/x) = 1$.
Example
Dividing by $2/3$ is the same as multiplying by its reciprocal $3/2$: $5/(2/3) = 5 \times (3/2) = 15/2 = 7.5$. This is the "keep-change-flip" method for fraction division.
Key Insight
The reciprocal converts division into multiplication - the most useful trick in fraction arithmetic. The deeper reason: division by $x$ is defined as multiplication by $x$'s multiplicative inverse, which must exist for division to be valid.
Definition
The reciprocal of a nonzero element $x$ in a field $F$ is its multiplicative inverse $x^{-1}$, the unique element satisfying $x \cdot x^{-1} = 1$. For $\mathbb{Q}$: $(a/b)^{-1} = b/a$. In a ring that is not a field, not every nonzero element has a multiplicative inverse (e.g., $2$ in $\mathbb{Z}$ has no reciprocal in $\mathbb{Z}$).
Example
In $\mathbb{Z}_7$ (integers mod $7$), the reciprocal of $3$ is $5$, since $3 \times 5 = 15 \equiv 1 \pmod{7}$. Finding modular inverses uses the extended Euclidean algorithm and is central to RSA encryption and modular arithmetic.
Key Insight
The field axioms require every nonzero element to have a multiplicative inverse. The existence of reciprocals is exactly what distinguishes a field from a ring, and it is why $\mathbb{Q}$ (field) supports division while $\mathbb{Z}$ (ring) does not - integer division leaves remainders precisely when the reciprocal does not exist in $\mathbb{Z}$.