Reciprocal

Flip it

$$\frac{3}{5} \quad\longrightarrow\quad \frac{5}{3}$$

A whole number sits over $1$, so its reciprocal is one over it:

$$4 = \frac{4}{1} \quad\longrightarrow\quad \frac{1}{4}$$

The defining property

A number times its reciprocal is always $1$.

$$\frac{3}{5} \times \frac{5}{3} = \frac{15}{15} = 1$$

That is the whole definition. The flipping is how you find it, not what it is.

Why division uses it

Dividing by a number is the same as multiplying by its reciprocal.

$$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{3}{2}$$

The reason is the counting question underneath division: dividing by $\tfrac{1}{2}$ asks how many halves fit, and each whole holds two of them. See dividing fractions.

Zero has none

There is no number you can multiply $0$ by to get $1$, so $0$ has no reciprocal. This is the same fact as “you cannot divide by zero”, seen from another angle.

A mixed number must be converted first

$2\tfrac{1}{3}$ does not flip to $3\tfrac{1}{2}$. Convert to $\tfrac{7}{3}$ first, and the reciprocal is $\tfrac{3}{7}$.