A QUICK REMINDER:
The reciprocal of a number is:
either: turn it upside down
or (equivalently): one divided by that number.
So the reciprocal of $\frac{2}{3}$ is $\frac{3}{2} = 1.5$ and the reciprocal of $5$ is $\frac{1}{5} = 0.2$.
Note that the first definition always works if you think of e.g. $5$ as being $\frac{5}{1}$.
SO WHAT’S THE POINT?
1) A Reciprocal is a multiplicative inverse: multiplying by a number and then by its reciprocal gets you back where you started. e.g. $17 \times \frac{3}{5} \times \frac{5}{3} = 17$. So the reciprocal does for multiplication what negative numbers do for addition (because e.g. $17 + 2.4 \space – \space 2.4 = 17$ so, again, right back where you started).
2) To divide by a fraction you multiply by its reciprocal
e.g. $\frac{2}{7} \div \frac{3}{7} = \frac{2}{\cancel{7}} \times \frac{\cancel{7}}{3} = \frac{2}{3}$
3) Indices: to the power of minus one means reciprocal
e.g. $6^{-1} = \frac{1}{6}$
There is even a reciprocal button on the calculator: labelled $x^{-1}$
4) If a is inversely proportional to b then a is proportional to the reciprocal of b. NOTE: this means that if you double one quantity you halve the other, such as speed and time over a fixed journey (travelling twice as fast halves the journey time).
5) In coordinate geometry, you can tell if two lines are perpendicular because their gradients will be negative reciprocals e.g. $y=\frac{5}{7}x+c$ is perpendicular to $y=\frac{7}{5}x+c$
AT A-LEVEL:
6) In transformations of curves,the graph of $y=f(kx)$ is the same as the graph of $y=f(x)$ but stretched in the x-direction by a scale factor not of k but of $\large{\frac{1}{k}}$ so the reciprocal of k. This means that $y=sin(2x)$ looks just like the graph of $y=sin(x)$ but “squished” in the horizontal direction to make it twice as thin (or half as wide).
As you can see, the reciprocal is very useful in maths so it’s well worth having a word for it!