Opposites and Reciprocals

Opposites

Two numbers are opposites of each other if they have the same absolute values, but not the same sign. The exception to this rule is that 0 is it's own opposite. Another way of thinking of this is that opposites are pairs of numbers that are both the same distance from 0 on the number line, but in opposite directions. Thus, pairs of opposites could be $3,-3\:or\:-5,5\:or\:17,-17.$ What is special about opposites is that when you add two numbers together that are opposites, you will ALWAYS get 0. For this reason opposites are sometimes called "Additive Inverses".

Reciprocals

The reciprocal of a number is just 1 divided by that number. So, the reciprocal of $7$ is $\frac{1}{7}$, and the reciprocal of $-5$ is $-\frac{1}{5}$. If the number you are taking the reciprocal of is a fraction, then you need to remember that taking 1 over a fraction is the same as exchanging the numerator and denominator. Thus, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, and the reciprocal of $-\frac{1}{9}$ would be $-9$. Similar to with opposites, except for with 0 (which has no reciprocal because it is not defined when you divide by 0), reciprocals always come in pairs. For example the reciprocal of $-\frac{7}{4}$ is $-\frac{4}{7}$, but the reciprocal of $-\frac{4}{7}$ would take us back to the number $-\frac{7}{4}$.

Reciprocals are special because when you multiply reciprocal pairs together, you ALWAYS get 1.  For example, $\left(-\frac{7}{4}\right)\left(-\frac{4}{7}\right)=1$. For this reason, reciprocals are sometimes called "Multiplicative Inverses".

You are now ready to complete Homework 1.2 on MyMathLab on the next page.