Exercise 55 Page 21

Multiplicative inverses are pairs of numbers whose product is $1.$ We can express this in symbols using integers $a$ and $b.$ Why does $0$ have no multiplicative inverse? There are two methods of thinking the answer to this question.

1. Multiplication by $0$ is always a $0$ product.
2. We cannot divide by $0.$

Multiplication by $0$ Is Always $0$

To explain this, let's first write $0$ as a fraction, $\frac{a}{b}.$ Then, according to the Multiplicative Inverse Property, we can substitute this into the algebraic definition of multiplicative inverses. We can tell that the number $\frac{b}{a}$ does not exist because any number multiplied by $0$ always gives a result of $0.$ Therefore, there is no fraction $\frac{b}{a}$ such that $0$ times $\frac{b}{a}$ is equal to $1.$

Dividing by $0$ Is Not Possible

Let's look at a specific example. We will take $a=0$ and $b=4,$ then write $0$ as a fraction, $\frac{0}{4}.$ According to the Multiplicative Inverse Property, there is a number $\frac{4}{0}$ such that the product of $\frac{0}{4}$ and $\frac{4}{0}$ is equal to $1.$ However, the number $\frac{4}{0}$ does not exist, since division by $0$ is not allowed! Therefore, $0$ has no multiplicative inverse.