6.3 Multiplication and Division with Rational Numbers

Multiplication of Rational Numbers

As we did with whole numbers and integers, multiplication is treated as repeated addition, as long as it makes sense to do so. For example, in the case of $7\cdot \frac{1}{2}$, we can easily just add $7$ copies of $\frac{1}{2}$

$$\HUGE \unicode{x25D6} \unicode{x25D6} \unicode{x25D6} \unicode{x25D6} \unicode{x25D6} \unicode{x25D6} \unicode{x25D6}$$

$$\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$$

When using a physical model, like our area model with these half circles, we can just regroup the pices and see how many are left over.

$$\HUGE \unicode{x25D6}\unicode{x25D7}\unicode{x25D6}\unicode{x25D7}\unicode{x25D6}\unicode{x25D7} \unicode{x25D6}$$ $$\frac{1+1+1+1+1+1+1}{2}=\frac{7}{2}$$

Then we can exchange our pieces for units by completing the circles.

$$\HUGE \unicode{x25CF} \unicode{x25CF} \unicode{x25CF} \unicode{x25D6}\normalsize = 3\frac{1}{2}$$

Where our model starts needs a new interpretation. Lets try to multiply something like $\frac{3}{5}\cdot \frac{1}{2}$. One way to interpret this is to take a unit and divide it into $\frac{3}{5}$ and then divide that same unit into $\frac{1}{2}$, but in a perpendicular direction. This visualizes us taking exactly $\frac{3}{5}\cdot \frac{1}{2}$.

Each of those pieces of $5$ are cut into $2$, so we have $5\cdot 2=10$ total pieces. And of these new pieces, we have $3$ rows of them from $\frac{3}{5}$ and we have $1$ column of them from the $\frac{1}{2}$, or $3\cdot 1 = 3$. Which is to say we have $3$ of the pieces of $10$, or $\frac{3}{10}$.

$$\frac{3}{5}\cdot \frac{1}{2}=\frac{3\cdot 1}{5\cdot 2}$$

This is why we say that to multiply fractions together, we "multiply straight across". It happens to work out that way, according to our area model.

Try interpreting this area model for multiplication.

MyOpenMath: Multiplying Area Model

MyOpenMath: Multiplying Rational Numbers

Properties of Multiplication with Rational Numbers

Rational multiplication inherits all of the properties of integer multiplication, but with one notable addition. All rational numbers (except $0$) have a multiplicative inverse. Recall that the additive inverse said that for all rational numbers, there is another such that

$$\frac{a}{b}+\frac{-a}{b}=0$$

The multiplicative inverse says that for all non-zero rational numbers there is another that you can multiply it with to get $1$.

This follows directly from our commutative property of integers and the fundamental law of fractions.

$$\frac ab\cdot\frac ba = \frac{ab}{ba}=\frac{ab}{ab}=\frac{a}{a}\cdot \frac{b}{b}=1$$

MyOpenMath: Multiplication Inverse vs Identity

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Mixed-Number Multiplication

When multiplying by mixed numbers, we can brute force the answer by using definitions of mixed numbers and our distributive property of multiplication. For example,

$$3\frac13\cdot2\frac57=\left(3+\frac{1}{3}\right)\left(2+\frac{5}{7}\right)=3\cdot 2 + 3\cdot \frac{5}{7} + \frac{1}{3}\cdot 2 + \frac{1}{3}\cdot \frac{5}{7}$$

Theres nothing wrong with this method, but it can be a bit tedious and allows many opportunities for mistakes. A better method would be to convert all of the mixed numbers to improper fractions first.

MyOpenMath: Multiplying mixed numbers

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Division of Rational Numbers

We used repeated addition to define multiplication. We can use repeated subtraction to define division. In terms of integers, $18\div 3=6$ because $6$ is the number of times we can subtract $3$ from $18$. $$18-3-3-3-3-3-3=0$$ We can use the same idea with rational numbers. For example, how could we interperet $3\div \frac{1}{2}$?

Where the repeated addition model starts to struggle is when our subtraction won't end with us being at exactly $0$. Back with integers, this means that we would have a remainder which is a piece of a subtraction. But with rational numbers, we have the models we need to take pieces of things.

Consider an area model, and what it might mean to evaluate an expression like $\frac{2}{3}\div \frac{1}{2}$.

We first have to make make each fraction have the same denominator so we can count accurately. Above is a representation of $\frac{2}{3}$ in the red and $\frac{2}{2}$ in green, both divided into $\frac{1}{6}$. Then we take the area for $\frac{2}{3}=\frac{4}{6}$ and line it up for comparison with the area for $\frac{1}{2}=\frac{3}{6}$.

So we have $4$ and we're removing $3$ of these new units. It is clear we can remove $1$ full group of $3$, but there is one left over. And since we were removing in groups of $3$, what is left is $\frac{1}{3}$. The model implies $$\frac{2}{3}\div \frac{1}{2}=1+\frac{1}{3}=\frac{4}{3}.$$

Getting this common denominator and using repeated subtraction to calculate divison and be simplifed greatly with an observation using unit fractions. Dividing something by $2$ is the same as multiplying it by $\frac{1}{2}$. In general, dividing something by $a$ is the same as multiplying it by the multiplicative inverse of $a$.

MyOpenMath: Division of rational numbers