Sums of rational and irrational numbers

Task

1. Explain why the sum and product of two rational numbers is always a rational number.

2. Kaylee says

   I know that $\pi$ is an irrational number so its decimal never repeats. I also know that $\frac{1}{7}$ is a rational number so its decimal repeats. But I don't know how to add or multiply these decimals so I am not sure if $\pi + \frac{1}{7}$ and $ \pi \times \frac{1}{7}$ are rational or irrational.

    

   Using part (a), help Kaylee decide whether or not $\pi + \frac{1}{7}$ and $\pi \times \frac{1}{7}$ are rational or irrational.

IM Commentary

The goal of this task is to examine sums and products of rational and irrational numbers. One important property of rational numbers is that their decimals always terminate or repeat: using a slightly different formulation, a rational number is a number whose decimal eventually repeats. It follows that irrational numbers are those numbers whose decimals never repeat. It turns out that this characterization of rational and irrational numbers does not lend itself well to understanding what happens when numbers are added and multiplied. The definition of rational and irrational numbers, in terms of fractions, resolves these questions nicely.

One important issue surrounding the idea of adding rational and irrational numbers is what happens when an irrational number is added to another irrational number. This issue is not addressed here but is treated in http://www.illustrativemathematics.org/illustrations/690. These two tasks are complementary and the goal of this task is to provide a detailed explanation for some of the patterns examined there.

This task provides an example of a common phenomenon in mathematics: equivalent statements (''$x$ is a rational number'' and ''$x$ has an eventually repeating decimal'' in this case) can vary greatly in terms of helping to solve a specific problem. This is one reason why it is important to be flexible and employ multiple strategies: in this case, how we think of ''rational number'' is a key to solving the problem. Kaylee is correct that it is difficult to see, using decimal expansions, what happens when you multiply a rational number by an irrational number and similarly for adding (in both cases because of carrying involved when the arithmetic is performed). Using the definition of a rational number, however, allows for a quick solution to this problem.

Solution

1. Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are two fractions. This means that $b$ and $d$ are non-zero integers and $a$ and $c$ are integers. We can calculate the sum by finding a common denominator: $$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}. $$ Since $b$ and $d$ are not zero, the denominator $bd$ is not zero. Moreover, sums and products of integers are integers so both the numerator $ad + bc$ and the denominator $bd$ are integers. This means that the sum $\frac{a}{b} + \frac{c}{d}$ is a rational number.

   The same argument works for the product and is quicker: $$ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}. $$ Since $b$ and $d$ are not zero, the denominator $bd$ is not zero. Products of integers are integers so the numerator $ac$ and the denominator $bd$ are both integers and the product $\frac{a}{b} \times \frac{c}{d}$ is a rational number.

2. The number $x = \pi + \frac{1}{7}$ is either rational or irrational. If $x$ were rational then from part (a) $x - \frac{1}{7}$ would also be rational. But $x - \frac{1}{7} = \pi$ is not rational. Since $x$ is not rational it must be irrational.

   A similar argument shows that $x = \pi \times \frac{1}{7}$ is irrational. Again, $x$ is either rational or irrational. If $x$ were rational then $7x$ would also be rational but $7x = \pi$ is irrational. So $x$ must be irrational.