Identifying Rational Numbers

Task

Decide whether each of the following numbers is rational or irrational. If it is rational, explain how you know.

1. $0.33\overline{3}$
2. $\sqrt{4}$
3. $\sqrt{2}= 1.414213…$
4. $1.414213$
5. $\pi = 3.141592…$
6. $11$
7. $\frac17=0.\overline{142857}$
8. $12.34565656\overline{56}$

IM Commentary

The task assumes that students are able to express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "8.NS Converting Decimal Representations of Rational Numbers to Fraction Representations ."

There is a lot of interesting mathematics behind deciding questions about irrationality. There are a variety of arguments demonstrating that $\sqrt{2}$ is irrational (some of which would be quite accessible to a motivated middle school student), the first of which were discovered somewhere around the 5th century BC. And yet the irrationality of $\pi$ was not proven until 1761, over two millenia later! Students who complete the task will probably be very close to being able to articulate the statement that a number is rational if and only if its decimal expansion is eventually periodic, in which case they could be posed problems like showing that the number

$$0.123456789101112131415161718192021....$$ is irrational. Note that even understanding the statement that $\sqrt{2}$ equals $1.414213…$ is non-trivial, and partly addresses the part of the standard that says “Understand informally that every number has a decimal expansion."

Solution

1. Since $$0.33\overline{3} = \frac13$$ $0.33\overline{3}$ is a rational number.
2. Since $$\sqrt{4} = 2 =\frac21$$ $\sqrt{4}$ is a rational number.
3. $\sqrt{2} = 1.414213…$ is not rational. In eighth grade most students know that the square root of a prime number is irrational as a "fact," but few 8th grade students will be able to prove it. There are arguments that 8th graders can understand if they are interested.

4. Since

   $$1.414213 = \frac{1414213}{100000},$$ $1.414213$ is a rational number.

5. $\pi = 3.141592…$ is not rational. In eighth grade most students know that $\pi$ is irrational as a "fact." The proof of this is quite sophisticated.

6. Since

   $$11=\frac{11}{1}$$ 11 it is rational.

7. $\frac17=0.\overline{142857}$ is already written in a way that makes it clear it is a rational number, although some students might say it is irrational, possibly because the repeating part of the decimal is longer than many familiar repeating decimals (like $\frac13$).

8. We have

   $$12.34565656\overline{56} = 12.34+.00\overline{56}=\frac{1234}{100}+\frac{56}{9900}=\frac{1234\cdot 99+56}{9900}=\frac{122222}{9900},$$ which is certainly rational.