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Properties of Rational Numbers versus Irrational Numbers

Differentiate between numbers that can be written as a fraction and numbers that can't be

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Properties of Rational Numbers versus Irrational Numbers
License: CC BY-NC 3.0

Justin accused his little brother of being irrational. Jason was telling a story about gnomes living in the cherry trees in front of their house and coming inside at night to raid the kitchen for flour and milk so they could make cherry pies to feed to the chickens… Jason went on and on forever. Was Justin correct?

In this concept, you will learn how to recognize rational and irrational numbers.

Recognizing Rational and Irrational Numbers

You may already know that a perfect square is a number whose square root is a whole number. Perfect squares have roots that are rational because whole numbers are rational. Most numbers are not perfect squares and have decimal roots.

Some decimal roots are rational, meaning that they end at a certain place value and/or can be written as fractions. Fractions are rational.


Some roots are negative. Whole or fractional negative numbers are rational.

\begin{align*}\sqrt{81}= \pm 9\end{align*}

Some roots are repeating decimals, as indicated by a line over the repeating digits. Repeating decimals are rational because they can be written as fractions.

\begin{align*}0.6666666 \ldots = 0.6 \bar{6} = \frac{2}{3}\end{align*}

More than one place value can be repeated.

\begin{align*}\frac{21}{55} = 0.38181818 \ldots = 0.3 \overline{18}\end{align*}

An ellipsis, three dots, following a series of digits indicates that the digits continue. Whether or not the value is terminated at some point down the line is unknown, and your calculator is limited to a certain number of places. Unless you can see the repeated digits, assume the number is irrational, and the digits continue forever.

When you use the ellipsis to indicate that the number goes on forever, you should not use the approximately equal to symbol, \begin{align*} \approx\end{align*}. The \begin{align*}\approx\end{align*} indicates that the decimal has been rounded.

For example:

\begin{align*}\sqrt{7} = 2.645751311 \ldots\end{align*} or \begin{align*}\sqrt{7} \approx 2.646\end{align*}

The most famous irrational number pi, or \begin{align*}\pi\end{align*}, is a ratio used in working with circles. \begin{align*}\text{Pi} = 3.141592653589 \ldots\end{align*}

Since pi cannot be expressed as a fraction, and the decimals go on forever, the value used for calculations is usually to the hundredths place.

It’s acceptable to say \begin{align*}\text{pi} = 3.14\end{align*}

Rational and irrational numbers make up the set of real numbers.

Let's look at one more example.

Is the \begin{align*}\sqrt{900}\end{align*} rational or irrational?

First, determine the root. You already know \begin{align*}\sqrt{9}=3\end{align*}, so you don’t even need to use your calculator.


Next, 30 is a whole number, and whole numbers are rational.

The answer is \begin{align*}\sqrt{900}\end{align*} is rational.


Example 1

Earlier, you were given a problem about Justin’s little brother, Jason, who was talking about gnomes making cherry pies.

Jason was not making any sense and going on forever with his story. Justin was correct in saying that Jason was being irrational. But someday, Jason might write a best seller!

Example 2

Is the root rational or irrational?

\begin{align*}\sqrt{50} \end{align*}

First, 50 is not a perfect square, so the square root is not a whole number.

Next, you can estimate the root to be between 7 and 8 because 49 and 64 are perfect squares.

Then, use a calculator.

\begin{align*}\sqrt{50} = 7.071067812 \ldots\end{align*}

You can’t see any repeating decimals, and you can’t tell if the decimals end there or go on forever.

The answer is \begin{align*}\sqrt{50}\end{align*} is irrational.

Example 3

Tell whether the solution is rational or irrational.


First, recognize the root of 8.7 as a decimal. The solution is not a perfect square.

Next, use your calculator.


Then, since the decimals end at the hundredths place, it can be written as a fraction.

\begin{align*}75.69=75 \frac{69}{100}\end{align*}

The answer is 75.69 is a rational number.

Example 4

Determine if the root is rational or irrational.


First, estimate. You know that \begin{align*}\sqrt{4}=2\end{align*}, so \begin{align*}\sqrt{400}=20\end{align*}. Your answer should be a bit more than 20.

Next, use your calculator.

\begin{align*}\sqrt{414} = 20.34698995 \ldots\end{align*}

Then, recognize that the decimal does not repeat.

The answer is \begin{align*}\sqrt{414}\end{align*} is irrational.

Example 5

Is the number -6.9 rational or irrational?

First, the number is negative. Negative numbers can be rational.

Next, the number contains a decimal. Decimals can be rational.

Then, the decimal is terminated at the tenths place.

The answer is that -6.9 is a rational number.


Identify each of the following numbers as rational or irrational.

  1. \begin{align*}.345 \ldots\end{align*}
  2. 2
  3. -9
  4. -122
  5. \begin{align*}3.456 \ldots\end{align*}
  6. \begin{align*}\sqrt{25}\end{align*}
  7. \begin{align*}\sqrt{16}\end{align*}
  8. \begin{align*}\sqrt{12}\end{align*}
  9. \begin{align*}\sqrt{38}\end{align*}
  10. -4.56
  11. \begin{align*}\pi\end{align*}
  12. \begin{align*}-\frac{4}{5}\end{align*}
  13. \begin{align*}9.8712 \ldots\end{align*}
  14. -19
  15. 2,345

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.6.


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approximate solution An approximate solution to a problem is a solution that has been rounded to a limited number of digits.
Irrational Number An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Perfect Square A perfect square is a number whose square root is an integer.
principal square root The principal square root is the positive square root of a number, to distinguish it from the negative value. 3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root.
rational number A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Square Root The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.
Tabular Interpolation Tabular interpolation is using a table to find approximate square roots.

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