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2.7: Square Roots and Real Numbers

Difficulty Level: At Grade Created by: CK-12
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Human chess is a variation of chess, often played at Renaissance fairs, in which people take on the roles of the various pieces on a chessboard. The chessboard is played on a square plot of land that measures 324 square meters with the chess squares marked on the grass. How long is each side of the chessboard?

To answer this question, you will need to know how to find the square root of a number.

The square root of a number \begin{align*}n\end{align*} is any number such that \begin{align*}s^2 = n\end{align*}.

Every positive number has two square roots, the positive and the negative. The symbol used to represent the square root is \begin{align*}\sqrt{x}\end{align*}. It is assumed that this is the positive square root of \begin{align*}x\end{align*}. To show both the positive and negative values, you can use the symbol \begin{align*}\pm\end{align*}, read “plus or minus.”

For example:

\begin{align*}\sqrt{81}=9\end{align*} means the positive square root of 81.

\begin{align*}-\sqrt{81}= -9\end{align*} means the negative square root of 81.

\begin{align*}\pm \sqrt{81} = \pm 9\end{align*} means the positive or negative square root of 81.

Example 1: The human chessboard measures 324 square meters. How long is one side of the square?

Solution: The area of a square is \begin{align*}s^2 = Area\end{align*}. The value of Area can be replaced with 324.


The value of \begin{align*}s\end{align*} represents the square root of 324.

\begin{align*}s= \sqrt{324}=18\end{align*}

The chessboard is 18 meters long by 18 meters wide.

Approximating Square Roots

When the square root of a number is a whole number, this number is called a perfect square. 9 is a perfect square because \begin{align*}\sqrt{9}=3\end{align*}.

Not all square roots are whole numbers. Many square roots are irrational numbers, meaning there is no rational number equivalent. For example, 2 is the square root of 4 because \begin{align*}2 \times 2 = 4\end{align*}. The number 7 is the square root of 49 because \begin{align*}7 \times 7 = 49\end{align*}. What is the square root of 5?

There is no whole number multiplied by itself to equal five, so the \begin{align*}\sqrt{5}\end{align*} is not a whole number. To find the value of \begin{align*}\sqrt{5}\end{align*}, we can use estimation.

To estimate the square root of a number, look for the perfect integers less than and greater than the value, then estimate the decimal.

Example 2: Estimate \begin{align*}\sqrt{5}\end{align*}.

Solution: The perfect square below 5 is 4 and the perfect square above 5 is 9. Therefore, \begin{align*}4<5<9\end{align*}. Therefore, \begin{align*}\sqrt{5}\end{align*} is between \begin{align*}\sqrt{4}\end{align*} and \begin{align*}\sqrt{9}\end{align*}, or \begin{align*}2< \sqrt{5}<3\end{align*}. Because 5 is closer to 4 than 9, the decimal is a low value. \begin{align*}\sqrt{5} \approx 2.2\end{align*}

Identifying Irrational Numbers

Recall the number hierarchy from Lesson 2.1. Real numbers have two categories: rational and irrational. If a value is not a perfect square, then it is considered an irrational number. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating). Although irrational square roots cannot be written as fractions, we can still write them exactly, without typing the value into a calculator.

For example, suppose you do not have a calculator and you need to find \begin{align*}\sqrt{18}\end{align*}. You know there is no whole number squared that equals 18, so \begin{align*}\sqrt{18}\end{align*} is an irrational number. The value is between \begin{align*}\sqrt{16}=4\end{align*} and \begin{align*}\sqrt{25}=5\end{align*}. However, we need to find an exact value of \begin{align*}\sqrt{18}\end{align*}.

Begin by writing the prime factorization of \begin{align*}\sqrt{18}\end{align*}. \begin{align*}\sqrt{18} = \sqrt{9 \times 2}= \sqrt{9} \times \sqrt{2}\end{align*}. The \begin{align*}\sqrt{9} = 3\end{align*} but \begin{align*}\sqrt{2}\end{align*} does not have a whole number value. Therefore, the exact value of \begin{align*}\sqrt{18} = 3 \sqrt{2}\end{align*}.

You can check your answer in the calculator by finding the decimal approximation for each square root.

Example 3: Find the exact value \begin{align*}\sqrt{75}\end{align*}.


\begin{align*}\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3}\end{align*}

Classifying Real Numbers

Example 4: Using the chart found in Lesson 2.1, categorize the following numbers:

a) 0

b) –1

c) \begin{align*}\frac{\pi}{3}\end{align*}

d) \begin{align*}\frac{\sqrt{36}}{9}\end{align*}


a) Zero is a whole number, an integer, a rational number, and a real number.

b) –1 is an integer, a rational number, and a real number.

c) \begin{align*}\frac{\pi}{3}\end{align*} is an irrational number and a real number.

d) \begin{align*}\frac{\sqrt{36}}{9} = \frac{6}{9} = \frac{2}{3}\end{align*}. This is a rational number and a real number.

Graphing and Ordering Real Numbers

Every real number can be positioned between two integers. Many times you will need to organize real numbers to determine the least value, greatest value, or both. This is usually done on a number line.

Example 5: Plot the following rational numbers on the number line.

a) \begin{align*}\frac{2}{3}\end{align*}

b) \begin{align*}-\frac{3}{7}\end{align*}

c) \begin{align*}\frac{\pi}{2}\end{align*}

d) \begin{align*}\frac{57}{16}\end{align*}


a) \begin{align*}\frac{2}{3} = 0.\overline{6}\end{align*}, which is between 0 and 1.

b) \begin{align*}-\frac{3}{7}\end{align*} is between –1 and 0.

c) \begin{align*}\frac{\pi}{2} \approx \frac{3.14159}{2} \approx 1.571\end{align*}

d) \begin{align*}\frac{57}{16} = 3.5625\end{align*}

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Square Roots and Real Numbers (10:18)

Find the following square roots exactly without using a calculator. Give your answer in the simplest form.

  1. \begin{align*}\sqrt{25}\end{align*}
  2. \begin{align*}\sqrt{24}\end{align*}
  3. \begin{align*}\sqrt{20}\end{align*}
  4. \begin{align*}\sqrt{200}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{\frac{1}{4}}\end{align*}
  7. \begin{align*}\sqrt{\frac{9}{4}}\end{align*}
  8. \begin{align*}\sqrt{0.16}\end{align*}
  9. \begin{align*}\sqrt{0.1}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}

Use a calculator to find the following square roots. Round to two decimal places.

  1. \begin{align*}\sqrt{13}\end{align*}
  2. \begin{align*}\sqrt{99}\end{align*}
  3. \begin{align*}\sqrt{123}\end{align*}
  4. \begin{align*}\sqrt{2}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{0.25}\end{align*}
  7. \begin{align*}\sqrt{1.35}\end{align*}
  8. \begin{align*}\sqrt{0.37}\end{align*}
  9. \begin{align*}\sqrt{0.7}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}

Classify the following numbers. Include all the categories that apply to the number.

  1. \begin{align*}\sqrt{0.25}\end{align*}
  2. \begin{align*}\sqrt{1.35}\end{align*}
  3. \begin{align*}\sqrt{20}\end{align*}
  4. \begin{align*}\sqrt{25}\end{align*}
  5. \begin{align*}\sqrt{100}\end{align*}
  6. Place the following numbers in numerical order, from lowest to highest. \begin{align*}\frac{\sqrt{6}}{2} && \frac{61}{50} && \sqrt{1.5} && \frac{16}{13}\end{align*}
  7. Use the marked points on the number line and identify each proper fraction.

Mixed Review

  1. Simplify \begin{align*}\frac{9}{4}\div 6\end{align*}.
  2. The area of a triangle is given by the formula \begin{align*}A= \frac{b(h)}{2}\end{align*}, where \begin{align*}b=\end{align*} base of the triangle and \begin{align*}h =\end{align*} height of the triangle. Determine the area of a triangle with base \begin{align*}= 3\end{align*} feet and height \begin{align*}= 7\end{align*} feet.
  3. Reduce the fraction \begin{align*}\frac{144}{6}\end{align*}.
  4. Write a table for the situation: Tracey jumps 60 times per minutes. Let the minutes be \begin{align*}\left \{0,1,2,3,4,5,6\right \}\end{align*}. What is the range of this function?

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