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2.7: Square Roots and Real Numbers

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

At the end of this lesson, students will be able to:

• Find square roots.
• Approximate square roots.
• Identify irrational numbers.
• Classify real numbers.
• Graph and order real numbers.

Vocabulary

Terms introduced in this lesson:

square root
principal square root
positive square root
perfect squares
prime factors
rational number
irrational number
approximation
non-repeating decimals
simplest form
integer
sub-intervals

Teaching Strategies and Tips

Contrary to what students think, there is no discrepancy between there being two possible values for b\begin{align*}b\end{align*}, given a positive x\begin{align*}x\end{align*}, such that b2=x\begin{align*}b^2 = x\end{align*} and only one value for x\begin{align*}\sqrt{x}\end{align*}. The positive number b\begin{align*}b\end{align*} is the principal square root. It is the value of the function.

Point out that ¯¯¯¯¯¯¯\begin{align*}\surd^{\overline{\;\;\;}}\end{align*} and 2\begin{align*}\sqrt[2]{}\end{align*} mean the same thing. The 2\begin{align*}“2”\end{align*} is understood.

In Example 1:

• Use factor trees to break down the radicands into as many perfect squares as possible.
• Primes which appear an even number of times constitute a perfect square.
• Any unpaired factors are left under the radical sign.
• The convention is to leave any irreducible radical in the form: (square root)(irreducible part)\begin{align*}\mathrm{(square\ root)} \sqrt{(\mathrm{irreducible \ part})}\end{align*}

Lots of practice early on makes it easier for students when the radicals become more complicated and involve variable expressions.

Use Example 3 to multiply, divide, and simplify radical expressions. State which rules were used in classroom examples.

In Example 4, students use a calculator and round their answer to three decimal places. Review rounding decimals.

Motivate irrational numbers in Example 5.

• Irrational numbers complete the set of real numbers.
• They cannot be expressed as ratio of two integers.
• They have an unending (non-terminating), seemingly random (non-repeating) decimal expansion.
• Some irrational numbers: π,1π,2,3,any prime\begin{align*}\pi, \frac{1} {\pi}, \sqrt{2}, \sqrt{3}, \sqrt{\mathrm{any \ prime}}\end{align*}

In Example 5, students identify the given numbers. A number is rational if it

• can be expressed as a fraction of integers.
• has a finite decimal expansion.
• has a repeating block of digits in its decimal expansion.

Have students check their calculator displays for repeating blocks of digits, not just patterns. For example, the two numbers below have obvious patterns, but no repeating blocks, and therefore are irrational:

• 0.010010001000010000010000001\begin{align*}0.010010001000010000010000001 \ldots \end{align*}
• 0.12345678910111213141516171819202122\begin{align*}0.12345678910111213141516171819202122 \ldots\end{align*}

Remind students that integers can be written as fractions with a 1\begin{align*}1\end{align*} in the denominator. See Example 6a and 6b. The strategy in Example 6e is to simplify first.

The strategy in Example 8 is to use a calculator to find the decimal expansions of the numbers rounded to as many places as is needed to identify each. Since all the numbers are between 3.1\begin{align*}3.1\end{align*} and 3.2\begin{align*}3.2\end{align*}, going out to three decimal places is sufficient.

Error Troubleshooting

In Problem 3 of the Review Questions:

• Students should find repeating blocks of digits before claiming that a number is rational.
• It is not sufficient to claim that the “unpredictable” decimals of a number on a calculator display belong to an irrational number. The decimal expansion of 2/19\begin{align*}2/19\end{align*} has 18\begin{align*}18\end{align*} seemingly random digits in its repeating block; therefore, the rational number 2/19\begin{align*}2/19\end{align*} would go “unchecked” on an ordinary calculator display because it could not show enough digits.

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