Classifying real numbers as rational and irrational in different situations

Subsection 1.1.2 Classifying real numbers as rational and irrational in different situations

Activity 1.1.2.

Working in groups, choose any set of natural numbers between \(1\) and \(10\) (e.g., \(2, 3, 5,7,..\))
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\(\bullet\) Use these numbers to create at least two fractions:
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One proper fraction (e.g., \(3/5\))
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One improper fraction (e.g., \(7/4\))
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Use a calculator to divide each of your fractions.
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Write down the decimal value of each.
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Example: \(7 ÷ 4 = 1.75\text{.}\)
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Choose any set of Natural numbers between \(1\) and \(20\text{.}\)
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Write each number as a square root.
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Example: \(\sqrt{9}= 3\)(a rational number),\(\sqrt{2} = 1.414...\)(an irrational number).
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Classify each number you have created (fractions, decimals, and square roots) as either rational or irrational.
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What do you notice about the decimal form of rational numbers compared to irrational numbers?
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Are there any patterns?
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Discuss your observations with your fellow learners.
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\(\textbf{ Key Takeaway}\)

\(\bullet\) \(\textbf{ Rational number} (\mathbb{Q}):\) A rational number is any number that can be written as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\text{.}\)

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Example: \(\frac{2}{3},-3,4 \)

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\(\bullet\) \(\textbf{ Irrational number:}\) An irrational number is any number that cannot be expressed as a fraction of two intergers.

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Example: \(\sqrt{7}, \sqrt{2}, \pi\)

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\(\bullet\) \(\textbf{Integers}\) consists of positive whole numbers, negative whole numbers and \(0\text{.}\)

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\(\bullet\) The decimal representation of a rational number either terminates (stops at some point) or repeats (continues but has a repeating pattern).

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Example: \(0.375, 3.45454545...\)

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\(\bullet\) The decimal representation of an irrational number neither terminates (does not stop) nor repeats (continues without a repeated pattern).

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Example: \(3.14285714..., 4.298103993...\)

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\(\bullet\) The square root of a perfect square is a rational number.

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Example: \(\sqrt{16} = 4\) and \(4\) can be written as \(\frac{4}{1}\)

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\(\bullet\) The square root of an imperfect square is an irrational number.

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Example: \(\sqrt{2} = 1.41421356237\) which is an irrational number.

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\(\textbf{ How to determine if a number is rational or irrational.}\)

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Rule: Check if the number is an integer or a fraction with an integer as the numerator and the denominator. If it is, then it is a rational number.
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Example: \(7\) or \(\frac{4}{5}\) are both rational numbers because they are either whole numbers or simple fractions.
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Rule: "If the number is in decimal form, check if the decimal stops at some point. If it stops then the number is a rational number."
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Example: \(3.25\) is rational because the decimal terminates.
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Rule: If the number is in decimal form, check if the decimal continues. If it continues with a repeated pattern then the number is a rational number and if it continues without a pattern then it is irrational.
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Example (rational): \(0.666… \)(repeating)
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xample (irrational): \(0.1010010001… \)(no pattern)
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Rule: "If the number is expressed as a square root, find the square root of the number first and identify if it is a perfect or an imperfect square. If it is a perfect square (results to a whole number) then it is rational and if it is an imperfect square, then it is irrational.
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Example (rational): \(\sqrt{49} = 7\)
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xample (irrational): \(\sqrt{2} ≈ 1.414213...\)
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Activity 1.1.3.

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Working in groups, write any \(5\) numbers between \(1\) and \(10\) (e.g., \(2, 3, 5,7,..\))
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Form at least \(3\) fractions using the numbers you formed above. Form both proper and improper fractions e.g.,\(\frac{2}{5},\frac{7}{3}\text{.}\)
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Divide each of the fractions you formed to express it as a decimal. Example: \(2 \div 5= 0.4, 7 \div 3 =2.\dot{3}\text{.}\)
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Express the numbers \(1\) to \(10\) as square roots. Example: \(\sqrt{1},\sqrt{2},...,\sqrt{10}\text{.}\)
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Classify the fractions, decimals, and square root numbers you have formed as rational and irrational numbers.
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What do you notice about the decimal forms of fractions?
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Discuss your work with fellow learners.
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Example 1.1.7.

Identify if the following numbers are rational or irrational.

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\(\displaystyle \pi\)
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\(\displaystyle \frac{2}{3}\)
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\(\displaystyle 3.75\)
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\(\displaystyle \sqrt{20}\)
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\(\displaystyle \frac{\sqrt{9}}{\sqrt{16}}\)
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Solution.

Since \(\pi\) is defined as \(3.1415926\text{.}\)
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We check if the decimal continues.
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Hence \(\pi\) is irrational because its decimal continues without having a repeated pattern.
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Since \(\frac{2}{3}\) is a fraction, we check if the fraction consist of integers with the denominator not equal to zero.
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Therefore \(\frac{2}{3}\) is a rational number.
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To identify if \(3.75\) is rational or irrational, we check the decimal.
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A rational number can have a terminating decimal or a repeating decimal. In this case, the decimal terminates, so the number is rational
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For \(\sqrt{20}\text{,}\) first we find its value.
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\(\sqrt{20} = 4.472135955\)
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\(\sqrt{20}\) has decimal which continues without a repeated pattern.
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Hence \(\sqrt{20}\) is irrational.
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The value of \(\frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}\)
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\(\frac{3}{4}\) is a fraction with integers on the numerator and the denominator.
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Therefore, \(\frac{\sqrt{9}}{\sqrt{16}}\) is rational.
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Example 1.1.8.

Joy is designing a square garden. She measures the total area of the garden to be \(50\) square meters and wants to find the length of one side. What is the exact length of one side of the garden? Classify the answer as a rational or irrational number.

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Solution.

Area of the garden = \(50 \text{ m}^2\)

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To find the length of one side of the square, we take the square root of its area.

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Side length = \(\sqrt{50}\text{.}\)

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Simplifying the side length \(\sqrt{50} = \sqrt{(25 \times 5)} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}\text{.}\)

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Since \(\sqrt{2}\) an irrational number, multiplying it by \(5\) still gives an irrational number.

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Therefore, the exact length of one side of the garden is \(5 \sqrt{2}\) meters, which is an irrational number.

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Checkpoint 1.1.9. Classifying Given Numbers as Irrational.

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Checkpoint 1.1.10. Classifying Rational Numbers from a List.

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\(\textbf{Exercise}\)

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1. Classify the following numbers as rational or irrational giving reasons.

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\(\displaystyle \sqrt{25}\)
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\(\displaystyle \pi\)
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\(\displaystyle \sqrt{2}\)
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\(\displaystyle \frac{7}{3}\)
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\(\displaystyle 0.25\)
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\(\displaystyle 0.121221222\)
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\(\displaystyle - 4.5\)
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\(\displaystyle \sqrt{2} + \sqrt{8}\)
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\(\displaystyle \pi - 3\)
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\(\displaystyle \frac{\sqrt{4}}{\sqrt{9}} \times 4\)
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\(\displaystyle 2 \times \sqrt{2}\)
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2. A square garden has a perimeter of \(8\) units. Find its area and identify if it is a rational or irrational number.

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3. For each of the following values of \(m\) state whether \(\frac{m}{16}\) is rational or irrational.

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\(\displaystyle 1\)
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\(\displaystyle -10\)
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\(\displaystyle \sqrt{2}\)
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\(\displaystyle \sqrt{25}\)
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\(\displaystyle \pi\)
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4. A car is moving at \(\sqrt{225}\) km/h. Is the speed rational or irrational? Explain your answer.

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5. Iregi a grade \(10\) student measures a triangular shelf in their home and found out its sides of length was \(\sqrt{12}\) meters, \(\sqrt{27}\) meters and \(5\) meters. He wants to find the perimeter of the triangle and identify if it is rational or irrational. Help Iregi to find out if the perimeter is rational or irrational explaining your workings.

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6. A rectangular garden has a length of \(4\) meters and a width of \(\sqrt{8}\) meters. Find the area of the garden and identify if it is rational or irrational.

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