Unit 3.1 Notes - 9A.pdf

Unit 3 – Rational Numbers Grade 9 Mathematics Section 3.1 – What is a Rational Number? Number sets are groups of numbers that were constructed by mathematicians in order to define and classify them. To better understand rational numbers,...

Unit 3 – Rational Numbers Grade 9 Mathematics Section 3.1 – What is a Rational Number? Number sets are groups of numbers that were constructed by mathematicians in order to define and classify them. To better understand rational numbers, we should first consider all the sets of numbers that make up the set of real numbers. Natural Numbers (ℕ): These are the counting numbers. {1, 2, 3, 4, …} Whole Numbers (𝕎): This set consists of the natural numbers plus zero. {0, 1, 2, 3, 4, …} Integers (𝕀): This set consists of positive and negative whole numbers. {…, -3, -2, -1, 0, 1, 2, 3, …} Rational Numbers (ℚ): Rational numbers are any numbers that can be written as a 𝑎 fraction, , where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0. 𝑏 1 Ex: 2, -4, 0.5, 3 Irrational Numbers ̅ ): Irrational numbers are numbers that cannot be written as a (ℚ fraction. These are non-terminating, non-repeating decimals. Ex: 𝜋 , √2 , √𝑥 Real Numbers (ℝ): This set includes both rational and irrational numbers. L. Clemens - Brenton Page |1 Unit 3 – Rational Numbers Grade 9 Mathematics As you can see, rational numbers include all integers, fractions, terminating decimals and repeating decimals. We can use a Venn diagram to show the relationship between the number sets. Real ℝ Rational ℚ Integers 𝕀 Whole 𝕎 Natural Irrational ℕ ̅ ℚ Example 1: Using any two integers, create 3 different fractions. Change each fraction to a decimal. Example 2: Which of the following numbers are rational? Justify your answer. −1 −4 √9 √75 𝜋 0.25 0.25786432 … 4 −9 L. Clemens - Brenton Page |2 Unit 3 – Rational Numbers Grade 9 Mathematics In previous grades, we learned a variety of strategies for comparing and ordering fractions and decimals. When working with decimals, we compare place value. Example: Which number is larger, 9.25 or 9.7? When working with fractions, we can: Use common denominators. If both fractions have the same denominator, the larger numerator represents the larger fraction. 3 2 Example: Which fraction is larger, or ? 8 5 Use common numerators. If both fractions have the same positive numerator, the fraction with the smallest denominator is larger. 1 2 Example: Which fraction is larger, or ? 4 7 Place the fractions on a number line with benchmarks. Example: Place these rational numbers in descending order. −3 7 3 , 0.5 , −1.8 , −5 , , 2 , −3. 3̅ , 1 4 4 3 L. Clemens - Brenton Page |3 Unit 3 – Rational Numbers Grade 9 Mathematics Example 3: Use > , < or = to determine which rational number is greater, where possible. 4 5 −3 −5 a. b. 7 9 8 8 2 2 −2 −2 c. d. 7 9 7 9 −3 3 −10 e. f. −2.8 4 4 4 The position of the negative sign does not affect the answer. −1 1 1 = = − 2 −2 2 L. Clemens - Brenton Page |4 Unit 3 – Rational Numbers Grade 9 Mathematics Example 4: Identify three decimal numbers between each pair of rational numbers. −1 −1 a. and 2 4 b. −0.25 and −0.26 Example 5: Identify a fraction between each pair of rational numbers. −2 −3 a. and 3 4 5 7 b. and 2 3 L. Clemens - Brenton Page |5