Download 4.2 Irrational Numbers Repeating Decimals

4.2 Irrational Numbers Repeating Decimals ‐ decimals which have a recurring pattern of digits Non‐Repeating Decimals ‐ decimal which have no recurring pattern of digits Terminating Decimals ‐ decimals with a finite number of digits Non‐Terminating Decimals ‐ decimals with an infinite number of digits eg. 0.666 … or 0. 6 eg. √2
eg. 0.75 or 1.4121 eg. 0.87641543... Determine Repeating/Non‐Repeating and Terminating/Non‐Terminating Decimal Example 1. State whether the decimal equivalent of each number is repeating or non‐repeating. Also state whether it is terminating or non‐terminating. a) √0.16 b) 0.94 c) √2 Repeating or Non‐Repeating Repeating or Non‐Repeating Repeating or Non‐Repeating Terminating or Non‐Terminating Terminating or Non‐Terminating Terminating or Non‐Terminating Try. State whether the decimal equivalent of each number is repeating or non‐repeating. Also state whether it is terminating or non‐terminating. b) √8 c) Repeating or Non‐Repeating Repeating or Non‐Repeating Repeating or Non‐Repeating Terminating or Non‐Terminating Terminating or Non‐Terminating Terminating or Non‐Terminating Rational and Irrational Numbers Decimal numbers which repeat or terminate can be converted into fractions and are called rational numbers, since they can be written as the ratio of two integers. Decimal number which are both non‐repeating and non‐terminating cannot be converted into fractions and are called irrational numbers. Identify Rational and Irrational Numbers Example 2. Identify each of the following numbers as rational or irrational. a) 1.493 b) √5 c) Try. Identify each of the following numbers as rational or irrational. a) √81 b) 0.234 c) – 8.11221112221111… 6 Natural Numbers: = { 1, 2, 3, 4, ... } Whole Numbers: W = { 0, 1, 2, 3, 4, ... } Integers: = { ..., –3, –2, –1, 0, 1, 2, 3, 4, ... } Rational Numbers: ℚ
where m and n are integers and n ≠ 0 . Irrational Numbers: ℚ = {non‐repeating and non‐terminating} The set of all rational numbers and set of all irrational numbers, when combined, form the set of real numbers. ℚ
ℚ Interrelationship Between the Sets of Numbers Example 3. Complete the Venn Diagram to show the interrelationship between the sets of numbers in the real number system. e π √2 √20 √0.16 – 3 – 2 – 1 0 1 2 3 Classify Numbers Example 4. For each of the following write all the sets of numbers to which the given number belongs. Write the answers from the largest set to the smallest set. a) −2 b) −2.1345218... c) 3.14159265 d) π e) f) √7 g) √36 h) 0 Try. For each of the following write all the sets of numbers to which the given number belongs. Write the answers from the largest set to the smallest set. a) 9 b) c) √5 d) – 7 7 Try. Determine whether each statement is true (T) or false (F). a) ___ All natural number are integers. b) ___ Real numbers consist of rational numbers and irrational number. c) ___ The set of integers is nested within the set of rational numbers. d) ___ All integers are rational numbers. e) ___ All irrational numbers are real. f) ___ The set is nested within the set . g) ___ The set ℚ is nested within the set W . h) ___ There is only one number in set W which is not also in set . When an irrational number is written as a radical, the radical is the exact value of the irrational number; for example, √2 and √ 5. We can use the square root and cube root keys on a calculator to determine approximate values of these irrational numbers. √2 = 1.414213562 √ 5 = – 1.709975947 Order Irrational Numbers on a Number Line Example 5. Use number line to order these numbers from least to greatest. √13 √18 √11 √27 √ 5 Try. Use number line to order these numbers from least to greatest. √2 √ 2 √6 √11 √30 8