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The Real Number System Real Numbers • Real numbers consist of all the rational and irrational numbers. • The real number system has many subsets: – Natural Numbers – Whole Numbers – Integers Natural Numbers • Natural numbers are the set of counting numbers. {1, 2, 3,…} Whole Numbers • Whole numbers are the set of numbers that include 0 plus the set of natural numbers. 0, 1, 2, 3, 4, 5,…} { Integers • Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…} Rational Numbers • Rational numbers are any numbers that can be expressed in the form of a , where a and b are integers, and b ≠ 0. b • They can always be expressed by using terminating decimals or repeating decimals. Terminating Decimals • Terminating decimals are decimals that contain a finite number of digits. • Examples: 36.8 0.125 4.5 Repeating Decimals • Repeating decimals are decimals that contain a infinite number of digits. • Examples: 0.333… 1.9 7.689689… FYI…The line above the decimals indicate that number repeats. Irrational Numbers • Irrational numbers are any numbers that cannot be expressed a as . b • They are expressed as non-terminating, nonrepeating decimals; decimals that go on forever without repeating a pattern. • Examples of irrational numbers: – 0.34334333433334… – 45.86745893… – (pi) – 2 MGSE9–12.N.RN.3 Explain why the sum or product of rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational. the sum and product of any two rational numbers is rational True or False Experiment with sum and product of two numbers from the following: 5, ½, -1/3, -7, 10, 5/4 the sum of a rational number and an irrational number is irrational True or False • Experiment with sums and products of two numbers from the following: Rational: 5, ½, -1/3, -7 Irrational: 2 5 the product of a nonzero rational number and an irrational number is irrational True or False Experiment with sums and products of two numbers from the following: Rational: 5, ½, -1/3, -7 Irrational: 2 5