Worth 1000 Words Real Numbers and Their Properties Learning Goals In this lesson, you will: Classify numbers in the real number system. Understand the properties of real numbers. Key Terms real number Venn diagram closure The word zero has had a long and interesting history so far. The word comes from the Hindu word sunya, which meant "void" or "emptiness." In Arabic, this word became sifr, which is also where the word cipher comes from. In Latin, it was changed to cephirum, and finally, in Italian it became zevero or zefiro, which was shortened to zero. The ancient Greeks, who were responsible for creating much of modern formal mathematics, did not even believe zero was a number! 5.3 Real Numbers and Their Properties 297
Problem 1 Picturing the Real Numbers Combining the set of rational numbers and the set of irrational numbers produces the set of real numbers. You can use a Venn diagram to represent how the sets within the set of real numbers are related. Look at the rectangles on the next page and follow the steps shown. 1. First, at the top of the large rectangle, write the label Real Numbers. This entire rectangle represents the set of real numbers. 2. Label the smaller rectangle at the right Irrational Numbers. 3. Label the top of the smaller rectangle at the left Rational Numbers. 4. Inside the rectangle that represents rational numbers, draw a large circle. Inside the circle, at its top, write the label Integers. 5. Inside the circle that represents integers, draw a smaller circle. Inside the circle, at its top, write the label Whole Numbers. 6. Inside the circle that represents the whole numbers, draw a smaller circle. Inside this circle, write the label Natural Numbers. Your Venn diagram that represents the real number system is complete. 7. Use your Venn diagram to decide whether each statement is true or false. Explain your reasoning. a. A whole number is sometimes an irrational number. The Venn diagram was introduced in 1881 by John Venn, British philosopher and mathematician. b. A real number is sometimes a rational number. c. A whole number is always an integer. 298 Chapter 5 The Real Number System
5.3 Real Numbers and Their Properties 299
d. A negative integer is always a whole number. e. A rational number is sometimes an integer. f. A decimal is sometimes an irrational number. 8. Omar A square root is always an irrational number. Explain to Omar why he is incorrect in his statement. 9. Robin A fraction is never an irrational number. Explain why Robin s statement is correct. 300 Chapter 5 The Real Number System
Problem 2 Properties of Real Numbers The real numbers, together with their operations and properties, form the real number system. You have already encountered many of the properties of the real number system in various lessons. Let s review these properties. Closure: A set of numbers is said to be closed under an operation if the result of the operation on two numbers in the set is a defined value also in the set. For instance, the set of integers is closed under addition. This means that for every two integers a and b, the sum a 1 b is also an integer. 1. Is the set of real numbers closed under addition? Write an example to support 2. Is the set of real numbers closed under subtraction? Write an example to support 3. Is the set of real numbers closed under multiplication? Write an example to support 4. Is the set of real numbers closed under division? Write an example to support Additive Identity: An additive identity is a number such that when you add it to a second number, the sum is equal to the second number. 5. For any real number a, is there a real number such that a 1 (the number) 5 a? What is the number? 6. Does the set of real numbers have an additive identity? Write an example to support 5.3 Real Numbers and Their Properties 301
Multiplicative Identity: A multiplicative identity is a number such that when you multiply it by a second number, the product is equal to the second number. 7. For any real number a, is there a real number such that a 3 (the number) 5 a? What is the number? 8. Does the set of real numbers have a multiplicative identity? Write an example to support Additive Inverse: Two numbers are additive inverses if their sum is the additive identity. 9. For any real number a, is there a real number such that a 1 (the number) 5 0? What is the number? 10. Does the set of real numbers have an additive inverse? Write an example to support Multiplicative Inverse: Two numbers are multiplicative inverses if their product is the multiplicative identity. You have been using these properties for a long time, moving forward you now know that they hold true for the set of real numbers. 11. For any real number a, is there a real number such that a 3 (the number) 5 1? What is the number? 12. Does the set of real numbers have a multiplicative inverse? Write an example to support 302 Chapter 5 The Real Number System
Commutative Property of Addition: Changing the order of two or more addends in an addition problem does not change the sum. For any real numbers a and b, a 1 b 5 b 1 a. 13. Write an example of the property. Commutative Property of Multiplication: Changing the order of two or more factors in a multiplication problem does not change the product. For any real numbers a and b, a 3 b 5 b 3 a. 14. Write an example of the property. Associative Property of Addition: Changing the grouping of the addends in an addition problem does not change the sum. For any real numbers a, b and c, (a 1 b) 1 c 5 a 1 (b 1 c). 15. Write an example of the property. Associative Property of Multiplication: Changing the grouping of the factors in a multiplication problem does not change the product. For any real numbers a, b, and c, (a 3 b) 3 c 5 a 3 (b 3 c). 16. Write an example of the property. Reflexive Property of Equality: For any real number a, a 5 a. 17. Write an example of the property. Symmetric Property of Equality: For any real numbers a and b, if a 5 b, then b 5 a. 18. Write an example of the property. The Transitive Property of Equality: For any real numbers a, b, and c, if a 5 b and b 5 c, then a 5 c. 19. Write an example of the property. 5.3 Real Numbers and Their Properties 303
Talk the Talk For each problem, identify the property that is represented. 1. 234 1 (2234) 5 0 2. 24 3 (3 3 5) 5 (24 3 3) 3 5 3. 224 3 15 224 4. 267 3 56 5 56 3 (267) 5. 2456 1 34 5 34 1 (2456) 6. 4 3 0.25 5 1 7. If 5 5 (21)(25) then (21)(25) 5 5. 8. If c 5 5 3 7 and 35 5 70 4 2, then c 5 70 4 2. 9. a 1 (4 1 c) 5 (a 1 4) 1 c 10. ( 2 3 4 ) ( 2 4 3 ) 5 1 11. 22 3 3 1 5 22 3 4 4 12. ( 2 3 4 ) 1 ( 4 3 1 5 ) 5 ( 2 3 1 4 4 3 ) 1 5 Be prepared to share your solutions and methods. 304 Chapter 5 The Real Number System