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Math 123 Section 1.3 – The Real Numbers - Page 1 Section 1.3 The Real Numbers I. Sets and Set Notation A. A set is a collection of objects. B. The objects of a set are called the elements. C. When a set is written in braces { }, with every element listed, it is written in roster form. D. Examples 1. A = {horses, pigs, cows} 2. B = {5, 10, 15} a. We say that 5 is an element of set B and write it: 5∈ B b. We say that 12 is not an element of set B and write it: 12 ∉ B E. Definitions 1. A finite set is a set that has a definite number of elements. 2. An infinite set has an indefinite number of elements. a. From above, A and B are finite sets. b. C = {2, 4, 6, 8, . . .} is an infinite set. 3. The Natural Numbers or Counting Numbers is the set {1, 2, 3, . . .} 4. Note that if we wanted to write the set of the first 100 counting numbers, we could write it as: {1, 2, 3, . . ., 100}. 5. The null set or empty set is the set with no elements. It is written: ∅ 6. A set is written in set builder notation when it is written as a collection of elements that must meet some rule to belong to the set. For example: a. The set of all natural numbers greater than 4 would be written: E = {x| x is a natural number > 4} b. The set of natural numbers between 2 and 200 would be written: F = {x| x is a natural number, 2 < x < 200} 7. A set G is a subset of a set H if every element of G is also in H. This is written: G⊆H 8. The set of Whole Numbers is the set {0, 1, 2, 3, 4, . . .} Note that the Natural Numbers are a subset of the Whole Numbers. 9. The Integers is the set {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}. Note that both the Natural Numbers and the Whole Numbers are a subset of the Integers. 10. The Rational Numbers is the set: p | p, q ∈ Integers, q ≠ 0 q In other words, the rationals are the terminating and repeating decimals. Note that to change a fraction to a decimal, you must divide the numerator by the denominator. 11. The Irrational Numbers are the set of all non-terminating, non-repeating decimals. 12. The Real Numbers is the union of the set of Rational and Irrational Numbers. 13. The number line is a line in which each point corresponds to a real number. 14. If a number m lies to the left of a number n on the number line, then we say that m < n, which is read “m is less than n”. 15. If a number m lies to the right of a number n on the number line, then we say that m > n, which is read “m is greater than n”. 16. The origin is 0. 17. The positive real numbers is the set {x: x > 0}. These are the numbers to the right of the origin. © Copyright 2009 by John Fetcho. All rights reserved Math 123 Section 1.3 – The Real Numbers - Page 2 18. The negative real numbers is the set {x: x < 0}. These are the numbers to the left of the origin. 19. The additive inverse or opposite of a real number is that number on the opposite side of 0 as the number, but the same distance from 0. We write the opposite of a number p as –p. 20. If x is a real number, then –(-x) = x . i.e. – A double negative gives a positive. 21. The absolute value of a number is the distance the number is from the origin. F. Examples - Graph each real number on a number line. 1. 5 Answer: −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 3 4 5 3 4 5 2. −2 Answer: 3. Now you try one: 3 1 2 Answer: −5 −4 −3 −2 1 3 2 7 3 Answer: 4. −5 −4 −3 −2 −1 0 1 2 7 3 5. Now you try one: Answer: −5 −4 −3 −1.8 −2 −1 0 1 2 −1.8 G. Examples – Simplify the following. 1. –(−35) Answer: 35 2. − − 11 = −(11) Answer: −11 3. Now you try this one: −8 Answer: −8 © Copyright 2009 by John Fetcho. All rights reserved Math 123 Section 1.3 – The Real Numbers - Page 3 H. Examples – List the elements of the set. 1. A = {x: x is an integer divisible by 5} Answer: A = {. . ., -10, -5, 0, 5, 10, . . .} 2. Now you try: B = {x: x is a natural number less than 10} Answer: B = {1, 2, 3, 4, 5, 6, 7, 8, 9} I. Examples – Write the set in set builder notation. 1. A = {3, 4, 5, 6, 7, 8} Answer: A = {x|x is a natural number between 2 and 9} 2. Now you try: B = {2, 4, 6, . . . } Answer: B = {x| x is a positive even integer} J. List all numbers from the set { } 1 −5.3, −5, − 3, −1, − ,0,1.2,1.8,3, 11 9 that are: 1. 2. 3. Natural Numbers: {3} Whole Numbers: {0, 3} Integers: {−5, −1, 0, 3} 4. Rational Numbers: {−5.3, −5, −1, 5. Irrational Numbers: { B. C. D. E. } 9 3, 11 } 1 Real Numbers: −5.3, −5, − 3, −1, − ,0,1.2,1.8,3, 11 9 Inequalities Not Equal 1. The symbol that we use for not equal is ≠ . 2. For example, 3 + 5 ≠ 10. Less Than 1. The symbol that we use for less than is <. 2. For example, 3 + 5 < 10. Greater Than 1. The symbol that we use for greater than is >. 2. For example, 10 > 3 + 5. For both B & C, we could also have stated that the expression was < or >. Examples - Insert either < or > in the blank between each pair of numbers to make a true statement. 1. 2 _____ −4 The main thing to remember is that the larger number is always farther to the right on the number line. Since any positive number is to the right of a negative number, we have: 6. II. A. {− 1 − , 0, 1.2, 1.8, 3} Answer: 2 > −4 © Copyright 2009 by John Fetcho. All rights reserved Math 123 Section 1.3 – The Real Numbers - Page 4 2. 3 _____ 2 On this one, we are getting a preview of coming attractions in this class! What is the opposite of square root? Squaring of course! So if we square a square root, we "undo" the square root. So squaring both sides, we get: 3 _____ 4 Since 3 is less than 4, we need to use the "<" sign. Answer: 3. 3 <2 Now you try one: Answer: −π _____ −3.5 −π > −3.5 © Copyright 2009 by John Fetcho. All rights reserved