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E - BOOK FOR COLLEGE ALGEBRA A1 King Fahd University of Petroleum & Minerals The Real Numbers System Set of Numbers Ordering of Real Numbers Absolute Value Interval of Real KFUPM - Prep Year Math Program (c) 2009 All Right Reserved E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Set of Numbers Natural Numbers The set of natural numbers (counting numbers) is {1, 2, 3, 4, 5 . . .} Whole Numbers The set of whole numbers (including 0 with the set of natural numbers) is {0, 1, 2, 3, 4, 5 . . .} Integers Integers are the expanded set of numbers obtained by including the negative of the counting numbers. {. . . –4, –3, –2, –1, 0, 1, 2, 3, 4 . . .} KFUPM - Prep Year Math Program (c) 2009 All Right Reserved E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Set of Numbers Rational Numbers Rational numbers are set of all numbers of the form a , b0 b where a (called numerator) and b (called denominator) are integers. Irrational Numbers Irrational numbers are real numbers that are not rational. Equivalently, they are non-terminating and 3 non-repeating decimals. For example , e, 2, 7 KFUPM - Prep Year Math Program (c) 2009 All Right Reserved King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Set of Numbers The Real Numbers The set of all the irrational numbers and the rational numbers constitutes the set of the real numbers. Irrational numbers Rational numbers 1−π −e e e−π Integers -1, -2, -3, - 4, .… Whole number 0 Natural numbers 1, 2, 3, …. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Ordering of Real Numbers The real number system has the property of being ordered. This means that for any two different real numbers a and b, one and only one of the following three statements must be true: a < b, a = b or a > b. This property allows the arrangement of the real numbers on any line, L, as shown in the figure below. −6 −5 −4 −3 −2 −1 KFUPM - Prep Year Math Program (c) 2009 All Right Reserved 0 1 2 3 4 5 6 King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Absolute Value The absolute value of a real number is the distance from the number to the origin O on the number line. The absolute value of x is denoted by │x│. Absolute Value Properties 1. x if x 0 x x if x 0 2. x 0 if and of if x 0 3. x y x y 4. xy x y Definition: We define the distance between two numbers a and b by │a − b│ KFUPM - Prep Year Math Program (c) 2009 All Right Reserved King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Intervals of Real Numbers Verbal Description Geometric Description All real numbers between a and b Notation ( a, b ) All real numbers between a and b inclusive a b a b ( a, b ] All real numbers between a inclusive and b [ a, b ) All real numbers between a inclusive and b inclusive a b a b [ a, b ] All real numbers that are greater than a ( a, ∞ ) a All real numbers that are greater than a inclusive [ a, ∞ ) a All real numbers that are less than a ( –∞, a ) a All real numbers that are less than a inclusive All real numbers ( –∞, a ] a -∞ KFUPM - Prep Year Math Program (c) 2009 All Right Reserved ∞ ( –∞, ∞ ) King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Use the definition of the absolute value to evaluate Example 1 because a) 5 5 b) 5 5 . c) .2 5 2 5 d) e) 5 2 5 0 because 5 0 because 2 . 5 5 5 because . 1 1 2 2 because 5 1 2 2 2 3 3 3 1 1 1 . 2 f) KFUPM - Prep Year Math Program (c) 2009 All Right Reserved because 3 5 King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA For 0<x<1, simplify each absolute value expression. Example 2 a) e 2x e 2x because 2 x 2 e 2.718 b) 3x 3x because 3x 3 c) .x3 x 2 x 2 x3 d) . . x x. 2 1 x because x3 x 2 for 0 x 1 x 2 x 1 x 2 1 x x2 x 1 x 1 x KFUPM - Prep Year Math Program (c) 2009 All Right Reserved . 2 because 0 x 1 King Fahd University of Petroleum & Minerals E - BOOK FOR COLLEGE ALGEBRA Example 3 a) Use the absolute value to describe each statement. the distance between k and 4 is 2 k4 2 b) the distance between x and –3 is less than 1 x 3 c) 1 x3 1 the city of Jeddah is closer to Makkah than it is to Madinah Jeddah Makkah Jeddah Madinah d) the city of Hail is more than 100 kilometers away from Buraidah Hail Buraida KFUPM - Prep Year Math Program (c) 2009 All Right Reserved 100 km