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MAT 111: Intermediate Algebra THE REAL NUMBER AND ABSOLUTE VALUE Set: A set is collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. There are several ways to represent a set and one of the most common one is using curly brackets. for examples: {1, 2, 3, 4}, {a, b, r,t, u, o}. Real Numbers Rational numbers can be written either one ( or more) of the following three forms: a (a and b are integers and b 6= 0) Terminating Decimal b 9 3 , 11.23 2 4 Repeating Decimal 7.12 Irrational numbers are those which are not rational. The decimal form of √an irrational number is non terminating and non repeating decimal number. Examples: 2.71828182846..., π and 2. √ 13 √ EX 1) Consider the set of numbers {3.56, − 5, − 58 , −1, , 0, 0.8, π, , 25, 111.12}, tell which numbers are 2 a Integers b Rational numbers c Irrational numbers We graph real numbers using a line called number line. Ex 2) Graph each number on the same number line. a. -2.5 b. 9 4 c. √ 2 d. 5 d Real numbers e. − 5 For any real number a, the additive inverse of a is −a. These numbers are at the same distance from 0, but on opposite sides of 0, on number line. Ex 3) Find the additive inverse of the following numbers. 1. −41 2. −8 21 3. 1.7 1 4. − 52 Written by B. Paudyal MAT 111: Intermediate Algebra The absolute value of a number is its distance from 0 on a number line. Absolute value of number is never negative. The notation uses two vertical lines: |number|. Some facts about absolute value: • absolute value of a positive number is the number • absolute value of a negative number is its additive inverse • absolute value of 0 is 0 • If |x| = a then either x = −a or x = a 2 b. |0| = Ex 4) Find the value. a. = 5 Some Symbols: = is equal to < is less then ≤ is less then or equal to 6= > ≥ c. | − 2.9| = d. − | − 9| = is not equal to is greater than is greater than or equal to Ex 5) Indicate whether the inequality is true or false. a. 0 < −6 c. b. −7 ≤ 0 10 5 ≥ 6 3 d. −1 > 0.3 1 Ex 6) Graph {−1.4, 3, − , −2.6} on the same number line. Then rank them in ascending and descending order. 2 Properties of Real Numbers: For any real numbers a, b, and c For Addition a+b = b+a Name of Property Commutative Property Associative Property (a · b) · c = a · (b · c) (2 · 3) · 4 = 2 · (3 · 4) (2 + 3) + 4 = 2 + (3 + 4) a+0 = a a·b = b·a 2·3 = 3·2 2+3 = 3+2 (a + b) + c = a + (b + c) For multiplication Identity Property a·1 = a 2+0 = 2 2·1 = 2 a + (−a) = 0 1 a· = 1, a 6= 0 a Inverse Property 2· 2 + (−2) = 0 2 1 2 =1 Written by B. Paudyal MAT 111: Intermediate Algebra The Distributive Property: a · (b + c) = a · b + a · c Example: 2 · (3 + 4) = 3 · 3 + 2 · 4 Zero factor property: a · 0 = 0 Example: 2·0 = 0 Division and 0: A number is NOT defined 0 Ex 7) Find each of the following. a. The additive inverse of 9.1 c. The multiplicative inverse of −5 b. The multiplicative inverse of 12 The additive inverse of −5 d. Ex 8) Indicate the definition, property, or number fact that justifies each statement. a. 5(−3) = −3(5) b. (3 + 13) + 27 = 3 + (13 + 27) c. 2 · 3 + 2 · 5 = 2(3 + 5) 0 = 6.3 + (−6.3) d. Addition, subtraction, multiplication and division EX 9) Add and Subtract 1. -3 + 6 2. -27 (-31) 3. -9 + (-25) + 4 To Multiply Real Two Numbers • positive · positive = positive • negative · negative = positive • positive · negative = negative 4. -7 2 + (-23) To Divide Two Real Numbers • positive ÷ positive = positive • negative ÷ negative = positive • positive ÷ negative = negative a Note • Different notations of division: a ÷ b, a/b and . b • Different notations of multiplication: a · b, a × b and (a)(b). • Decimal notation (.) and dot product notation (·). EX 10) Multiply and divide 1. 4 · (−5) 4. −12 4 2. 3 · (−4) · (−1) 5. −1 0 1 3. ( ) · (−1)(1)(−3)(−1)(−1) 3 6. 0 −800008 3 Written by B. Paudyal