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Unformatted text preview: Section 1.1 Real Numbers Types of Real Numbers 1. Natural numbers ( N ): 1 , 2 , 3 , 4 , 5 , . . . 2. Integer numbers ( Z ): , ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , . . . REMARK: Any natural number is an integer number, but not any integer number is a natural number. 3. Rational numbers ( Q ): r = m n , where m ∈ Z , n ∈ N EXAMPLES: 1 2 7 3 11 53 2 = 2 1 . 2 = 2 10 = 1 5 . 222 . . . = 0 . 2 = 2 9 . 999 . . . = 0 . 9 = 1 REMARK: Any integer number is a rational number, but not any rational number is an integer. 4. Irrational Numbers . These are numbers that cannot be expressed as a ratio of integers. EXAMPLES: √ 2 √ 3 √ 2 + √ 3 1 + √ 5 2 3 √ 2 π √ π 2 √ 2 The set of all real numbers is denoted by R . The real numbers can be represented by points on a line which is called a coordinate line , or a real number line , or simply a real line : 1 Every real number has a decimal representation. If the number is rational, then its correspond ing decimal is repeating. For example, 1 2 = 0 . 500 . . . = 0 . 5 0 = 0 . 4 9 , 2 3 = 0 . 66666 . . . = 0 . 6 157 495 = 0 . 3171717 . . . = 0 . 3 17 , 9 7 = 1 . 285714285714 . . . = 1 . 285714 If the number is irrational, the decimal representation is nonrepeating: √ 2 = 1 . 414213562373095 . . . π = 3 . 141592653589793 . . . Operations on Real Numbers Real numbers can be combined using the familiar operations of addition, subtraction, multi plication, and division. When evaluating arithmetic expressions that contain several of these operations, we use the following conventions to determine the order in which the operations are performed: 1. Perform operations inside parentheses first, beginning with the innermost pair. In dividing two expressions, the numerator and denominator of the quotient are treated as if they are within parentheses....
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This note was uploaded on 01/23/2012 for the course MATH 8650 taught by Professor Kiryltsishchanka during the Spring '12 term at NYU.
 Spring '12
 KIRYLTSISHCHANKA
 Calculus, Algebra, Real Numbers, Natural Numbers

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