SECTION A.1 Algebra Essentials A3 Figure 3 A B Universal set (b) A B union (a) A B Universal set A B intersection Universal set A (c) A complement A Figure 2 A B Universal set (b) A ¨ B = [ disjoint sets Universal set subset A B (a) A 8 B Figures 3(a), 3(b), and 3(c) use Venn diagrams to illustrate intersection, union, and complement, respectively. Real Numbers Real numbers are represented by symbols such as 25, 0, 3, 1 2 , 5 4 , 0.125, 2, , 2, 0.666 . . . 3 π − − − The set of counting numbers , or natural numbers , contains the numbers in the set 1, 2, 3, 4, . . . . { } (The three dots, called an ellipsis , indicate that the pattern continues indefinitely.) The set of integers contains the numbers in the set . . . , 3, 2, 1, 0, 1, 2, 3, . . . . { } − − − A rational number is a number that can be expressed as a quotient a b of two integers, where the integer b cannot be 0. Examples of rational numbers are 3 4 , 5 2 , 0 4 , and 2 3 . − Since a a 1 = for any integer a , every integer is also a rational number. Real numbers that are not rational are called irrational . Examples of irrational numbers are 2 and π (the Greek letter pi), which equals the constant ratio of the circumference to the diameter of a circle. See Figure 4. Real numbers can be represented as decimals . Rational real numbers have decimal representations that either terminate or are nonterminating with repeating blocks of digits. For example, 3 4 0.75, = which terminates; and 2 3 0.666 . . . , = in which the digit 6 repeats indefinitely. Irrational real numbers have decimal representations that neither repeat nor terminate. For example, 2 1.414213 . . . = and 3.14159 . . . . π = In practice, the decimal representation of an irrational number is given as an approximation. We use the symbol ≈ (read as “approximately equal to”) to write 2 1.4142 ≈ and 3.1416. π ≈ Two frequently used properties of real numbers are given next. Suppose that a , b , and c are real numbers. Distributive Property a b c ab ac a b c a c b c ( ) ( ) ⋅ + = + + ⋅ = ⋅ + ⋅ Figure 4 C d π = C d
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