Section 1.1 The Set of Real Numbers 3 To denote that a number is or is not a member of a set, we use the notation shown in the following table. Symbol Meaning Example Verbal Statement ∈ Element of 4 ∈ A “4 is an element of A.” or “4 is a member of A.” ∉ Is not an element of 6 ∉ A “6 is not an element of A.” or “6 is not a member of A.” Numbers belong to certain sets. The next table defines these sets. Natural numbers (or counting numbers) = {1, 2, 3, 4, 5, . . . } Whole numbers = {0, 1, 2, 3, 4, 5, . . . } Integers = {. . . , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . . } Rational numbers = e p q ` p, q with q 2 0 f Irrational numbers = {numbers that are not rational} Real numbers = {rational or irrational numbers} The set of rational numbers is given in set-builder notation. It is read as “the set of numbers p over q, such that p and q are integers with q not equal to 0.” That is, the values of p and q can be replaced with any integer except that q cannot be replaced with zero. So, a rational number is the quotient, or ratio, of two integers whose denominator is not zero. It is impossible to list all of the rational numbers, since there are infinitely many of them. Some examples of rational numbers are 10 1 = 10, 37 7 = 5 2 7 3 5 , - =-0.6, 1 3 = 0.3333 . . . Rational numbers include integers, mixed numbers, decimals that terminate, and repeating decimals. Note all integers are rational numbers since each integer can be written as its value divided by 1, as illustrated by 10 = 10 1 . The set of irrational numbers arose out of the study of geometry, specifically triangles and circles. A simple definition of an irrational number is a number that is not rational. This means that the number cannot be written as a quotient of two integers. ������������ Irrational numbers are numbers whose values are nonterminating and nonrepeating decimals. The number π (pi) may be the most commonly known irrational number. We often use the value 3.14 for π but this is only a decimal approximation. The exact value of π is 3.1415926535 . . . . This value continues indefinitely with no repeating pattern. Another irrational number used in advanced math classes is the number e, Euler’s number. It is used in many important formulas, such as those involving population growth. Its value is approximately 2.72. Other examples of irrational numbers involve square roots. The square root of a number is the number that must be multiplied by itself to get the original number. For instance, 19 = 3 since 3 · 3 = 9. The square root of 9 is a rational number since its value is equivalent to 3, which can be expressed as a quotient of two integers, 3 1 . However, not all square roots are rational numbers.
hendricks_beginning_algebra_1e_ch1_3
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