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Integers, Rational, and Real Numbers A set is a collection of objects. Typically in math we will refer to collections of numbers as sets. The objects in a set are called elements. In math, we represents our sets as numbers in squiggly brackets. {2, 6, 9} is a set of numbers. The integers include the zero, the counting numbers, and the counting numbers “signed” negative. {. . . , -3, -2, -1, 0, 1, 2, 3, . . .} is one way to write the integers in set notation. Goal – to introduce 3 sets of numbers The counting numbers is a set of numbers. We could represent the counting numbers is set notation as {1, 2, 3, . . .}. Math need lots more numbers that the counting numbers. Zero is one of the most important numbers, but is not a counting number! A number line is often labeled with integers. The size of the integer indicates how far from zero the number is; the sign of the integer indicates whether the integer is to the left (negative) or to the right (positive) of zero. -4 -3 -2 -1 0 1 2 3 4 The size of an integer, or the distance from zero of that integer along a number line is called the absolute value of that integer. | 1 | = 1 because 1 is one unit away from zero on a number line, but | -1 | = 1 also, because -1 is also one unit away from zero on a number line! In fact, 1 and -1 are the only numbers with absolute value of 1. All rational numbers can be written as decimals. Some of the decimal representations of rational numbers are very familiar. Examples: 1 1 = 0.5, = 0.25 2 4 The decimal representation of all rational numbers are terminating or repeating decimals. The set of rational numbers is the set of all number that can be formed as quotients, or fractions, using integers. This set can be denoted a { b such that a and b and integers, b ≠ 0} or a { | a and b and integers, b b≠0} Where | stand for “such that”. If you don’t know the decimal representation of a rational number you can find it by division by hand or by calculator. Example: .11 9 1.00 1 9 9 10 1 = .111..., or 0.1 9 Example: Label the rational numbers on the number line: 3.5, -2.7, 1 3 -4 -3 -2 -1 0 1 2 3 4 Some numbers have decimal representations that are nonrepeating and nonterminating. These numbers are called irrational. Real Rational Irrational Integers All of the sets of numbers discussed so far fall into a larger set of numbers called the real numbers. Counting numbers All numbers on the number line are real numbers. The number line places the real numbers in order. To describe the order of the real numbers, we use the symbols: Finally absolute value works on any real number, and describes the distance from that real number to zero on the number line. > meaning greater than, and Examples: < meaning less than. Example: 10 > 8 -3.7 < -3 | 0.1 | = 0.1 | -3.7 | = 3.7 − 10 10 = 3 3