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Rational and Irrational Numbers Most of us are somewhat familiar with integers and rational numbers: Integers: Z = {. . . , −2, −1, 0, +1, +2, . . .} (positive and negative counting numbers) Rationals: Q = {m/n | m, n ∈ Z, n 6= 0} (fractions of the integers) We should all remember how to add, subtract, multiply, and divide these numbers. For fractions, there are some simple rules that always hold: Addition: an + mb a m + = b n bn Subtraction: a m an − mb − = b n bn Multiplication: Division: a m am · = b n bn an a m ÷ = b n bm The integers are a subset of the rationals, because every integer can be written as a number divided by 1. For example, −4 = (−4)/1. Fractions of integers can always be represented as a decimal expansion that either terminates after a certain number of decimal places, or ends in an infinitely repeating sequence of some block of digits. (This can be proven using the division algorithm, learned in elementary school). For example... 123/1000 = 0.123 74/9 = 8.22222 . . . 4/11 = 0.36 36 36 . . . 7/13 = 0.538461 538461 538461 . . . 5 8 + 1 1100 = 5508/8800 = 0.625 90 90 90 . . . 1 Any decimal expansion can be written as an infinite sum of fractions as in the following... 1.123 123 123 . . . = 1 + 1 2 3 1 + + + + ... 10 100 1000 10000 If we wanted, we can also form sums like this which are not terminating and contain no repeating blocks. Such numbers are still well-defined, however they are not rational: we call them irrational. Some examples... 1.09 009 0009 00009 . . . √ 3 ≈ 1.732050807568877 . . . Two of the most important numbers in all of mathematics are the following irrational numbers: The number Pi The Natural Base π ≈ 3.14159 e ≈ 2.71828 It is important to note that any irrational number can always be approximated by a decimal number in almost any mathematical expression. And, the more decimal places you add, the better the approximation gets. For example, we know √ 2 that 2 = 2, exactly. √ 2 2 ≈ (1.4)2 = 1.96 √ 2 2 ≈ (1.414)2 = 1.999396 √ 2 2 ≈ (1.41421)2 = 1.9999899241 As we add more decimal places, we get a sequence of numbers that gets closer and closer to 2, the exact value. The set of all rational and irrational numbers is denoted by R. 2