Appendix A: The Real Line and the Cartesian Plane
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چکیده
A natural number is a non-negative whole number. We use the set of natural numbers, IN, for counting. It is useful to know that the sum or product of two natural numbers is again a natural number. We say that IN is closed with respect to addition and multiplication. However IN is not closed with respect to subtraction, since for example 2 − 4 = −2 and −2 is not a non-negative whole number. An integer is a whole number. The set of integers, ZZ, contains the set of natural numbers. It is also closed with respect to addition and multiplication, and has the added advantage of being closed with respect to subtraction. It is not closed with respect to division, since for example 12÷ 5 = 2 5 is not a whole number. The set of rational numbers, Q (q for quotient), consists of all ratios of integers with non-zero denominators. Q is closed with respect to addition, multiplication, and subtraction. Also as long as we don’t divide by zero, the ratio of two rational numbers is again a rational number. In fact Q is the smallest set containing IN where we can perform all four basic binary operations, barring only dividing by zero. End of story? NO. The ancient Greeks knew that certain necessary numbers were not rational. Fact: √ 2 is not rational. Proof: Suppose √ 2 ∈ Q. Then √ 2 = p q , where p, q ∈ ZZ, and q 6= 0. We may also suppose
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