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Intro to Maths for CS: Rational and Irrational Numbers John Barnden School of Computer Science, University of Birmingham Term 1, 2014–15 Rational and Irrational Numbers A rational number is one that is equal to some fraction m/n where m and n are whole numbers (m positive, negative or zero; n positive or negative but non-zero). All other numbers are irrational. The word “rational” comes from “ratio” — it’s not a synonym of “logical” or “not crazy”! ASIDE: but look up the psychological sense of “rational” in a dictionary that gives etymologies (word origins/histories). In particular, all integers are rational. It can be shown that π is irrational. Many square roots (and cube roots, fourth roots, ...) are irrational. In particular, the square root of any whole number that is not a perfect square is irrational. For instance, take five .... Barnden (SoCS) Intro to Maths for CS: Rational and Irrational Numbers Term 1, 2014–15 2/1 ... Take 5 Proof that √ 5 is irrational. √ SUPPOSE it’s rational, so 5 = m/n for some integers m, n. Clearly, we can ensure that m and n are both bigger than 1. Square both sides, to get m2 = 5n2 . Consider the prime factorizations of m and n. The equation tells us that the power of 5 in m2 is one more than the power of 5 in n2 . But this is IMPOSSIBLE, because the power must be even in both m2 and n2 . So our supposition was WRONG, and we have our proof. Barnden (SoCS) Intro to Maths for CS: Rational and Irrational Numbers Term 1, 2014–15 3/1 ... and Now Take φ Remember that the golden ratio φ (related to the Fibonacci sequence—see Fun Quiz) is √ 1+ 5 2 We now know that φ is irrational. Why? Because if it were rational, then √ m 1+ 5 = 2 n from which we get √ 5 = 2× m −1 n = 2m − n n which is rational. Barnden (SoCS) Intro to Maths for CS: Rational and Irrational Numbers Term 1, 2014–15 4/1 Irrationals via Rationals It can be shown that any irrational number z can be approximated arbitrarily closely by rational numbers. That is, for any number ǫ > 0, we can find a rational number r whose difference from z is less than ǫ. In fact, we can show that any irrational (and non-”complex”) number can be expressed as a decimal number with infinitely many digits after the decimal point (where the digits after the point never form a recurring pattern). What this really means is that ... Barnden (SoCS) Intro to Maths for CS: Rational and Irrational Numbers Term 1, 2014–15 5/1 Irrationals via Rationals, contd √ ... for instance, in the case of√ 5 there is a a never-ending sequence of rational numbers that get closer and closer to 5, where the first few are: 2 2.2 2.23 2.236 2.23606 2.236067 2.2360679 2.23606797 2.236067977 2.2360679774 2.23606797749 2.236067977499 2.2360679774997 2.23606797749979 Barnden (SoCS) Intro to Maths for CS: Rational and Irrational Numbers Term 1, 2014–15 6/1