Written homework problem 3 - extra-credit Assigned 1/13 and due 1/19 A number x is called rational if x = a/b, where a and b are whole numbers (which can be positive or negative). We say that x is irrational if x√is not rational. For example, 0, −1, 3/5, √ 12/29 are rational numbers. The numbers 2, 2 − 2, and π are irrational (although this is not obvious). Facts: If x and y are numbers, then (i) If x and y are rational, then so are x + y and xy. (ii) If x 6= 0 is rational and y is irrational, then x + y and xy are both irrational. Let ( 1 f (x) = 0 if x is rational if x is irrational . (a) Use definition 2 on page 299 to show that the area under f (x), 0 ≤ x ≤ √ 2 is 0. (b) Use definition 2 to show that the area under f (x) for 0 ≤ x ≤ 1 is 1. √ (c) Use definition 2 to show that the area under f (x) for 1 ≤ x ≤ 2 is 0. (These three facts show that if we defined the integral as limn→∞ Rn , then not all the properties of integrals we want would be true.) 1