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