- Suppose we have an irrational number and we display the first few digits of its endless
nonrepeating decimal expansion. If we multiply this number by 10, is the result still
irrational? What if we multiply it by 100, or 1,000, or any natural-number power of
10? Will the results always be irrational? - What is the cardinality of
(a) The set of even natural numbers?
(b) The set of naturals divisible by 10 without a remainder?
(c) The set of naturals divisible by 100 without a remainder?
(d) The set of naturals divisible by any natural power of 10 without a remainder?- Consider this equation in real variables x and y:
36 x+ 48 y= 216How can we simplify this so it has the minimum possible number of symbols (variables
and digits)?
- The nonnegative square root of 18 can be simplified, or resolved, into a product of a
natural number and an irrational number. What are these numbers? - The nonnegative square root of 83 cannot be resolved into a product of a natural
number and an irrational number, other than the trivial case 1 × 831/2. How can we tell? - The numbers 501/2 and 21/2 are both irrational. But the ratio of 501/2 to 21/2 is a natural
number. What natural number? - Using the sum of quotients rule, add the fractions 7/11 and −5/17, and then reduce this
sum to lowest terms. - Using the product of sums rule, multiply out the product (x+y)(x−y).
- Using the rules we have learned so far, derive a formula for multiplying out the
following expression, where u, v, w, x, y, and z are real numbers:
(u+v+w)(x+y+z)Practice Exercises 141