Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not
represent the only way a problem can be figured out. If you think you can solve a particular
problem in a quicker or better way than you see there, by all means try it!
- Evaluate and compare these two sums:
a= |− 3 + 4 + (−5)+ 6|
and
b= |−3|+ |4| + |−5|+ |6|
What general fact can you deduce from the results?
- To illustrate the importance of the placement of parentheses in a mixed sum and
difference, evaluate the following two expressions. In long strings of sums and
differences, you should first perform the operations inside the parentheses from left to
right, and then perform the operations outside the parentheses from left to right. Here
are the expressions:
(3+ 5) − (7 + 9) − (11 + 13) − 15
and
3 + (5 − 7) + (9 − 11) + (13 − 15)
- Using the rules explained in the previous exercise, how should you evaluate the string of
sums and differences if there are no parentheses at all? Here it is:
3 + 5 − 7 + 9 − 11 + 13 − 15
- Suppose someone tells you that there was a significant trend in the mid-winter average
temperatures in the town of Hoodopolis during the period 1998 through 2005. You
want to find out if this is true. You come across some old heating bills from the utility
company that show how much warmer or cooler a given month was, on the average,
Table 4-1. Here is a proof that shows how you can reverse
the order in which three integers a,b, and c are added, and
get the same sum. As you read down the left-hand column,
each statement is equal to all the statements above it.
Statements Reasons
a+b+c Begin here
a+ (b+c) Group the second two integers
a+ (c+b) Commutative law for the sum of b and c
(c+b)+a Commutative law for the sum of a and (c+b)
c+b+a Ungroup the first two integers
Q.E.D. Latin Quod erat demonstradum, translated into
English as “Which was to be proved”
Practice Exercises 63
