Algebra Know-It-ALL

We aren’t allowed to get away with trivial solutions, such as letting all the integers be equal

to 1 or letting them all be equal to −1. But suppose that a= 7, b= 5, c= 14, and d= 10.

(This isn’t the only example we can use, but it should give you the general idea.) Then

ad / bc= (7 × 10) / (5 × 14)

= 70/70

= 1

and

bc / ad= (5 × 14) / (7 × 10)

= 70/70

= 1

Let’s “plug in” the values a= 7, b= 5, c= 14, and d= 10 to the original equation and see

what we get:

(a/b) / (c/d)= (c/d) / (a/b)

therefore

(7/5) / (14/10) = (14/10) / (7/5)

Note that 7/5 and 14/10 actually represent the same rational number. All we’ve really

done here is show that 1 = 1, in a roundabout way.

Chapter 7

1. Figure A-1 shows a number line that covers the range of positive rational numbers from
   10 up to 100,000. To find the number of orders of magnitude, subtract the powers of 10.

10

5

100,000

104 10,000

103 1,000

10

2

100

(^10110)
Figure A-1 Illustration for the solution
to Prob. 1 in Chap. 7.
Chapter 7 607