Algebra Know-It-ALL

608 Worked-Out Solutions to Exercises: Chapters 1 to 9

Here, 100,000 = 105 and 10 = 101. We have 5 − 1, or 4, so there are four orders of

magnitude in this span. Another way to see this is to count the number of intervals between

“hash marks” on the number line.

2. Figure A-2 shows a number line that covers the range of positive rational numbers from
   30 up to 300,000. In this case, we can divide the larger number by the smaller, and
   then count the number of ciphers in the quotient. We get

300,000 / 30 = 10,000

These two numbers differ by four orders of magnitude, the same extent as the two

numbers differ in Prob. 1. We can also count the number of intervals between hash

marks, just as in Prob. 1.

3. In this situation, the larger number is 75,000,000 and the smaller number is 330. If we
   divide the larger by the smaller, we get 227,272 and a fraction. That’s between a factor
   of 100,000 (or 10^5 ) and 1,000,000 (or 10^6 ). We can therefore say that 75,000,000 is
   between five and six orders of magnitude larger than 330. Right now, that’s the best we
   can do. (It’s possible to come up with a more precise value, but that involves logarithms,
   which we have not yet studied.)


4. The answers, along with explanations, are as follows:
   (a) The number 4.7 is equivalent to 4-7/10. The fractional part here is in lowest terms,
   because the numerator, 7, is prime.
   (b) The number −8.35 is equivalent to −8-35/100. The fractional part can be reduced to
   7/20, so the entire expression becomes −8-7/20. Note the difference in appearance, as
   well as the difference in purpose, between the minus sign and the dash!
   (c) The number 0.02 has no integer portion. The fractional part is 2/100, which
   reduces to 1/50. Therefore, the entire expression is 1/50.
   (d) The number −0.29 has no integer portion. The fractional part is −29/100, which is
   already in lowest terms because the absolute value of the numerator, 29, is prime and
   is not a prime factor of the denominator. Therefore, the entire expression is −29/100.

3 × 105

3 × 104

3 ×^103

10

2

3 ×

10

1

3 ×

300,000

30,000

3,000

300

30

Figure A-2 Illustration for the solution

to Prob. 2 in Chap. 7.