Now that you know how a rational number can be expressed as a ratio of two integers, let’s
look at the other common way these numbers are symbolized. Since the middle of the twenti-
eth century, calculators and computers have replaced “manual” methods of calculation. These
machines use decimal fractions.Powers of 10
Apositive-integer power is a quantity multiplied by itself a certain number of times. If a non-
zero quantity is divided by itself once, it is said to be “raised” to the zeroth power, and the result
is always 1. A negative-integer power is a nonzero quantity divided by itself more than once.
Powers are denoted by exponents. Decimal notation is based on integer powers of 10. The
number 10 is called the exponential base. It can also be called simply the base or the radix.Orders of magnitude
Figure 7-1 is a number line showing the powers of 10, from 10^5 (the largest number in the
illustration) down to 10−^5 (the smallest). Each multiple of 10 is called an order of magnitude.
For example, 10^3 is one order of magnitude larger than 10^2 , and 10−^2 is three orders of mag-
nitude larger than 10−^5.
This number line differs from the ones you’ve seen so far. All the values here are positive. As
you go upward on the line, the numerical value increases faster and faster, so you race off toward
“infinity” more rapidly than you do on a conventional number line. As you go downward, the
value decreases at a slower and slower rate, “closing in” on 0 but never quite getting there.
If you expand on this idea, you can “build” any positive rational number by taking single-
digit multiples of powers of 10, and adding them up. Every positive number in this form has
its negative “twin.” To show the negative rational numbers, you can make up a separate num-
ber line for them. In order to account for 0, you can give it a special point that isn’t on either
line. Figure 7-2 portrays all the rational numbers using this system. “Infinity” and “negative
infinity” are not entitled to points here, because neither of them is a rational number!95CHAPTER7 Decimal Fractions
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