Algebra Know-It-ALL

98 Decimal Fractions

Starting at the decimal point and working toward the right:

(7 × 10 −^1 )+ (7 × 10 −^2 )+ (3 × 10 −^3 )+ (5 × 10 −^4 )

= 7/10 + 7/100 + 3/1,000 + 5/10,000 = 0.7735

When we add the whole number to the decimal fraction, we get

362 + 0.7735 = 362.7735

Are you confused?

You will sometimes hear scientists talk rather loosely about orders of magnitude. They mean to say that

theabsolute value of one quantity is some power of 10 times bigger or smaller than the absolute value of the

other quantity. Here are a few examples:

• 45,300 is one order of magnitude larger than 4,530


• 0.56 is two orders of magnitude smaller, in absolute terms, than − 56


• −0.565 is three orders of magnitude smaller, in absolute terms, than 565


• −88,888 is four orders of magnitude larger, in absolute terms, than −8.8888

If one or both quantities is negative, you should be sure to include the phrase “in absolute terms” so there’s

no confusion about the meanings of “smaller” or “larger.”

When you want to portray a decimal number greater than or equal to 0 but smaller than 1, it is custom-

ary to write a single numeral 0 to the left of the decimal point. If the number is greater than −1 but smaller

than 0, you should write the minus sign first, then a single 0, and then the decimal point.

Here’s a challenge!

Express the scheme for “building” a decimal numeral in general terms, rather than merely providing

examples.

Solution

Imagine two sets of single-digit numerals, called set A and set B. Suppose A has m elements and B has

n elements, named as follows:

A= {a 1 ,a 2 , a 3 , ...,am}

and

B= {b 1 ,b 2 ,b 3 , ...,bn}

Now imagine these single-digit numerals arranged around a decimal point like this:

am ... a 3 a 2 a 1 .b 1 b 2 b 3 ... bn

The string of numerals to the left of the point represents the sum

SA= (am × 10 m−^1 )+ ··· + (a 3 × 102 )+ (a 2 × 101 )+ (a 1 × 100 )