CHAPTER 5
DECIMALS
The origin and meaning of the word "decimal" were
discussed in chapter 1 of this course.
Also discussed in chapter 1 were the concept
of place value and the use of the number ten
as the base for our number system. Another
term which is frequently used to denote the
base of a number system is RADIX. .For
example, two is the radix of the binary
system and ten is the radix of the decimal
system. The radix of a number system is
always equal to the number of different digits used in the system. For example,
the decimal system, with radix ten, has ten
digits: 0 through 9.
DECIMAL FRACTIONS
A decimal fraction is a fraction whose denominator is 10 or some power of 10,
such as 100, 1,000, or 10,000. Thus, 7/10
13/100 and 215 /1000 are decimal fractions. Decimal fractions have special
characteristics that make computation much
simpler than with other fractions.
Decimal fractions complete our decimal system
of numbers. In the study of whole numbers, we found that we could proceed to the
left from the units place, tens, hundreds,
thousands, and on indefinitely to any larger
place value, but the development stopped with
the units place. Decimal fractions complete
the development so that we can proceed to the
right of the units place to any smaller
number indefinitely. Figure 5l
(A) shows how decimal fractions complete the
system. It should be noted that as we proceed
from left to right, the value of each place
is onetenth the value of the preceding place,
and that the system continues uninterrupted with the decimal fractions. Figure
5l (B) shows the system again, this time
using numbers. Notice in (A) and (B) that the
units place is the center of the system and that
the place values proceed
to the right or left of it by powers of ten.
Ten on the left is balanced by tenths on the
right, hundred6 by hundredths, thousands by
thousandths, etc. Notice that l/10 is one
place to the right of the units digit, 1/100
is two places to the right, etc. (See fig.
5l.) If a marker is placed after the units
digit, we can decide whether a decimal digit
is in the tenths, hundredths, or thousandths position
by counting places to the right of the marker.
In some European countries, the marker is a
comma; but in the Englishspeaking countries,
the marker is the DECIMAL POINT.
Thus, 3/10 is written 0.3. To write 3/100 is necessary
to show that 3 is in the second place to the
right of the decimal point, so a zero is inserted
in the first place. Thus, 3/100 is written
Similarly, 3/1000. can be written by inserting zeros in the first two places to
the right of the
decimal point. Thus, 3/1000 is written 0.003. In
the number 0.3, we say that 3 is in the first decimal
place; in 0.03, 3 is in the second decimal place; and in 0.003, 3 is in the
third decimal place. Quite frequently decimal fraction6 are
simply called decimals when written in this shortened
form.
WRITING DECIMALS
Any decimal fraction may be written in the shortened
form by a simple mechanical process. Simply
begin at the righthand digit of the numerator and count off to the left as many
places as there are zeros in the denominator.
Place the decimal point to the left of the
last digit counted. The denominator may then
be disregarded. If there are not enough digits, as many
placeholding zero6 as are necessary are added
to the left of the lefthand digit in the numerator.
Thus, is 23/10000 beginning with the digit 3, we
count off four places to the left, adding two O’s
as we count, and place the decimal point to the
extreme left. (See fig. 52.) Either form is
read "twentythree tenthousandths." When
a decimal fraction is written in the shortened
form, there will always be as many decimal
places in the shortened form as there
Figure 52.–Conversion of
a decimal fraction to shortened
form.
Figure 53.–Steps in the conversion
of a decimal fraction to
shortened form. are zeros in the
denominator of the fractional form.
Figure 53 shows the fraction 24358/100000 and what is meant when it is
changed to the shortended form.
This figure is presented to show further
that each digit of a decimal fraction holds
a certain position in the digit sequence and
has a particular value.
By the fundamental rule of fractions, it should
be clear that Writing 5/10 = 50/100 = 500/1000 the
same values in the shortened way, we have 0.5
= 0.50 = 0.500. In other words, the value of a
decimal is not changed by annexing zeros at the
righthand end of the number. This is not true
of whole numbers. Thus, 0.3, 0.30, and 0.300
are equal but 3, 30, and 300 are not equal. Also
notice that zeros directly after the decimal point
do change values. Thus 0.3 is not equal
to either 0.03 or 0.003. Decimals
such as 0.125 are frequently seen. Although
the 0 on the left of the decimal point is
not required, it is often helpful. This is particularly true in
an expression such as 32 + 0.1. In
this expression, the lower dot of the division symbol
must not be crowded against the decimal point;
the 0 serves as an effective spacer. If any
doubt exists concerning the clarity of an expression
such as .125, it should be written as 0.125.
Practice problems. In problems 1 through 4,
change the fractions to decimals. In problems
5 through 8, write the given numbers as decimals:
1. 8/100
2. 5/1000
3. 43/1000
4. 32/10000
5. Four hundredths
6. Four thousandths
7. Five hundred one tenthousandths
8. Ninetyseven thousandths
Answers:
1. 0.08
2. 0.005
3. 0.043
4. 0.0032
5. 0.04
6. 0.004
7. 0.0501
8. 0.097

