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16 ⏐⏐⏐ INTRODUCTION
EXAMPLE 1.9
a. (0.00003) 3 � (3 � 10 �5 ) 3 � (3) 3 � (10 �5 ) 3
� 27 � 10 �15
b. (90,800,000) 2 � (9.08 � 10 7 ) 2 � (9.08) 2 � (10 7 ) 2
� 82.4464 � 10 14
� S
I
In particular, remember that the following operations are not the
same. One is the product of two numbers in the powers-of-ten format,
while the other is a number in the powers-of-ten format taken to a
power. As noted below, the results of each are quite different:
(10 3 )(10 3 ) � (10 3 ) 3
(10 3 )(10 3 ) � 10 6 � 1,000,000
(10 3 ) 3 � (10 3 )(10 3 )(10 3 ) � 10 9 � 1,000,000,000
Fixed-Point, Floating-Point, Scientific,
and Engineering Notation
There are, in general, four ways in which numbers appear when using a
computer or calculator. If powers of ten are not employed, they are
written in the fixed-point or floating-point notation. The fixed-point
format requires that the decimal point appear in the same place each
time. In the floating-point format, the decimal point will appear in a
location defined by the number to be displayed. Most computers and
calculators permit a choice of fixed- or floating-point notation. In the
fixed format, the user can choose the level of precision for the output as
tenths place, hundredths place, thousandths place, and so on. Every output
will then fix the decimal point to one location, such as the following
examples using thousandths place accuracy:
1
1
2300
� � 0.333 � � 0.063 � � 1150.000
3
16
2
If left in the floating-point format, the results will appear as follows
for the above operations:
1
1
2300
� � 0.333333333333 � � 0.0625 � � 1150
3
16
2
Powers of ten will creep into the fixed- or floating-point notation if the
number is too small or too large to be displayed properly.
Scientific (also called standard) notation and engineering notation
make use of powers of ten with restrictions on the mantissa (multiplier)
or scale factor (power of the power of ten). Scientific notation requires
that the decimal point appear directly after the first digit greater than or
equal to 1 but less than 10. A power of ten will then appear with the
number (usually following the power notation E), even if it has to be to
the zero power. A few examples:
1
1
2300
� � 3.33333333333E�1 � � 6.25E�2 � � 1.15E3
3
16
2
Within the scientific notation, the fixed- or floating-point format can
be chosen. In the above examples, floating was employed. If fixed is
chosen and set at the thousandths-point accuracy, the following will
result for the above operations: