Selected Key Terms
Scientific A system for representing any number as a
notation number between 1 and 10 times a power of ten.
Exponent The number to which a base is raised.
Engineering A system for representing any number as a one-,
notation two-, or three-digit number times a power of ten
with an exponent that is a multiple of three.
Metric prefix A symbol that is used to replace the power of
ten in numbers expressed in scientific or
engineering notation.
Scientific and Engineering Notation
Very large and very small numbers are
represented with scientific and
engineering notation. This eliminates
leading and trailing zeros.
47,000,000 = 4.7 x 107 (Scientific Notation)
= 47. x 106 (Engineering Notation)
Scientific Notation
In scientific notation, each quantity is expressed as
a product of a positive number (usually between 1
and 10) and a power of ten. The power of ten is
expressed as an exponent of base 10.
Scientific Notation
1 0 7 = 1 0 ,0 0 0 ,0 0 0
1 0 6 = 1 ,0 0 0 ,0 0 0
1 0 5 = 1 0 0 ,0 0 0
1 0 4 = 1 0 ,0 0 0
1 0 3 = 1 ,0 0 0
102 = 100
1 01 = 1 0
100 = 1
1 0 − 1 = 0 .1
1 0 − 2 = 0 .0 1
1 0 − 3 = 0 .0 0 1
1 0 − 4 = 0 .0 0 0 1
Scientific Notation
Numbers (less than one) are expressed in scientific
notation with negative powers of ten.
.0035 = 3.5 x 10-3
Large numbers (greater than one thousand) are
expressed in scientific notation with positive
powers of ten.
3500 = 3.5 x 10+3
Scientific Notation
0.000027 = 2.7 x 10-5
0.605 = 6.05 x 10-1
8300 = 8.3 x 10+3
756000 = 7.56 x 10+5
Rules for Calculating
Addition requires that all added numbers be expressed
in the same power of ten. Convert numbers so they
have the same power of ten. Then add the numbers
without their powers of ten to get the sum. The power
of ten of the sum is the same as for the added
numbers.
(1 × 105 ) + (2 × 105 ) = (1 + 2 ) × 105 = 3 × 105
(3.7x10-5) + (62x10-6) = (37x10-6) + (62x10-6) = (37+62) x 10-6=99x10-6
Rules for Calculating
Subtraction also requires that all numbers be
expressed in the same power of ten. Subtract
the numbers without their powers of ten to get
the difference. The power of ten of the results
is the same as for the individual numbers.
(25 × 10−6 ) − (130 × 10−7 ) = (250 × 10−7 ) − (130 × 10−7 ) =
= (250 − 130) × 10− 7 = 120 × 10− 7
Rules for Calculating
Multiplication does not require any prior
calculations. The numbers are multiplied
directly without their powers of ten. Then the
powers of ten must be added algebraically.
( 3 × 10 6 ) × ( 2 × 10 4 ) = ( 3 × 2 ) × 106 + 4 = 6 × 1010
Rules for Calculating
Division does not require any prior calculations.
The numbers are divided directly without their
powers of ten. Then the power of ten of the
denominator is subtracted algebraically from
the power of ten of the numerator.
3
8
10
(8 × 103 ) ÷ (4 × 10 2 ) = × 2 = 2 × 103− 2 = 2 × 101
4 10
Engineering Notation
Engineering notation is an application of
SCIENTIFIC NOTATION in which the powers
of ten are limited to the multiples of three such
as 103; 109; 10-6;10-3. The reason for this is that
in engineering and technology all units are
expressed with prefixes that symbolize relevant
powers of ten. These are called metric prefixes
Engineering Metric Prefixes
1015
peta
P
1015
1012
tera
T
trillion
109
giga
G
billion
106
mega
M
million
103
kilo
k
thousand
Engineering Metric Prefixes
10-3
milli
m
one-thousandth
10-6
micro
µ
one-millionth
10-9
nano
n
one-billionth
10-12
pico
p
one-trillionth
10-15
femto
f
femto
Engineering Notation
0.000027 = 27 x 10-6 = 27
0.605 = 605 x 10-3 = 605m
8300 = 8.3 x 10+3 = 8.3k
756000 = 756 x 10+3 = 756k
0.0285 = 28.5 x 10-3 = 28.5m
Metric Conversions
When converting from a larger unit to a smaller unit,
move the decimal point to the right. Remember, a
smaller unit means the number must be larger.
Smaller unit
0.47 MΩ = 470 kΩ
Larger number
Metric Conversions
When converting from a smaller unit to a larger unit,
move the decimal point to the left. Remember, a larger
unit means the number must be smaller.
Larger unit
10,000 pF = 0.01 µF
Smaller number
Quiz Scientific and Engineering Notation
1.
a.
b.
c.
d.
Express in scientific notation: 77200
7.72 × 104
7.72 × 10–7
772 × 102
7.72 × 10–4
2.
a.
b.
c.
d.
Express in scientific notation: 0.000447
4.47 × 104
447 × 106
4.47 × 10–4
447 × 10–6
3.
a.
b.
c.
d.
Express in decimal notation: 2.58 × 105.
0.000258
258000
0.000000258
25800
Quiz Scientific and Engineering Notation
4.
a.
b.
c.
d.
Express in decimal notation: 9.09 × 10–4
9090
0.00909
0.000909
90900
5.
a.
b.
c.
d.
Express in decimal notation: 29300 µs
29.3 s
29300000 s
29300 s
0.0293 s
6.
a.
b.
c.
d.
Express in decimal notation: 552 ms
0.552 s
0.000552 s
55200000 s
552000 s
Quiz Scientific and Engineering Notation
7.
a.
b.
c.
d.
Express in decimal notation: 91.6 k meters (km)
0.0916 meters
0.0000916 meters
91600 meters
91600000 meters
8.
a.
b.
c.
d.
Express in decimal notation: 22.9 Mm
0.0000229 m
0.0229 m
22900 m
22900000 m
9.
a.
b.
c.
d.
Express in engineering notation: 88200 meters
88.2 µm
88.2 mm
88.2 km
88.2 Mm
Quiz Scientific and Engineering Notation
10 .
a.
b.
c.
d.
Express in engineering notation: 0.000771 meters
771 mm
771 µm
771 km
771 Mm
11 .
a.
b.
c.
d.
Convert to engineering notation. 11.8 × 10–4 meters
118 mm
1.18 Mm
1.18 km
1.18 mm
12 .
a.
b.
c.
d.
Convert to engineering notation. 63.9 × 104 meters
6.39 mm
639 km
63.9 km
639 mm
Quiz Scientific and Engineering Notation
13 .
a.
b.
c.
d.
Evaluate: 2.3 × 103 + 0.42 × 104 =
6.5 × 104
6.5 × 103
650
65000
14 .
a.
b.
c.
d.
Evaluate: 7800 µA + 0.23 mA =
10.1 µA
8.03 mA
8.03 µA
10.1 mA
15 .
a.
b.
c.
d.
Evaluate: (2.2 x 10-9)(7 x 10-6)
15.4 x 10-3
9.2 x 10-3
1.54 x 10-14
5.2 x 10+15