5
Powers of Ten
30
5.0 Powers of Ten
Powers of ten may be used when working with
very small numbers.
For
example
very large or
654900000 is a very large number, but using powers of ten, it can
be written as 6.549 x 108.
0.0000000032 is a very small number. Using powers of ten it can be
written as 3.2 x 10-9.
A power of ten, is the number ten multiplied by itself a number of
times, according to the power – the little superscript number.
power
104
So 104 is ten multiplied by itself 4 times:
10 x 10 x 10 x 10= 10000
Complete these:
Power?
1
10 x 10 x 10 x 10 x 10 = 10
= 10 0000
Power?
2
3.0 x 10 x 10 x 10 = 3.0 x 10
= 3000
Power?
3
2.5 x 10 x 10 x 10 x 10 = 2.5 x 10
= 25000
notice how many places the decimal point has moved to the right:
it should correspond to the power of ten.
Convert these powers of ten into ordinary numbers:
4
5.6 x 103 =
5
8.27 x 106 =
31
?
What about negative powers, like 10-1?
10-1
= 1 = 0.1
10
In the same way, 10-4
=
1
10 x 10 x 10 x 10
= 0.0001
decimal point moved 4
places to the left
and 2.0 x 10-4
= 2.0 x 0.0001
= 0.0002
Again, notice how the power of ten gives the number of places
moved by the decimal point: this time to the left because the power
is negative.
Convert these powers of ten into ordinary numbers:
-3
6
3.0 x 10
=
7
6 x 10
8
2.4 x 10
9
7.34 x 10
10
645.56 x 10 =
-9
=
-2
=
-5
=
-3
32
5.2 Powers of ten and your calculator
Using a basic scientific calculator, powers of ten can be input by
using the EXP (EE on Texas calculators) key.
For example, to input 6 x 105 follow this sequence:
1
key in 6
display
6.
2
press EXP key
display
6.00
The EXP stands for
exponent, another word
for power
This display represents
6 x 105
3
key in 5
display
6.05
To input 3 x 10-2 follow this sequence:
1
key in 3
display
2
press EXP
display
3
press +/- key
display
3.
3.00
3.-00
This display
represents 3 x 10-2
4
key in 2
display
3.-02
Use a calculator for these exercises:
/ is divided by
11
2 x 102 x 4 x 103
=
12
4 x 105 x 6 x 102
=
13
5.2 x 103 / 2.56 x 109
=
14
2.0 x 10-3 x 4.1 x 10-7
=
15
9.98 x 10-6 x 1.54 x 103
=
5.3 SI units and prefixes
33
The international system of units, SI (systeme internationale),
includes metre (m), kilogram (kg), second (s), volt (V), ampere (A),
hertz (Hz), joule (J), farad (F), becquerel (Bq), watt (W), newton (N).
Each unit may be prefixed by a symbol, such as kV representing
“kilovolts”, which multiplies the value by a power of ten from the
following table:
symbol
T
G
M
k
h
da
d
c
m
n
p
f
value
1012
109
106
103
102
101
10-1
10-2
10-3
10-6
10-9
10-12
10-15
The prefix represents a power of ten.
prefix
tera
giga
mega
kilo
hecto
deka
deci
centi
milli
micro
nano
pico
femto
For
example
60 kV = 60 x 103 V = 60 x 1000 = 60 000 V
Notice, the decimal point moves 3 places to the right …
Another
example
:
55 m = 55 x 10-6 m= 55 x 0.000001= 0.000055 m
The decimal point has moved 6 places to the left …
Convert these into ordinary numbers:
16
6 mm =
m
17
1.54 km =
m
18
4.5 MHz =
Hz
19
0.3 s =
s
20
20 kHz =
Hz
34
5.4 Changing between prefixes
Imagine the prefixes arranged on a number scale:
10-15 10-12 10-9
f
p
10-6
n
10-3
100
103
106
109
1012
m
base
k
M
G
T
To change between prefixes, for example from 60 m to mm,
calculate the difference in the powers of ten:
from micro to milli is from 10-6 to 10-3, a difference of 10-3.
So 60 m
=
or
60 x 10-3 mm
0.06 mm
taking the decimal point 3
places left
To change from 350mV to kV, from milli to kilo is from 10-3 to 103,
the difference in powers of ten is 10-6.
Up the scale,
decimal point
moves left
so 350 mV
=
350 x 10-6 kV
or
0.00035 kV
decimal point moved 6 places
to left
To change from 2 MHz to Hz, from mega to base is from 10 6 to 100:
the difference in powers of ten is 106.
Down the
scale, decimal
point moves
right
so 2 MHz
=
or
2 x 106 Hz
2 000 000 Hz
decimal point moved 6 places
to right
Try these:
21
0.4 mm to m
22
30 s to s
23
0.000002 MW to mW
24
50 kJ to J
25
20 000 Hz to MHz
5.5 Answers to exercises
35
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
105
3.0 x 103
2.5 x 104
5600
8270000
0.003
0.000000006
0.024
0.0000734
0.64556
800000 or 8 x 105
2.4 x 108
2.03125 x 10-6
8.2 x 10-10
0.0153692 or 1.53692 x 10-2
0.006 m
1540 m
4500000 Hz
0.0000003 s
20000 Hz
36