THE SUN-EARTH SYSTEM
APPENDIX I
Scientific Notation
Science deals with very large and very small
numbers, numbers that are inconvenient to
write in the everyday “long form.” If we need
to write a number like one hundred fifty, it’s
easy and quick—150. But suppose, instead, we
are dealing with a number like one hundred
fifty million, the number of kilometers between
the Sun and the Earth. That is a little harder—
150,000,000. In fields of science like astronomy,
even this is a relatively small number. Numbers
like a million million million million are not
uncommon. We see, then, that it would be
helpful to have a “shorthand” way to write
very large (and very small) numbers. Scientific
notation is a convenient way to write such
numbers. It involves expressing numbers in
powers of ten, using a superscript or exponent.
The exponent gives the number of zeros to add
after 1. For example,
such as 1/1,000,000. We may write this as
0.000001. In scientific notation this number is
written as 1.0 x 10-6. When the exponent is negative, we move the decimal point to the left instead of the right.
Multiplying and dividing numbers written in
scientific notation is especially easy, when you
get the hang of it. Simply multiply or divide the
numbers in front of the tens and add or subtract, respectively, the exponents. As an example, let us multiply 4.0 x 108 and 2.0 x 10-4.
(4.0 x 108) x (2.0 x 10-4) = (4.0 x 2.0) x (108+ (-4))
= (8.0) x (104)
= 8.0 x 104.
Now let us divide the same numbers.
(4.0 x 108)/(2.0 x 10-4) = (4.0/2.0) x (108-(-4))
= (2.0) x (108+4)
= 2.0 x 1012.
101 = 10
102 = 100
103 = 1,000
1012 = 1,000,000,000,000.
You might want to check these results by
writing the numbers out in the long form. If
you do you will see how much easier and
quicker using scientific notation can be.
If you need to add or subtract numbers written in scientific notation, you must first write
the numbers so that they are given in the same
power of ten. Then just add or subtract as you
normally would. For example, to add 2.46 x 103
to 5.23 x 104 we would do the following:
This system can be applied to any number.
Suppose, for example, we wanted to write the
number 5,280, the number of feet in a mile.
Since 5,280 is 5.280 times 1,000, we may write it
as 5.280 x 103. (Always place the decimal point
between the first and second number when
using scientific notation.) The exponent tells us
how many places to move the decimal point to
the right to express the number in the long
form.
We may also need to write fractions, numbers less than one, that are extremely small,
2.46 x 103 + 5.23 x 104 = 2.46 x 103 + 52.3 x 103.
Now we have expressed both numbers to the
same power of ten, namely 3.
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APPENDIX I
The result is 54.76 x 103. We would then
round this number to 54.8 x 103 and, to follow
our convention, move the decimal point to between the first and second number and raise the
power of 10 by 1, that is, to 4. This would finally
give us the answer 5.48 x 104. The complete
process looks like this:
Scientists also use prefixes to indicate powers
of ten in describing large and small numbers.
For instance, we use nanometer in talking about
wavelengths. A nanometer is a meter times 10-9.
The following table lists some common prefixes
and a few familiar examples.
2.46 x 103 + 5.23 x 104 = 2.46 x 103 + 52.3 x 103
= 54.76 x 103
= 54.8 x 103
= 5.48 x 104.
Table 3
Scientific Prefixes for Common Powers of Ten
Factor
Prefix
Symbol
1012
109
tera
giga
T
G
106
103
102
101
10–1
10-2
10-3
10-6
10-9
10–12
10–15
10–18
mega
kilo
hecto
decka
deci
centi
milli
micro
nano
pico
femto
atto
M
k
h
da
d
c
m
µ
n
p
f
a
Example
GHz – gigahertz (Hz, hertz, means cycles
per second)
MHz – megahertz
km – kilometer
cm – centimeter
mm – millimeter
mm – micrometer
nm – nanometer
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