Scientific Notation
Astronomy deals with big numbers. Really big numbers. It’s impossible
to talk about the distance to the Sun or the speed of light without thinking
about the tremendously huge. But big numbers intrude on all aspects of our
lives and as responsible citizens, we ought to have a way of dealing with
them. You need only drive past a McDonald’s restaurant to see that numbers
larger than a billion (or however many they’ve served by now) are
commonplace in our society. On a more serious note, a person must have an
understanding of the word "trillions" in order to comprehend the size of
the National Debt.
Mathematicians and scientists use scientific
notation to handle large numbers. Scientific notation is necessary
when discussing astronomical quantities, unless we want to write out a lot
of zeros ("Excuse me, Dr. Sagan, did you say the Universe began 15,000,000,000
years ago or 150,000,000,000 years ago?"). Scientific notation is also essential
for dealing with extremely small numbers, such as the mass of a Hydrogen
atom or the wavelength of visible light.
The following table and tips are intended to be used as a reference:
Words |
Decimal
Representation |
Scientific
Notation |
Metric
Prefix |
Symbol |
| one billionth |
0.000000001 |
1 x
10-9 |
nano |
n |
| one
hundred-millionth |
0.00000001 |
1 x
10-8 |
|
|
| one ten-millionth |
0.0000001 |
1 x
10-7 |
|
|
| one millionth |
0.000001 |
1 x
10-6 |
micro |
m |
| one
hundred-thousandth |
0.00001 |
1 x
10-5 |
|
|
| one ten-thousandth |
0.0001 |
1 x
10-4 |
|
|
| one thousandth |
0.001 |
1 x
10-3 |
milli |
m |
| one hundredth |
0.01 |
1 x
10-2 |
centi |
c |
| one tenth |
0.1 |
1 x
10-1 |
deci |
d |
| one |
1 |
1 x
100 |
|
|
| ten |
10 |
1 x
101 |
|
|
| one hundred |
100 |
1 x
102 |
|
|
| one thousand |
1,000 |
1 x
103 |
kilo |
k |
| ten thousand |
10,000 |
1 x
104 |
|
|
| one hundred
thousand |
100,000 |
1 x
105 |
|
|
| one million |
1,000,000 |
1 x
106 |
Mega |
M |
| ten million |
10,000,000 |
1 x
107 |
|
|
| one hundred
million |
100,000,000 |
1 x
108 |
|
|
| one billion |
1,000,000,000 |
1 x
109 |
Giga |
G |
| ten billion |
10,000,000,000 |
1 x
1010 |
|
|
| one hundred
billion |
100,000,000,000 |
1 x
1011 |
|
|
| one trillion |
1,000,000,000,000 |
1 x
1012 |
Tera |
T |
Hints
-
A number written in scientific notation consists of a
coefficient
(the part before the times sign) and an
exponent (the power of 10 by which the
coefficient is multiplied. For example, in 4.3 x 106
(which equals 4,300,000; four million three hundred thousand),
4.3 is the coefficient and 6 is the exponent. Sometimes the "times" symbol
"x" is replaced by a dot, for example
4.3.106.
-
When you multiply two numbers, you multiply the coefficients
and add the exponents. For example,
4.3 x 106 x 2 x 102 =
8.6 x 108
4.3 x 106 x 2 x 10-2 = 8.6
x 104
-
When you divide two numbers, you divide the coefficients and
subtract the exponents. For example,
4.2 x 106 / 2 x 102 = 2.1
x 104
4.2 x 106 / 2 x 10-2 = 2.1
x 108
-
When you move the decimal place in the coefficient one position to the
left (i.e. you divide the coefficient by 10), you add one to
the exponent. For example,
42 x 106 = 4.2 x 107
4200 x 106 = 4.2 x 109
42 x 10-6 = 4.2 x 10-5
-
When you move the decimal place in the coefficient one position to the
right (i.e. you multiply the coefficient by 10), you subtract one
from the exponent. For example,
0.42 x 106 = 4.2 x 105
0.000043 x 106 = 4.3 x
101
0.42 x 10-6 = 4.2 x
10-7
Always adjust the decimal place in the coefficient so that the coefficient
is always greater than one but less than ten. Mathematically it doesn't make
any difference, but that is the standard practice, and it does make a number
easier to read.
-
When you add two numbers, you need to make their exponents equal.
Take the number with the smaller exponent and move the decimal point to the
left until its exponent matches the larger. Then add the coefficients
and keep the (matching) exponent. For example,
4.2 x 106 + 6.4 x 105 =
4.2 x 106 + 0.64 x 106 = 4.84 x 106
4.2 x 10-6 + 6.4 x 10-5 =
0.42 x 10-5 + 6.4 x 10-5 = 6.82 x 10-5
9.2 x 1011 + 9.4 x 1010 =
9.2 x 1011 + 0.94 x 1011 = 10.14 x 1011
= 1.014 x 1012
-
When you subtract two numbers, you again need to make their exponents
equal. Take the number with the smaller exponent and move the decimal
point to the left until its exponent matches the larger. Then subtract
the coefficients and keep the (matching) exponent. Note that you
might have to adjust the exponent when you are done to get into "standard
form." For example,
4.2 x 106 - 6.4 x 105 =
4.2 x 106 - 0.64 x 106 = 3.56 x 106
4.2 x 10-6 - 6.4 x 10-5 =
0.42 x 10-5 - 6.4 x 10-5 = -6.38 x
10-5
1.2 x 1011 - 9.4 x 1010 =
1.2 x 1011 + 0.94 x 1011 = 0.26 x 1011 =
2.6 x 1010
-
WARNING TO PEOPLE WHO USE CALCULATORS: Many
calculators handle scientific notation. The exponent is usually displayed
all the way on the right, with a space between it and the coefficient. To
enter a number in scientific notation, you enter the coefficient, press the
EXP key (one some calculators it is labeled
EE) and enter the exponent. For example,
4. 05 means 4 x 105 . To enter the
number 103, you have to enter 1. EXP
03. DO NOT enter 10. EXP 03,
since that equals 10 x 103 or 1 x 104. This is a common
mistake. See your calculator instruction booklet for more help.