NATS 1750
'The Earth and Its Atmosphere'
Tutorial #1
Scientific Notation
1. Powers of Ten
Throughout NATS 1750 we are going to have to deal with some very small numbers and
some very large numbers.
You have already seen some examples of this, e.g. time since the Big Bang is 15 billion
years - 15,000,000,000 years; and at the other extreme the fraction of carbon dioxide in
our atmosphere today is 0.00038.
To save time with writing out all of these zeros scientists use a shorthand system called
'Scientific Notation'
Instead of writing 15,000,000,000 years we just write 15x109 years
Instead of writing 10,000,000,000 C we just write 1x1010 C
And instead of saying "15 billion years" or "15 thousand million years" or "15 zero zero
zero zero zero zero zero zero zero years " (all of which are the same) we could say:
"1.5 times 10 to the10" years. - written as 1.5x1010
For example
100 is "1 times 10 to the 2" or 1x102
200 is "2 times 10 to the 2" or 2x102
300 is "3 times 10 to the 2" or 3x102
1,000 is "1 times 10 to the 3" or 1x103
3,000 is "3 times 10 to the 3" or 3x103
20,000,000,000 is "2 times 10 to the 10" or 2x1010
The expression 10n is called the nth power of ten and represents the number that is equal
to 10 multiplied by itself n-1 times. The number n here is called the exponent. In other
words:
101 = 10
102 = 10 x 10 =100
103 = 10 x 10 x 10 =1000
104 = 10 x 10 x 10 x 10 =10000
Note that 100 is equal to 1 (but that is not perhaps very obvious!)
2. Powers of Ten with Negative Exponents
Numbers smaller than one can be represented using negative exponents. For example,
0.1, which is the same as 1/10 or 1/101, can be written as 10-1. In other words:
0.1 = 1/10 = 1/101 = 10-1
0.01 = 1/100 = 1/102 = 10-2
0.001 = 1/1000 = 1/103 = 10-3
0.0001 = 1/10000 = 1/104 = 10-4
Q. What about 0.000000001? How would we write that?
Answer. The rule is: count the number of times you have to move the decimal point to
the right to pass the first non-zero digit - this gives you the value of the negative
exponent.
Therefore 0.000000001 can be written as 10-9 and spoken as "10 to the minus 9"
3. Scientific Notation
Any number can be written as a decimal number between 1 and 10 (called the digit term)
multiplied by a power of 10.
For example 357 can be written as 3.57 x 102 (I know this is actually more writing but we
will see why it is useful shortly)
Similarly 357.6 can be written as 3.576 x 102.
Note that a nice number like 300 can be written as either 3.0 x 102 or just 3 x 102 without
the decimal point.
The exponent required to write 357.6 in scientific notation is found by counting how may
times you have to move the decimal point to the left in order to make it a number
between 1 and 10 (or really between 1 and 9.999999 ...... but let's not worry about that!)
We can do the same thing with numbers smaller than 1 by using a negative power after it.
For example 0.000054 can be written as 5.4 x 10-5. In this case the rule for finding the
value of the negative exponent is to count the number of times you have to move the
decimal point to the right to get a number between 1 and 10.
4. Multiplication with Powers of Ten
The product of any two powers of ten is another power of ten:
102 x 103 is 100 x 1000 which equals 100000 and which we can write as 1 x 105 using the
rules above.
Note that the value of the exponent of the product is just the sum of the exponents of the
things we are multiplying - in this case 2 + 3 = 5 and we only have to know how to add
not multiply!!!
Similarly:
103 x 106 = 109
103 x 10-1 = 102
102 x 10-3 = 10-1
i.e. 3 + 6 = 9
i.e. 3 + (-1) = 2
i.e. 2 + (-3) = -1
5. Division with Powers of Ten
Division of powers of ten can be carried out in a similar way. The result of the division
(called the quotient) is again a power of ten but in this case the exponent of the quotient
is the difference between the exponents of the number on top and the number below, i.e.
the exponent of the number on top minus the exponent of the number on the bottom.
For example:
105/102 is 10000/100 which equals 1000 and which can be written as 103.
Similarly:
106/102
104/10-2
10-2/103
10-2/10-3
= 104
= 106
= 10-5
= 101
i.e. 6 - 2 = 4
i.e. 4 - (-2) = 6
i.e. (-2) - 3 = -5
i.e. (-2) - (-3) = +1
6. Calculations with Scientific Notation
Multiplication of two numbers written in scientific notation is carried out as follows:
First multiply their two digit terms as you normally would and convert that product to
scientific notation. Then multiply their two powers of ten as described above. Then
combine the two powers of ten. Got it ? Perhaps the example below is better.
What is 6.75 x 103 multiplied by 7.2 x 104 ?
(6.75 x 103) x (7.2 x 104)
= (6.75 x 7.2) x (103 x 104)
= 48.6
x (103 x 104)
= 4.86 x 101 x (103 x 104)
= 4.86 x 101 x (107)
= 4.86 x (101 x 107)
= 4.86 x 108
For division you do something similar.
First divide their two digit terms as you normally would and convert that quotient to
scientific notation. Then divide their two powers of ten as described earlier. Then
combine the two powers of ten.
For example:
(4.86 x 108)/(7.2 x 104)
= (4.86/7.2) x (108/104)
= 0.675
x (108/104)
= 6.75 x 10-1 x (108/104)
= 6.75 x 10-1 x ( 104)
= 6.75 x (10-1 x 104)
= 6.75 x 103
Addition and subtraction in scientific notation is actually harder than multiplication and
division!
The procedure is as follows:
If the two numbers do not have the same power of ten undo the scientific notation of the
bigger number until it has the same power of ten as the smaller one. Add (or subtract) the
new digit numbers as you normally would. Convert this to proper scientific notation and
combine the two powers of ten. Again an illustration is probably better!
For example, what is 6.75 x 103 plus 7.2 x 104 ? Answer:
(6.75 x 103) + (7.2 x 104)
= (6.75 x 103) + (72 x 103)
= 6.75 + 72 x (103)
= 78.75
x (103)
1
= 7.875 x10 x (103)
= 7.875 x (101x 103)
= 7.875 x 104
Exercises: Divide 1.8 x 10-7 by 4.8 x 10-5; Subtract 1.32 x 106 from 2.6 x 107
7. Metric Prefixes
All scientific units for things like distance or weight can be written with prefixes that
reduce the need to express these quantities in scientific notation.
For example 1000 metres, which is 1x103 metres, can be expressed as 1 kilometre or
1000 grams can be expressed as 1 kilogram where kilo is a prefix.
Prefixes can be used for things which are much smaller than the basic unit as well as for
things that are much larger than the basic unit. For example one millionth of a metre is
called a micrometre and one thousandth of a gram is called a milligram. These prefixed
units also have their own additional shorthand symbols. For example 1 micrometre can
be written as 1 m.
The list of scaling factors, prefixes and symbols is as follows
Factor Prefix Symbol
10-9
10-6
10-3
10-2
10-1
nano
micro
milli
centi
deci
n

m (not to be confused with m for metre ! Sorry it wasn't my idea!)
c
d
101
102
103
106
109
deca
hecto
kilo
mega
giga
da
h
k
M
G
Don't worry - we will only be using a few of these mostly micro, kilo, mega and giga.
Some more examples:
1 megatonne = 1x106 tonnes; 1 gigawatt = 1 GW =1x109 watts
1 millilitre = 1ml = 1x10-3 litre
For a neat video visit : http://www.youtube.com/watch?v=0fKBhvDjuy0