Scientific Notation
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Often in astronomy numbers get too big to be conventionally represented.
For example the distance from the Sun to the Andromeda galaxy is about
12,000,000,000,000,000,000 miles. The scientific notation gives us a simple
way of simplifying the way these large numbers are represented.
The reverse can also be true, that is a number can be too small to represent
easily using standard notation. Scientific notation takes care of these
problems.
In the scientific notation all numbers are represented as a number which
lies between 1 and 9.99999999.... times a power of ten. For example the
number 2,500 can be written as 2.5 x 1000. The last factor is then written
as 1000 = 10x10x10 = 10^{3}. Thus in scientific notation 2,500
= 2.5 x 10^{3}, or "two point five times ten to the power of three".
The first number (in this case 2.5) is known as the mantissa, and the power
of ten (in this case 3) as the exponent.

Large numbers For large numbers the power is always positive, as
in the above example. The power is equal to the number of places in the
first expression after the first (non zero) digit. 40,000,000 has 7 zeroes
after the four, and so is the number 4 x 10^{7}. 315,000,000,000
has 11 places after the 3, and so is 3.15 x 10^{11}.

Small numbers The same principle can be applied to small numbers,
such as 0.00000000045. In this case the power is a negative number, equal
to the number of zeroes in the number after the decimal point and before
the first nonzero digit, plus one. The above number has 9 zeroes
after the '.' and before the '4', and so remembering to add 1 to this,
we get 4.5 x 10^{10}.
Try these:
What are the following in scientific notation? (Click the blue cross for
the answer.)

6,000,000,000,000,000,000,000,000 X

34,599,800,000 X

12.56 X

0.3456789 X

0.0000000332 X

0.000000000000000000000000000000123456789 X

6 trillion X
What are the following in standard notation?

3 x 10^{5} X

2.44 x 10^{7} X

6.02 x 10^{23} X

6.62 x 10^{34} X

1.5 x 10^{11} X

1.44 x 10^{1} X
Multiplication
When muliplying numbers in scientific notation, remember the following
rules: a) multiply the mantissas, and b) add the powers. Using this rule
4 x 10^{7} times 2 x 10^{4} is equal to (4 times 2) x 10^{(7+4)}
= 8 x 10^{11}.
If the new mantissa is now greater than ten, adjust by moving the decimal
point to the left, and increase the power by one. For example 3.45 x 10^{9}
times 6.7 x 10^{7} = (3.45 times 6.7) x 10^{(9+7)} = 23.115
x 10^{16} = 2.3115 x 10^{17}.
The rule works equally well if one or both of the powers is negative.
2 x 10^{6} times 3 x 10^{7} = (2 times 3) x 10^{(6+(7))}
= 6 x 10^{13}.
3 x 10^{12} times 4 x 10^{8} = (3 times 4) x 10^{(12+(8))}
= 12 x 10^{4} = 1.2 x 10^{5}.
Division
Division is similar, except you divide the mantissas and subtract the powers.
For example, 5 x 10^{8} divided by 2 x 10^{3} = (5 divided
by 2) x 10^{(83)} = 2.5 x 10^{5}
If the new mantissa is now less than one, move the decimal point to
the right, and decrease the power by one, as in 3 x 10^{6} divided
by 7.8 x 10^{11} = (3 divided by 7.8) x 10^{(611)} = 0.3846
x 10^{5} = 3.836 x 10^{6}.
Try these:

2 x 10^{9} times 4 x 10^{5} X

6.7 x 10^{16} times 8.44 x 10^{5} X

55.63 times 4 x 10^{5} (remember to converet the 55.63 to scientific
notation first) X

2.55 x 10^{7} times 6 x 10^{6} X

2.55 x 10^{7} divided by 6 x 10^{6} X

0.000000074 divided by 555,666,777,888 X

5 x 10^{5} divided by 5 x 10^{9} X
Addition
In adding numbers you must first make sure that all the terms have the
same power. If they do not then you will need to convert. With terms of
the same power, add the mantissas to get the new mantissa by the same power
that the original terms had. Thus, 3.00 x 10^{8} + 4.00 x 10^{8}
+ 3.66 x 10^{8} = (3.00 + 4.00 + 3.66) x 10^{8} = 10.66
x 10^{8}, which would then be written as 1.066 x 10^{9}.
On the other hand the sum 4.22 x10^{6} + 3 x 10^{4} cannot
be performed at present since the powers are different. First you must
convert one, say the latter, to rewrite the sum as 4.22 x10^{6}
+ 0.03 x 10^{6}, which can then be added to get (4.22+ 0.03) x
10^{6} = 4.25 x 10^{6}.
Subtraction is the same except you subtract the mantissas.
Try these:

2 x 10^{9} + 4 x 10^{9} X

6.7 x 10^{16} + 8.44 x 10^{16} X

5.563 x 10^{7} + 4 x 10^{5} X

2.55 x 10^{7} + 6 x 10^{8} X

2.55 x 10^{7}  6 x 10^{8} X

0.000000074  0.000000555666777888 X

5 x 10^{5}  5 x 10^{9} X
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