Scientific Notation

The numbers encountered in this class are, by definition,
astronomical.
The nearest star is 41,000,000,000,000 kilometers
away. The mass of the sun is
2,000,000,000,000,000,000,000,000,000,000,000
grams. When written out completely, as above, these numbers
are unwieldy and difficult to digest at first glance. Scientists
have devised a more compact notation for dealing with such
numbers called ``scientific notation". If you think about
it, there are really only two important parts to each of the
numbers above

- The leading digits - which establish the precision of the
number itself.
- The number of digits - which sets the size or
magnitude of the number.

Scientific notation strips a number down to these two constituent
parts. In this notation the two numbers above are
4.1 x 10^{13} and 2.0 x 10^{33} .
The digits to the left of the `` " establish the
precise value of the number. The exponent of the 10
tells you how many decimal places to move the decimal point
either to the right (if the exponent
is positive) or to the left (if the exponent is negative).

The `` x " is, as it appears, a multiplication.

- 10
^{2} = 10 x 10 = 100
- 10
^{3} = 10 x 10 x 10 = 1,000
- 10
^{4} = 10 x 10 x 10 x 10 = 10,000
- 10
^{5} = 10 x 10 x 10 x 10 x 10 = 100,000
- 10
^{6} = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 ...

so 4.5 x 10^{4} = 4.5 x 10 x 10 x 10 x 10 = 45,000.
The trick above (moving the decimal place) is just a simple
way of carrying out the multiplication.

Multiplying numbers expressed in scientific notation is
not difficult. Adding, however, may take a little thought. The
best way to see how it works is to write things out as above.

Consider multiplying 10^{3} by 10^{5} (actually 1.0 x 10^{3} and 1.0 x 10^{5}, but we can ignore the multiplication by 1 x 1).

10^{3} x 10^{5} = (10 x 10 x 10) x (10 x 10 x 10 x 10 x 10) = 10^{8}

The result of that long string of multiplication is that
we **add** the exponents of the 10's. 3+5=8.

As mentioned above, addition requires a little more attention as
can be seen from the example below using the same pair of numbers.

10^{3} + 10^{5} = 10 x 10 x 10 + 10 x 10 x 10 x 10 x 10 = 1,000 + 100,000 = 101,000 = 1.01 x 10^{5}

W