### 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 1013 and 2.0 x 1033 .

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.

• 102 = 10 x 10 = 100
• 103 = 10 x 10 x 10 = 1,000
• 104 = 10 x 10 x 10 x 10 = 10,000
• 105 = 10 x 10 x 10 x 10 x 10 = 100,000
• 106 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 ...
so 4.5 x 104 = 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 103 by 105 (actually 1.0 x 103 and 1.0 x 105, but we can ignore the multiplication by 1 x 1).

103 x 105 = (10 x 10 x 10) x (10 x 10 x 10 x 10 x 10) = 108

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.

103 + 105 = 10 x 10 x 10 + 10 x 10 x 10 x 10 x 10 = 1,000 + 100,000 = 101,000 = 1.01 x 105

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