Scientific Notation
How do we write and manipulate large numbers? Consider the value of 6,800 Km we encountered as an estimate of the radius of the Earth by Erathostenes. He couldn't have measured the Earth's radius down to the last Km because of the imprecision of his measurement tools. It is awkward to carry all the zeros around; let's simplify the writing of numbers by using the scientific notation: 6,800 can be written as 6.8 x 103, which means 6.8 multiplied by 1,000. The exponent gives the number of zeros after the digit "1" (and NOT after the number "10"). Often the multiplication sign is omitted: 6.8 103.Suppose you want to multiply the number 12,000 by 300.
12,000 x 300 = 3,600,000 or 1.2 104 x 3 102 = (1.2 x 3) 10(4+2) = 3.6 106.
Notice that the exponents (4 and 2) are added to each other. We say: 3.6 "times ten to the power of six", or more briefly, 3.6 "ten to the 6".
The division of numbers proceeds in the same way:
1.2 104 / 3 102 = (1.2/3) 10(4-2) =0.4 102 = 4 10(-1+2)= 4 101 =40.
Here we subtracted the exponents since we performed a division. For example, the number 0.4 can be written as 4/10 or 4 10-1 etc. What is 100? It is equal to 1. So, 4.0 = 4 100
Suppose you have this number: 4,567,000,000. How can you write it in scientific notation? We want to write it as: digit (first digit from left rather than "0") - followed by decimal point - followed by whatever digit rather than zero - multiplied by 10number of digits after the first digit. Count the number of digits after the first digit: there are nine. So: 4.567 109.
One more example. How can I write 0.0003005 in scientific notation?
The first non-zero digit from the left is "3". Thus: 3.005 10number of digits. Count the number of places you have to move the decimal point: there are 4 places. Because 0.0003005 is less than 1, the exponent will be negative, i.e., -4. Thus, 3.005 10-4.
Notice that these rules of addition/subtraction of exponents work only in the case you multiply/divide two numbers. If you have to add/substract two numbers, you have to do it the usual way, by writing out the numbers with all the zeros. For example: 1.23 + .00000056 = 1.23000056