Scientific Notation
When we attempt to write very large and very small numbers, it becomes difficult to write out all the zeros present. Numbers like 602,000,000,000,000,000,000,000 and 0.000000000000000000162 are difficult to write and are not easy to make out. Therefore, scientists and mathematicians have developed a way to list very large and very small numbers with just the significant figures, omitting the insignificant zeros, and then multiply the number by a factor of 10. The process is quite simple.
Example 1:
1) For a number like 103,000,000 we write the significant digits, leaving out any insignificant zeros.
Example 1:
1) For a number like 103,000,000 we write the significant digits, leaving out any insignificant zeros.
103
2) Then, we place a decimal after the first significant digit:
1.03
3) Then, to make 1.03 look like 103,000,000 it must be multiplied by 100,000,000, or 108 (10 to the 8th power). Therefore, our number in scientific notation is:
1.03 x 108
Another way of looking at this is to say 1.03 and 103,000,000 have all the same significant numbers. They differ in where the decimal place is. Well, if I wanted to make 1.03 into 103,000,000 all I would really need to do is move the decimal. If I move the decimal 8 times to the right on 1.03, I get 103,000,000. Whenever I move the decimal, I actually multiply or divide by 10. In this case, I have multiplied the number 1.03 by 10 eight times.
When you are dividing the number, then the exponent becomes negative.
When you are dividing the number, then the exponent becomes negative.
Example 2: 0. 0.00045 becomes 4.5 x 10-4, because we need to divide 4.5 by 10,000 to get it to become 0.00045.
To use the shortcut way of doing this, because we are multiplying or dividing the sig figs by a factor of 10 would be to move the decimal. Every time you move the decimal one place, you multiply or divide by 10. In example 1, to get the decimal between the 1 and 0, it was necessary to move the decimal 8 times to the left, hence the exponent 8 on the factor of 10. In example 2, we moved the decimal 4 times to the right, but since we are going right instead of left (dividing instead of multiplying), the exponent is a -4.
Scientific Notation can always be transformed into standard notation by the opposite process. Hence, the number 1.24 x 105, becomes 124,000 in standard notation. We have simply multipied by 100,000 or moved the decimal 5 places to the right.
Ex. 3: 9.003 x 10-7 becomes 0.0000009003
Ex. 4: 6.07 x 100 becomes 6.07
Ex. 4: 6.07 x 100 becomes 6.07