1.2 Exponents and scientific notation  (Page 6/9)

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

Converting from scientific to standard notation

To convert a number in scientific notation    to standard notation, simply reverse the process. Move the decimal $n$ places to the right if $n$ is positive or $n$ places to the left if $n$ is negative and add zeros as needed. Remember, if $n$ is positive, the value of the number is greater than 1, and if $n$ is negative, the value of the number is less than one.

Using scientific notation in applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around $1.32\times {10}^{21}$ molecules of water and 1 L of water holds about $1.22\times {10}^4$ average drops. Therefore, there are approximately $3\cdot \left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^4\right)\approx 4.83\times {10}^{25}$ atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!