3.7 Scientific notation

Overview

• Standard Form to Scientific Form
• Scientific Form to Standard Form
• Working with Numbers in Scientific Notation

Standard form to scientific form

Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik’s cube) and very small numbers such as $0.000000000000000000000340$ (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Such numbers can be expressed more conveniently by writing them as part of a power of 10.

To see how this is done, let us start with a somewhat smaller number such as 2480. Notice that

$\begin{array}{ll}\underset{\text{Standard}\text{form}}{\underbrace{2480}}\hfill & =248.0\times {10}^1\hfill \\ \hfill & =24.80\times {10}^2\hfill \\ \hfill & =\underset{\text{Scientific}\text{form}}{\underbrace{2.480\times {10}^3}}\hfill \end{array}$

Scientific form

$\begin{array}{lllllll}0.00059\hfill & =\hfill & \frac{0.0059}{10}\hfill & =\hfill & \frac{0.0059}{{10}^1}\hfill & =\hfill & 0.0059\times {10}^{-1}\hfill \\ \hfill & =\hfill & \frac{0.059}{100}\hfill & =\hfill & \frac{0.059}{{10}^2}\hfill & =\hfill & 0.059\times {10}^{-2}\hfill \\ \hfill & =\hfill & \frac{0.59}{1000}\hfill & =\hfill & \frac{0.59}{{10}^3}\hfill & =\hfill & 0.59\times {10}^{-3}\hfill \\ \hfill & =\hfill & \frac{5.9}{10,000}\hfill & =\hfill & \frac{5.9}{{10}^4}\hfill & =\hfill & 5.9\times {10}^{-4}\hfill \end{array}$

There is one nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the right .

Scientific notation

Writing a number in scientific notation

1. Move the decimal point so that there is one nonzero digit to its left.
2. Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Sample set a

Write the numbers in scientific notation.