Scientific Notation
In science one often needs to work with very large or very small numbers. These can be written more easily (and more compactly) in scientific notation, in a general form which we will look at in this lesson.
Scientific Notation
In science one often needs to work with very large or very small numbers. These can be written more easily (and more compactly) in scientific notation, in the general form:
\(N × 10^n\)
where \(N\) is a decimal number between 0 and 10 that is rounded off to a few decimal places. \(n\) is known as the exponent and is an integer. If \(n > 0\), it represents how many times the decimal place in \(N\) should be moved to the right. If \(n < 0\), then it represents how many times the decimal place in \(N\) should be moved to the left. For example 3.24 × 103 represents 3240 (the decimal moved three places to the right) and 3.24 × 10−3 represents 0.003 24 (the decimal moved three places to the left).
If a number must be converted into scientific notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 (i.e. the value of \(n\)) and what this number between 1 and 10 is (the value of \(N\)). We do this by counting the number of decimal places the decimal comma must move.
For example, write the speed of light (299,792,458 ms−1) in scientific notation, to two decimal places. First, we find where the decimal comma must go for two decimal places (to find \(N\)) and then count how many places there are after the decimal comma to determine \(n\).
In this example, the decimal comma must go after the first 2, but since the number after the 9 is 7, \(N = 3.00\). \(n = 8\) because there are 8 digits left after the decimal comma. So the speed of light in scientific notation, to two decimal places is 3.00 × 108 ms−1.
We can also perform addition, subtraction, multiplication and division with scientific notation. The following two worked examples show how to do this:
Example 1: Addition and Subtraction with Scientific Notation
Question
\(1.99 × 10^{-26} + 1.67 × 10^{-27} \; - \; 2.79 × 10^{-25}\)
Answer
To add or subtract numbers in scientific notation we must make all the exponents the same:
1.99 × 10−26 = 0.199 × 10−25 and
1.67 × 10−27 = 0.0167 × 10−25
Now that the exponents are the same we can simply add or subtract the \(N\) part of each number:
\(0.199 + 0.0167 \; - \; 2.79 = -2.5743\)
Write the final answerTo get the final answer we put the common exponent back:
\(−2,5743 × 10^{-25}\)
Note that we follow the same process if the exponents are positive. For example \(5.1 × 10^3 + 4.2 × 10^4 = 4.71 × 10^4.\)
Example 2: Multiplication and Division with Scientific Notation
Question
\(1.6 × 10^{-19} × 3.2 × 10^{-19} ÷ 5 × 10^{-21}\)
Answer
For multiplication and division the exponents do not need to be the same. For multiplication we add the exponents and multiply the \(N\) terms:
\(1.6 × 10^{-19} × 3.2 × 10^{-19} = (1.6 × 3.2) × 10^{-19 + (-19)} = 5.12 × 10^{-38}\)
For division we subtract the exponents and divide the \(N\) terms. Using our result from the previous step we get:
\(2.56 × 10^{-38} ÷ 5 × 10^{-21} = (5.12 ÷ 5) × 10^{-38 - (-19)} = 1.024 × 10^{-18}\)
The answer is: \(1.024 × 10^{-18}\)
Note that we follow the same process if the exponents are positive. For example: \(5.1 × 10^{-3} × 4.2 × 10^4 = 21.42 × 10^7 = 2.142 × 10^8\)
This lesson is part of:
Skills for Science