2.02 Compare and order numbers in scientific notation
Introduction
When we compare any numbers, it's easiest to compare them when they're in the same form.
For example, if we are asked to compare the numbers \dfrac{1}{2} and 0.2, it will be easiest to compare them if they are either both fractions or both decimals. Let's convert \dfrac{1}{2} to be a decimal:
| \displaystyle \dfrac{1}{2} | \displaystyle = | \displaystyle 0.5 |
Now that our two numbers are both in the same, decimal form, we can more easily compare them by inspection and determine that 0.5 > 0.2 or \dfrac{1}{2} > 0.2.
We'll now consider how to compare and order numbers in scientific notation.
Order and compare numbers in scientific notation
We can compare numbers in scientific notation by looking at their powers of 10. The bigger the power, the bigger the number. Alternatively, the smaller the power, the smaller the number. In the event that the powers of 10 are equal, we can then compare the decimal number.
Let's compare the following two numbers: 8 \times 10^{6} and 6 \times 10^{4}.
Since 8 \times 10^{6} has a 10 to the power of 6, which is greater than a power of 4 as seen in 6 \times 10^{4}, it means that 8 \times 10^{6}>6 \times 10^{4}.
Now let's compare the following two numbers: 3400000 and 3.2 \times 10^{6}.
As discussed in the introduction, when comparing numbers, it's easiest to compare them when they're in the same form. In this case, we can choose to either look at both numbers in standard form, or scientific notation. Let's convert 3400000 into scientific notation.
| \displaystyle 3400000 | \displaystyle = | \displaystyle 3.4 \times 10^{6} |
Our two numbers are: 3.4 \times 10^{6} and 3.2 \times 10^{6}. Since the powers of 10 are both 6, we will compare the decimals. We know that 3.4 > 3.2, so therefore 3.4 \times 10^{6} > 3.2 \times 10^{6}.
When asked to order numbers, we will follow the same comparison procedures, followed by listing the numbers either in ascending or descending order.
Examples
Example 1
Which of the following numbers is larger:
2.7 \times 10^{4} \enspace \text{or} \enspace 3.4 \times 10^{3}
Example 2
Order the following numbers from least to greatest:
7.23 \times 10^{7}, \, 7.1 \times 10^{6}, and \, 5.6 \times 10^{7}.
When ordering and comparing numbers in scientific notation, we can do the following:
- If the powers of 10 differ, the larger the power, the larger the number, and the smaller the power, the smaller the number.
- If the powers of 10 are equal, we will then compare the decimal numbers. The larger decimal number will be the larger number and vise versa.
Calculating comparison values
When comparing and ordering numbers in scientific notation, we may need to take our calculations a step further and determine by how much the numbers differ by.
For example, we may need to determine how much larger one population is compared to another. As numbers such as population are generally big numbers, they are often represented in scientific notation.
Let's look at an example of calculating a comparison between numbers in scientific notation:
Examples
Example 3
The population of the United States in 2022 is about 3.3 \times 10^{8}, whereas the population of Australia in 2022 is about 2.6 \times 10^{7}.
How much bigger is the population of the United States compared to Australia? Round your answer to one decimal place.
We can apply the operations and properties that we use with other numbers to compare numbers in scientific notation.