Scientific notation can be used to express very large and very small numbers by multiplying a decimal number by a power of 10. By very large number, we mean a number containing several digits to the left of the decimal, such as 23,000,000 (23 million). By small number, we mean a number with zero in the ones place, with several decimal digits behind it, such as 0.000023 (23 millionths).
Generally, numbers written in scientific notation can be expressed as a • 10^n, where a is a decimal number, and 10^n represents a power of ten.
In proper scientific notation, there are a couple of restrictions to what “a” is allowed to be. “a” can only contain one non-zero digit to the left of the decimal. Here are some examples of numbers that might look like they are in scientific notation, but they violate this rule for what “a” can be:
In cases where there is a zero or more than one digit to the left of the decimal, we can move the decimal and adjust the exponent. Let’s change our previous examples into proper scientific notation:
When moving the decimal in scientific notation, movement to the left increases the power of 10, and movement to the right decreases the power of 10.
Be care when increasing and decreasing negative powers of 10. For example, decreasing -3 by 1 makes the number more negative: -4. Similarly, increasing -3 by 1 makes the number less negative: -2.
Adding and Subtracting with Common Powers of 10
How does this help us add and subtract numbers in scientific notation? We can easily add and subtract numbers in scientific notation if they are expressed using the same power of 10. This is illustrated in the following example:
We can confirm this is a valid process by showing the addition in standard notation:
Sometimes when you add the decimal numbers, you will get a sum that does not fit the rules for proper scientific notation. Remember to always check that a single non-zero digit is to the left of the decimal. If necessary, move the decimal and adjust the exponent.
When the powers of 10 are the same, we can add and subtract numbers in scientific notation following the rule a x 10n + b x 10n = (a+b) x 10n
Adding and Subtracting with Unlike Powers of 10
What if we wish to add or subtract numbers in scientific notation that have unlike powers of 10? We can use our handy trick of moving the decimal and adjusting the exponent to express the numbers with a common power of 10. This is illustrated in the following examples:
Note that it does not matter which number you choose to rewrite. We would get the same result if we chose to rewrite the other number, although we might give ourselves more work to do. Let’s go through the same example, but in a different way:
As you can see in the second example, we needed to make a final adjustment to our solution, because -16.9 x 10^3 is not in proper scientific notation. No matter which number you decide to rewrite, always be sure to make any necessary movements to the decimal, and increase or decrease your exponent accordingly.
When adding or subtracting numbers in scientific notation with unlike powers of 10, we can rewrite one of the numbers by moving the decimal and increase or decreasing the power of 10. This creates two numbers with a common power of 10, which can be added or subtracted by combining the decimal numbers and keeping the common power of 10.
This tip is worth repeating, especially when adding or subtracting numbers in scientific notation. During the process, you will probably work with numbers that are not written in proper scientific notation. This is okay, because it is required to create a common power of 10. Just be sure that you make final adjustments to your expression, so that the decimal number contains only a single non-zero digit to the left of the decimal. Remember, decimal movement to the left increases the power of 10, and decimal movement to the right decreases the power of 10.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html