Addition and Subtraction in Scientific Notation

1. Addition and Subtracting in Scientific Notation

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:

EXAMPLE

A zero to the left of the decimal is not allowed

EXAMPLE

More than 1 digit to the left of the decimal is not allowed

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.

EXAMPLE

	A zero to the left of the decimal is not allowed
	Decimal moved right 1 place, exponent decreased by 1

EXAMPLE

	More than one digit to the left of the decimal is not allowed
	Decimal moved left 1 place, exponent increased by 1

big idea

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.

hint

Be careful 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.

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.

EXAMPLE

	Add the decimal numbers together, keep the power of 10
	Simplify
	Our Solution

We can confirm this is a valid process by showing the addition in standard notation:

EXAMPLE

	Write in standard notation
	Add the numbers
	Convert to scientific notation
	Our Solution

hint

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.

big idea

When the powers of 10 are the same, we can add and subtract numbers in scientific notation following the rule: .

3. 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.

EXAMPLE

	Rewrite one numbe to have a common power of 10
	Group the decimal numbers, keep power of 10
	Subtract 21 from 4.1
	Write in proper scientific notation with only one number in front of the decimal, adjust exponent
	Our Solution

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:

EXAMPLE

	Rewrite one number to have a common power of 10
	Group the decimal numbers, keep power of 10
	Subtract 2.1 from 0.41
	Our Solution

​As you can see in the first 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.

hint

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.