Adding and Subtracting Complex Numbers

1. Imaginary and Complex Numbers

To review, the square root of a negative number is a non-real, or imaginary, number. The imaginary unit i is defined as the square root of -1.

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Imaginary Number

A complex number is a value in the form below, in which a and b are real numbers, and i is the imaginary unit. In a complex number, a is the real part, and b times i is the imaginary part.

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Complex numbers are used in fields such as engineering and physics.

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Complex Number

A value of the form , where a and b are real numbers and i is the imaginary unit

2. Adding and Subtracting Imaginary Numbers

You may recall that the product property for square roots states that the square root of a times b is equal to the square root of a times the square root of b.

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Product Property of Square Roots

You can apply the product property of square roots to solve equations involving the square root of a negative number, so that you are able to simplify your solution using imaginary numbers.

EXAMPLE

You can rewrite the square root of -25 as follows, then apply the product property for square roots. Then you are able to simplify to arrive at your solution, which is an imaginary number.

The product property of square roots can also be applied when adding and subtracting imaginary numbers.

EXAMPLE

Suppose you are solving the equation:



Applying the product property for square roots, you can simplify to:



Now, 2i, 7i, and 3i are all like terms. Therefore, you can combine them together by adding or subtracting their coefficients, to arrive at your final answer:

3. Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers is similar to combining like terms. You can add or subtract the real parts together, and add or subtract the coefficients of the imaginary parts together. You can add or subtract complex numbers in this way because of the commutative property of addition.

EXAMPLE

Suppose you want to add the complex numbers:



You would start by adding your real parts, 4 and 2, together. Then you would add your imaginary parts, 8i and 3i, together.



Combining your real parts together and your imaginary parts together gives you the final answer.

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Consider the following expression:

Combine like terms to subtract the complex numbers.

Start by combining and subtracting your real parts, 11 minus 7, then combine and subtract your imaginary parts, -6i minus 9i, to arrive at your final answer.