Problem 59 - College Algebra Essentials

Question

Explain how to add complex numbers. Provide an example with your explanation.

Step-by-Step Solution

Answer

To add complex numbers, add the real parts and the imaginary parts separately. For example, the sum of \(3 + 2i\) and \(1 + 4i\) is \(4 + 6i\).

Step 1: Understanding Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is a solution of the equation \(x^2 = -1\). 'a' is the real part and 'bi' is the imaginary part of the complex number. For example, \(3 + 2i\) is a complex number, where 3 is the real part and 2 is the imaginary part.

Step 2: Adding real parts

When adding complex numbers, start by adding the real parts together. For example, if we have two complex numbers \(3 + 2i\) and \(1 + 4i\), add the real parts 3 from \(3 + 2i\) and 1 from \(1 + 4i\), to get a total real part of 4.

Step 3: Adding imaginary parts

Next, add the imaginary parts together. Following our example, add the imaginary parts 2 from \(3 + 2i\) and 4 from \(1 + 4i\), to get a total imaginary part of \(6i\).

Step 4: Combine the real and imaginary parts

Finally, combine the total real and imaginary parts to get the sum of the complex numbers. In our example, this is \(4 + 6i\).