Problem 59 - College Algebra

Question

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

Step-by-Step Solution

Answer

Complex numbers are added by adding the real components together and the imaginary components together. For example, given the complex numbers \(3+2i\) and \(1+4i\), the sum is \(4+6i\).

Step 1: Definition and Components of a Complex Number

A complex number has two parts: a real portion and an imaginary portion. It is generally written in the form \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the unit imaginary number, satisfying the equation \(i^2 = -1\). The real part is \(a\) and the imaginary part is \(b\).

Step 2: Adding Complex Numbers

When adding complex numbers, like real numbers, similar components are added together. This means the real parts of the numbers are added together and the imaginary parts are added together. The result is another complex number. If the initial complex numbers were \(a+bi\) and \(c+di\), the resulting complex number after addition would be \((a+c) + (b+d)i\).

Step 3: Example

As an example, let us consider the two complex numbers \(3+2i\) and \(1+4i\). For adding these numbers, the real parts will be added together: \(3+1 = 4\), and the imaginary parts will also be added together: \(2i+4i = 6i\). Thus the sum of \(3+2i\) and \(1+4i\) is \(4+6i\).