Complex Numbers Problems and Solutions

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This is one in a series of Math resources you’ll find in this site. In this article, we hope to clear up common complex numbers problems. They aren’t called complex for no reason but, as with all other math problems, these have solutions.

What are complex numbers?
First, it’s best to be cleared about complex numbers. Not too long ago, mathematicians of all time devised a complete number system that includes both real and imaginary. Numbers in this system are called complex numbers. These numbers is composed of all sums a + bi where a and b are real numbers and i is imaginary.

Real numbers belong to the complex number system because a = a + Oi.

I acts as would any other variable, such as:

Ex.
Problem: 7i + 9i
Solution: combine like terms.
16i

Like real numbers, complex numbers may be equal, as in the example: a + bi = c + di, where a and c and b and d must be equal.

Ex.
Find the values of x and y in 3x + yi = 5x + 1 + 2i

Solution: From the above definition for complex numbers equality, group the real numbers equal and do the same for the complex numbers to get the answer.

3x = 5x + 1          yi = 2i
-2x = 1                 y = 2
X = -(1/2)

Both examples show how to approach complex numbers problems.

Complex Number Overview

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From wikipedia:  A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis.

This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

From first grade math to the Algebra 2 course in high school, students work with real numbers. Real numbers are numbers that appear on the traditional number line; i.e., the positive and negative integers, the rational numbers, and the irrational numbers. These numbers are used in everyday applications. The traditional number line used in most math courses is illustrated below:

When a student takes Algebra 1 in high school, he is told that taking square roots of negative numbers cannot be done. Take a look at −9. What is the square root of −9? The square root of 9 is 3.