Complex Numbers: Definition, Terminology, Visualization, and Operations

MATH5103 M2 Application
Student Full Name
Institutional Affiliation
Course Full Title
Instructor Full Name
Due Date
MATH5103 M2 Application
A difficult mathematical concept that others find challenging to understand, exceptionally high school learners taking algebra classes, is complex numbers. Complex numbers can be challenging because they deal with "imaginary" numbers, something most learners have an aversion to. Besides, basic operations involving complex numbers require a fundamental understanding of vector algebra, polar coordinates, and trigonometry, particularly when plotting points on a complex plane. This intersection of algebraic, trigonometric, and geometric methods can seem confusing and overwhelming to those encountering complex numbers for the first time. However, complex numbers are not that challenging to grasp once someone understands the definition, terminology, visualization, and operations of the same.
Complex numbers refer to the sum of a real number and an imaginary number, not intricate numbers, as most students interpret. Real numbers are those values contained in a number system, including rational numbers, positive numbers, zero, negative numbers, fractions, integers, and irrational numbers. For instance, 5, 12, 2.8, -45, 0, 1/7, and etcetera are real numbers. On the other hand, imaginary numbers are those numbers that are not contained in a number system. For example, -5, -11, -1, and similar others are all imaginary numbers. Complex numbers were introduced to solve quadratic equations. For instance, the equation x2 + 1 = 0 requires one to find the root of -1, and yet no such real roots exist. In quadratic equations, we have imaginary roots where the imaginary square root -1 is represented by the symbol ‘i’ or iota. Complex numbers take the form of a + ib and are usually characterized by z. Although a and b are real numbers, i is an imaginary number. Therefore the value a is the real part of the complex number while ib, taken together, represents the imaginary part.
The real part is represented as Re(z) while the imaginary part is denoted by Im(z). Consequently, the equation Z = a + ib has two parts, a as the real part and ib as the imaginary part: the only real values are a and b while i is the imaginary number. A more visual example of complex numbers includes 4 + 3i, ½ + 6i, 5 + 4/9i, 7/3 + 2/8i, 0.4 +7.4i, -12 + 4i, 5 + 3i and etcetera. The alphabet i, also known as iota, is essential to finding the square root of negative numbers: i = -1 and so i2 = -1 and therefore the -9 can be written as i29 = +/-3i. On the whole, the square of i2 = -1 is the central characteristic of complex numbers. Further examples of the compounding powers of i include: i3 = -1, i4 = ((-1) 2)2 = (-1)2 = 1, i 4n = 1, i 4n + 1 = i, i 4n + 2 = -1, i 4n + 3 = -i, and etcetera. After understanding the increasing powers of i, the next thing to grasp about complex numbers is graphing them: the real and imaginary parts of a complex number can be plotted on an Euclidean plane using Re(z) and Im(z) as coordinate points. Th...