Summary of Complex Numbers: Introduction

Goals

1. Understand what a complex number is and identify its real and imaginary parts.

2. Classify numbers as real, pure imaginary, or simply imaginary.

Contextualization

Complex numbers are pivotal in various areas of science and engineering. They come into play when trying to solve quadratic equations that don’t have real solutions. For instance, the equation x² + 1 = 0 has no real solutions, but in the domain of complex numbers, it offers two solutions: x = i and x = -i. These numbers are particularly useful in electronics, where alternating current can be modelled using complex numbers, as well as in fractal graphics, such as the well-known Mandelbrot Set.

Subject Relevance

To Remember!

Definition of Complex Numbers

A complex number takes the form a + bi, where 'a' and 'b' are real numbers and 'i' represents the imaginary unit, which is defined as the square root of -1. Here, 'a' is known as the real part and 'b' is referred to as the imaginary part of the complex number.

• Complex numbers are represented as a + bi.

• The real part is denoted by 'a'.

• The imaginary part is denoted by 'b'.

• The imaginary unit 'i' satisfies the equation i² = -1.

Real Part and Imaginary Part

In a complex number expressed as a + bi, 'a' signifies the real part and 'b' symbolizes the imaginary part. The real part can be located on the traditional number line, while the imaginary part is a multiple of the imaginary unit 'i'.

• The real part ('a') is a real number.

• The imaginary part ('b') is a real number that multiplies 'i'.

• Both the real and imaginary parts can be positive, negative, or zero.

Real Numbers, Pure Imaginary Numbers, and Simply Imaginary Numbers

Real numbers are those that can be represented without the imaginary unit 'i'. Pure imaginary numbers have a zero real part, whereas simply imaginary numbers contain both non-zero real and imaginary parts.

• Real Numbers: Do not include the imaginary unit 'i'. E.g., 5, -3, 0.

• Pure Imaginary Numbers: Possess a zero real part and a non-zero imaginary part. E.g., 0 + 3i, 0 - 2i.

• Simply Imaginary Numbers: Have non-zero real and imaginary components. E.g., 3 + 4i, -2 + 5i.

Practical Applications

• Electrical Engineering: Using complex numbers to analyze alternating current (AC) circuits.

• Quantum Physics: Employing complex numbers in Schrödinger's equations to describe subatomic particle behavior.

• Computer Graphics: Utilizing complex numbers to produce realistic visual effects and simulations in 3D rendering.

Key Terms

• Complex Numbers: Numbers of the format a + bi, where 'a' is the real part and 'b' is the imaginary part.

• Real Part: The component 'a' of a complex number formatted as a + bi.

• Imaginary Part: The component 'b' of a complex number formatted as a + bi, multiplied by the imaginary unit 'i'.

• Imaginary Unit (i): A number that satisfies the equation i² = -1.

• Real Numbers: Numbers that do not contain the imaginary unit 'i'.

• Pure Imaginary Numbers: Complex numbers having a zero real part.

• Simply Imaginary Numbers: Complex numbers characterized by non-zero real and imaginary parts.

Questions for Reflections

• How has the incorporation of imaginary numbers in mathematics broadened the scope of solving previously thought unsolvable problems?

• In which aspects can we observe the application of complex numbers in everyday technologies, like smartphones and video games?

• How does the expansion of mathematical knowledge through complex numbers compare to other scientific and technological advancements? What parallels can we draw?

Drawing Complex Numbers on the Cartesian Plane

In this mini-challenge, you will graphically illustrate complex numbers on the Cartesian plane and identify their real and imaginary components.

Instructions

• Get some graphing paper, a ruler, a pencil, and some coloured pens.

• Draw a horizontal axis (real axis) and a vertical axis (imaginary axis) on the graphing paper to create a Cartesian plane.

• Select five different complex numbers. For example: 2 + 3i, -1 + 4i, 3 - 2i, -2 - 3i, 0 + 5i.

• For each complex number, determine the real part and the imaginary part.

• Plot each complex number on the Cartesian plane, clearly marking the point and using a string to connect it to the origin (0,0).

• Use various colours to differentiate between the real part and the imaginary part of the complex numbers.

• Label each point with the corresponding complex number and classify it as real, pure imaginary, or simply imaginary.