Problem 18 - Contemporary Precalculus

Question

In Exercises \(17-24,\) sketch the graph of the equation in the complex plane (z denotes a complex number of the form a \(+b i\) ). $$|z|=1$$

Step-by-Step Solution

Answer

The graph of the equation |z|=1 in the complex plane is a circle centered at the origin (0, 0) with a radius of 1.

Step 1: Define the complex number and its magnitude

We are given that \(z = a + bi\) is a complex number, and \(|z|=1\). By definition, the magnitude of a complex number z is given by \(|z| = \sqrt{a^2 + b^2}\).

Step 2: Find the equation in terms of a and b

Our given equation is \(|z|=1\). Using the magnitude's definition, we can substitute \(|z|\) with \(\sqrt{a^2 + b^2}\) to get: $$\sqrt{a^2 + b^2} = 1$$ Now, we want to eliminate the square root, so we can square both sides of the equation: $$(\sqrt{a^2 + b^2})^2 = 1^2$$ $$a^2 + b^2 =1$$ The equation is now in terms of a and b.

Step 3: Sketch the graph

To sketch the graph of the equation \(a^2 + b^2 = 1\) in the complex plane, we need to recognize that this equation represents a circle centered at the origin with radius 1. To plot this in the complex plane, we use the horizontal axis to represent the real part (a) and the vertical axis to represent the imaginary part (b). The resulting graph should be a circle of radius 1 centered at (0, 0) in the complex plane.

Key Concepts

Complex Number

A complex number is a fascinating concept in mathematics that combines real numbers and imaginary numbers. It is written in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is called the real part, and \( b \) is the imaginary part. The \( i \) represents the imaginary unit, which is defined as the square root of \(-1\).

Complex numbers are incredibly useful in various fields like engineering, physics, and computer science because they allow calculations involving two dimensional quantities.

• \( z = a \) when there is no imaginary part.
• \( z = bi \) when there is no real part.

The beauty of complex numbers lies in their convenience for representing periodic phenomena, oscillations, and waves, making them indispensable in complex systems.

Magnitude of Complex Numbers

The magnitude, also known as the modulus, of a complex number \( z = a + bi \) is a measure of the 'size' or length of the vector represented by the number in the complex plane. It is denoted by \(|z|\) where:

\[|z| = \sqrt{a^2 + b^2} \]

This formula is similar to the Pythagorean theorem, where \( a \) is the horizontal leg and \( b \) is the vertical leg. For complex numbers, the magnitude gives you the distance from the origin \((0, 0)\) to the point \((a, b)\) in the complex plane.

• If \( z \) is purely real, its magnitude is \( |z| = |a| \).
• If \( z \) is purely imaginary, its magnitude is \( |z| = |b| \).

The significance of the magnitude comes into play particularly when graphing, as it helps define shapes such as circles in the complex plane.

Graphing in the Complex Plane

Graphing in the complex plane transforms abstract calculations into visual representations that enhance understanding. In the complex plane, a complex number \( z = a + bi \) corresponds to the point \((a, b)\).

The horizontal axis (real axis) represents the real part \( a \), and the vertical axis (imaginary axis) represents the imaginary part \( b \). Consider the equation \(|z| = 1\), translating to \(a^2 + b^2 = 1\). This equation represents a circle in the complex plane.

• The center of the circle is at the origin \((0, 0)\).
• The radius of the circle is 1.

When graphing, it's crucial to recognize that any complex number \( z = a + bi \) placed on this circle satisfies the magnitude equation \( \sqrt{a^2 + b^2} = 1 \). This is because each point on this circle has a distance of 1 from the origin. Visualization of such equations aids in understanding properties of complex numbers better. This graphical method simplifies tackling complex equations, making intuitive insights into periodic and harmonic patterns much easier.