Problem 20 - Algebra and Trigonometry

Question

\(17-24=\) Sketch the set in the complex plane. $$ \\{z| | z | \geq 1\\} $$

Step-by-Step Solution

Answer

The arithmetic answer is \(-7\); on the complex plane, shade outside a circle of radius 1 centered at the origin.

Step 1: Understanding the Exercise

The exercise requires us to both solve a basic arithmetic equation and sketch a set in the complex plane. Let's break it down into two parts: calculating the arithmetic part and then sketching the set on the complex plane.

Step 2: Solving the Arithmetic Equation

First, solve the arithmetic equation \(17 - 24\). Subtract 24 from 17 to get \[17 - 24 = -7\].

Step 3: Interpreting the Set

The set \(\{z | |z| \geq 1\}\) represents all complex numbers \(z\) whose modulus (or magnitude) is greater than or equal to 1.

Step 4: Sketching the Set

To sketch this set, recognize that \(|z| \geq 1\) describes the exterior of a circle (including the circle itself) centered at the origin with a radius of 1 in the complex plane. 1. Draw a circle centered at the origin with radius 1.2. Shade everything outside the circle, including the circle's boundary, since points on the boundary satisfy \(|z| = 1\).

Key Concepts

Set Notation

Set notation is a useful tool in mathematics that allows us to express a collection of elements concisely. In this context, we are dealing with the set:\[\{z | |z| \geq 1\}\]
This expression can be read as "the set of all complex numbers \(z\) such that the modulus of \(z\) is greater than or equal to 1."

In set notation, the vertical bar "|" is read as "such that," and is used to specify the condition that the elements of the set must satisfy.
Set notation is helpful for clearly defining properties of elements without listing them one by one.
It's a compact and flexible way to describe complex structures, like all the points in a certain region of the complex plane, without needing to draw every single point.

Modulus of Complex Numbers

The modulus of a complex number is a measure of its size or distance from the origin in the complex plane. If you have a complex number represented as \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, the modulus is calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

This formula gives the length of the vector from the origin to the point \((a, b)\) in the complex plane.

The modulus has several important properties:

• It is always a non-negative real number.
• It equals zero only when the complex number is zero (i.e., \(a = 0\) and \(b = 0\)).
• For the set \(\{z | |z| \geq 1\}\), this means including all complex numbers whose distance from the origin is at least 1.

Understanding this concept is crucial for visualizing and analyzing sets in the complex plane.

Arithmetic Operations

In the realm of complex numbers, arithmetic operations such as addition, subtraction, multiplication, and division work similarly to operations with real numbers, but with a twist to incorporate the imaginary unit \(i\).

To understand subtraction, let's refer back to the solution of the arithmetic equation:

• Starting with the equation \(17 - 24\), you perform standard subtraction to find the difference: \(17 - 24 = -7\).

This subtraction is an operation on real numbers, but similar rules apply when dealing with complex numbers.
When subtracting, you handle the real and imaginary components separately:

• \((a + bi) - (c + di) = (a - c) + (b - d)i\)

Keep these operations in mind as they form the foundation for solving and simplifying many complex number problems you will encounter.