Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$|z|=7$$

Step-by-Step Solution

Answer

The graphical representation of all complex numbers with a magnitude of 7 is a circle centered at the origin with a radius of 7.

Step 1. Understanding the Problem

The exercise is asking for a graphical representation of all complex numbers with a magnitude of 7. That means we need to understand that the magnitude of a complex number is its distance from the origin on the complex plane. So, we are looking for all points that are at a distance of 7 from the origin.

Step 2. Graphing the Set

Given that the magnitude of a complex number represents the distance from the origin on the complex plane, all complex numbers with a magnitude of 7 will form a circle centered at the origin with a radius of 7. So, the graph will be a circle centered at the origin and crossing the vertical and horizontal axis at \(+7i\), \(-7i\), \(+7\) and \(-7\).

Step 3. Summary

All complex numbers with a given magnitude share the same distance from the origin, so they form a circle on the complex plane. In this case, since the magnitude is 7, we get a circle of radius 7 centered at the origin.

Key Concepts

Magnitude of Complex Number

The magnitude of a complex number, often referred to as the modulus, measures how far the number is from the origin on the complex plane. If we have a complex number represented as \(z = a + bi\), where \(a\) and \(b\) are real numbers, its magnitude is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). This formula is derived from the Pythagorean theorem. In simpler terms, the magnitude is the 'length' of the vector in the complex plane that goes from the origin to the point \((a, b)\).

• Magnitude represents distance.
• For real numbers, the magnitude is the absolute value.
• Magnitude is always non-negative.

Understanding the magnitude is crucial because it allows us to visualize the position of complex numbers in the complex plane and to solve problems involving distances.

Complex Plane

The complex plane is a geometric representation that facilitates the visualization of complex numbers. It is a two-dimensional plane where each point represents a complex number. The horizontal axis (often called the real axis) represents the real part of the complex number, while the vertical axis (often called the imaginary axis) represents the imaginary part.

• Real axis corresponds to real numbers.
• Imaginary axis corresponds to imaginary numbers.
• Each complex number is a unique point on the plane.

Understanding the complex plane is important because it provides a powerful visual way to comprehend operations on complex numbers, like addition and multiplication, and properties such as magnitude and argument.

Graphing Complex Numbers

Graphing complex numbers involves plotting them on the complex plane based on their real and imaginary components. For a complex number \(z = a + bi\), the number is placed at the point \((a, b)\), where \(a\) is the coordinate on the real axis, and \(b\) is the coordinate on the imaginary axis. For example, to graph the set of complex numbers with magnitude 7, you plot all points whose distance from the origin is exactly 7. This creates a circle centered at the origin with a radius of 7. The circle includes points like \(+7i\), \(-7i\), \(+7\), and \(-7\) among many others.

• Complex numbers have both direction and magnitude.
• A set of points sharing the same magnitude forms a circle on the plane.
• Circle's radius equals the common magnitude of the set.

Graphing is not just about plotting, but also about understanding the spatial relationships and distances on the complex plane.