Problem 23 - Algebra and Trigonometry

Question

Sketch the set in the complex plane. $$ \\{z| | z |=3\\} $$

Step-by-Step Solution

Answer

The set describes a circle centered at the origin with radius 3 in the complex plane.

Step 1: Understanding the Problem

We need to sketch the set of all complex numbers \(z\) in the complex plane such that the modulus \(|z| = 3\). This means we are looking for complex numbers with a constant distance from the origin.

Step 2: Identify the Geometric Representation

The modulus \(|z|\) of a complex number \(z = x + yi\) is given by \(\sqrt{x^2 + y^2}\). If \(|z| = 3\), then \(\sqrt{x^2 + y^2} = 3\). Squaring both sides, we have the equation \(x^2 + y^2 = 9\). This is the equation of a circle in the complex plane.

Step 3: Sketch the Circle

In the complex plane, the equation \(x^2 + y^2 = 9\) represents a circle centered at the origin \((0, 0)\) with radius 3. To sketch it, draw a circle with its center at the origin and ensure the circle passes through points 3 units away from the origin along any direction.

Step 4: Mark Key Points on the Circle

The circle will pass through key points such as \((3, 0)\), \((-3, 0)\), \((0, 3)\), and \((0, -3)\) on the complex plane. These points can be used as reference points to ensure the circle is accurately drawn.

Step 5: Final Review

Review the sketch to ensure the circle is properly drawn with the described radius and center. Ensure it appears symmetrical and centered around the origin as the equation suggests.

Key Concepts

Complex Plane

The complex plane is a two-dimensional space where every point represents a complex number. Think of it like a flat map for visualizing complex numbers, which consist of a real part and an imaginary part. In the complex plane:

• The horizontal axis (x-axis) represents the real part of the complex number.
• The vertical axis (y-axis) represents the imaginary part.

For example, the complex number 3 + 4i can be visualized as the point (3,4) in the complex plane. This makes it easier to see relationships and operations involving complex numbers, such as addition, subtraction, and even multiplication. Just as we might plot points or draw lines on a Cartesian plane, we can do the same operations on this complex graph to visualize complex numbers and their relationships easily.

Modulus of a Complex Number

The modulus of a complex number is essentially its distance from the origin on the complex plane. It’s a measure of the 'size' or 'magnitude' of the number regardless of direction.
For a complex number in the form of \[ z = x + yi,\] its modulus is calculated using the formula:\[ |z| = \sqrt{x^2 + y^2}\]where x is the real part and y is the imaginary part.
Looking at this equation, it's easy to notice the similarity to the Pythagorean theorem; thus, this modulus tells us the hypotenuse length of a right-angled triangle formed by x and y.

• If the modulus |z| = 3, then the distance of this complex number from the origin is 3 units.

The modulus therefore helps us easily determine how far a point is from the origin, which is crucial in identifying sets like those forming a circle on the complex plane.

Geometric Representation in Mathematics

Geometric representation is a powerful tool for visualizing and understanding complex numbers. It helps bridge the gap between abstract mathematical concepts and visual intuition.
In this scenario, a set of complex numbers \[ \{ z | |z| = 3 \}\]can be geometrically represented as a circle.

• This circle is centered at the origin (0, 0).
• Every point on this circle is exactly 3 units away from the origin.

This representation comes from equating the modulus |z| with a constant (3 in this case), which corresponds to points in the complex plane that lie on a circle centered at the origin.
Therefore, geometric representation in mathematics isn't just about drawing shapes; it’s about using these shapes to understand and solve algebraic equations or sets involving complex numbers more easily. This approach turns abstract numbers into something you can see and interpret visually, making math not just numbers on a page, but a map of relationships and patterns we can explore and analyze.