Further complex numbers

Welcome to Further Complex Numbers!

In your previous studies, you met the Argand diagram and learned how to draw simple circles and lines. Now, we are stepping into Further Pure Mathematics 2 territory. We are going to explore more advanced "loci" (the paths that points follow) and how to shade specific regions on the diagram.

Don't worry if this seems a bit abstract at first! Think of these equations like a set of GPS instructions. Instead of telling a point exactly where to sit, we are giving it a rule it must follow. By the end of these notes, you’ll be able to visualize these rules as beautiful geometric shapes.

1. The Ratio of Distances: \( |z - a| = k|z - b| \)

This looks a bit scary, but let's break it down. Remember that \( |z - a| \) just means "the distance from the point \( z \) to the point \( a \)".

What is it?
This equation describes a set of points where the distance to point \( a \) is exactly \( k \) times the distance to point \( b \).

The Two Scenarios:
• If \( k = 1 \): This is the "Fair Fight." The distance to \( a \) is the same as the distance to \( b \). This results in a perpendicular bisector (a straight line exactly halfway between them).
• If \( k \neq 1 \): This is more interesting! This creates a circle. In mathematics, we call this the Circle of Apollonius.

How to solve it step-by-step:
If you are asked to find the Cartesian equation (the \( x \) and \( y \) version), follow these steps:
1. Replace \( z \) with \( x + iy \).
2. Write out the modulus for both sides: \( \sqrt{(x - a_{real})^2 + (y - a_{imag})^2} = k\sqrt{(x - b_{real})^2 + (y - b_{imag})^2} \).
3. Crucial Step: Square both sides immediately to get rid of those square roots!
4. Expand the brackets, move everything to one side, and "complete the square" for \( x \) and \( y \) to find the center and radius of the circle.

Quick Tip: Always double-check your squaring. If you have \( 2|z - b| \), when you square it, it becomes \( 4|z - b|^2 \). Forgetting to square the \( k \) value is the most common mistake students make!

Key Takeaway: Whenever you see one modulus equal to a multiple of another modulus (where that multiple isn't 1), you are looking at a circle.

2. Angles and Arcs: \( arg\left(\frac{z - a}{z - b}\right) = \beta \)

This is arguably the trickiest part of the chapter, but we can use a clever analogy to make it simple.

The Concept:
The expression \( arg\left(\frac{z - a}{z - b}\right) \) is equivalent to \( arg(z - a) - arg(z - b) \). Geometrically, this represents the angle between two lines meeting at point \( z \), where one line comes from \( a \) and the other from \( b \).

What does it look like?
This represents an arc of a circle passing through points \( a \) and \( b \).
• Think of it like a "viewing angle." If you are standing at point \( z \) looking at two statues at \( a \) and \( b \), the angle between your arms stays the same as you walk along this specific arc.

Major vs. Minor Arcs:
• If \( \beta \) is acute (less than \( \pi/2 \) or 90°), it is a Major Arc (more than half a circle).
• If \( \beta \) is obtuse (more than \( \pi/2 \)), it is a Minor Arc (less than half a circle).
• If \( \beta = \pi/2 \), it is a semi-circle!

Common Mistake to Avoid:
Do not include the end points \( a \) and \( b \) themselves. The argument is undefined at those points because you can't have an angle if you are standing directly on the statue!

Key Takeaway: This equation creates an "arc" between two points. Use the size of the angle \( \beta \) to decide how much of the circle to draw.

3. Shading Regions in the Argand Diagram

Sometimes, instead of a line or a curve, we want to talk about a whole area. This is where inequalities come in.

A. Real and Imaginary Boundaries

The syllabus mentions regions like \( p \leq Re(z) \leq q \).
• This is like a vertical hallway. If \( 1 \leq Re(z) \leq 3 \), you shade everything between the vertical lines \( x = 1 \) and \( x = 3 \).
• Similarly, \( p \leq Im(z) \leq q \) would be a horizontal hallway between two \( y \)-values.

B. Argument Sectors

Regions like \( \alpha \leq arg(z - z_1) \leq \beta \) look like a spotlight or a slice of pizza.
• The point \( z_1 \) is the "source" of the light.
• The angles \( \alpha \) and \( \beta \) are the edges of the beam.
• You shade the area inside the beam of light.

C. Combined Regions

You might be asked to shade a region that satisfies two rules at once, such as \( |z - a| \leq |z - b| \).
• Step 1: Draw the boundary line (the perpendicular bisector).
• Step 2: Decide which side to shade. Since the "distance to \( a \)" is less than the "distance to \( b \)", you shade the side closer to point \( a \).

Did you know?
In many engineering fields, these regions are used to define "stability zones." For example, if a complex number representing a bridge's vibration stays within a certain "safe" region on the Argand diagram, the bridge won't collapse!

Key Takeaway: For regions, always draw the boundary first (use a solid line for \( \leq \) or \( \geq \), and a dashed line for \( < \) or \( > \)), then pick a "test point" to see which side to shade.

Quick Review Box

Equation: \( |z - a| = k|z - b| \) (with \( k \neq 1 \))
Shape: A Circle.

Equation: \( arg\left(\frac{z - a}{z - b}\right) = \beta \)
Shape: An Arc of a Circle.

Region: \( \alpha \leq arg(z - z_1) \leq \beta \)
Shape: A "wedge" or "pizza slice" starting at \( z_1 \).

Region: \( p \leq Re(z) \leq q \)
Shape: A vertical strip between \( x = p \) and \( x = q \).

Don't be discouraged if the algebra for the Apollonius Circle takes a few tries to get right. It’s mostly just careful expanding of brackets! Keep practicing, and you'll find the patterns.