Further complex numbers

Welcome to Further Complex Numbers!

In your earlier studies, you learned how complex numbers work and how to draw simple shapes like circles and lines on an Argand diagram. Now, we are going to dive deeper. We will look at more complex shapes (loci) and learn how to "transform" an entire plane into a new one using mathematical functions. Think of it like a digital filter that stretches and twists a picture – but with algebra!

1. Advanced Loci: The Circle of Apollonius

You already know that \( |z - a| = |z - b| \) represents a perpendicular bisector (the line exactly halfway between point \( a \) and point \( b \)). But what happens if one side is multiplied by a constant \( k \)?

The Equation: \( |z - a| = k|z - b| \)

When \( k \neq 1 \), this locus is always a circle. This is known as the Circle of Apollonius.

Analogy: Imagine two radio towers, \( a \) and \( b \). Tower \( b \) is much stronger than tower \( a \). The locus represents all the points where the signal from tower \( b \) is exactly \( k \) times as strong (or weak) as tower \( a \). Instead of a straight line, the points where these signals balance out form a circle around the weaker tower.

How to solve these:
1. Replace \( z \) with \( x + iy \).
2. Square both sides to get rid of the square roots: \( (x - a_1)^2 + (y - a_2)^2 = k^2[(x - b_1)^2 + (y - b_2)^2] \).
3. Expand everything and move it to one side.
4. Complete the square for both \( x \) and \( y \) to find the center and radius of the circle.

Common Mistake: Forgetting to square the \( k \)! If the equation is \( |z - 3| = 2|z - i| \), when you square both sides, the \( 2 \) must become a \( 4 \).

Key Takeaway: If \( k = 1 \), it’s a line. If \( k \neq 1 \), it’s a circle!

2. The "Angle" Locus

This is often the trickiest part of the chapter, but there is a beautiful geometric trick to it.

The Equation: \( \arg\left(\frac{z - a}{z - b}\right) = \beta \)

This represents the arc of a circle. Specifically, it is the set of points \( z \) such that the angle between the lines connecting \( z \) to \( a \) and \( z \) to \( b \) is exactly \( \beta \).

Think of it like this:
Imagine \( a \) and \( b \) are two goalposts on a pitch. If you want to stand somewhere so that the angle between your view of the two posts is always \( 30^\circ \), you would have to move along a specific curved path—an arc of a circle!

Important Rules:
- The arc starts at \( b \) and ends at \( a \).
- If \( \beta \) is acute (less than \( 90^\circ \)), the arc is a major arc (more than half a circle).
- If \( \beta \) is obtuse (more than \( 90^\circ \)), the arc is a minor arc (less than half a circle).
- If \( \beta = \pi/2 \) (\( 90^\circ \)), the arc is a semi-circle.

Memory Aid: The "top" number (\( a \)) is the finish point, and the "bottom" number (\( b \)) is the start point. The angle is measured anticlockwise from the line segment \( zb \) to \( za \).

Key Takeaway: This locus is an arc, not a full circle. Don't forget to exclude the end points \( a \) and \( b \) themselves, as the angle isn't defined right on top of the posts!

3. Regions in the Argand Diagram

Sometimes we aren't looking for a line, but a whole area. This is shown using inequalities.

Horizontal/Vertical Strips: \( p \le \text{Re}(z) \le q \) represents all complex numbers whose real part is between \( p \) and \( q \). This looks like a vertical shaded corridor on your diagram.

Angular Wedges: \( \alpha \le \arg(z - z_1) \le \beta \) represents a "pizza slice" shape starting from the point \( z_1 \). The "crust" goes on forever!

Quick Review:
- Solid line means \( \le \) or \( \ge \) (includes the boundary).
- Dashed/Dotted line means \( < \) or \( > \) (excludes the boundary).

4. Transformations: Moving from the z-plane to the w-plane

In this section, we take a point \( z \) in one Argand diagram and use a formula to map it to a new point \( w = u + iv \) in a different Argand diagram.

Type 1: \( w = z^2 \)

This transformation is a "doubler."
- It squares the modulus: if \( |z| = r \), then \( |w| = r^2 \).
- It doubles the argument: if \( \arg(z) = \theta \), then \( \arg(w) = 2\theta \).

Type 2: The Möbius Transformation \( w = \frac{az + b}{cz + d} \)

This is a very powerful transformation. One of its "superpowers" is that it generally maps circles and lines to other circles and lines.

Step-by-Step process for transformation questions:
Example: Find the image of the circle \( |z| = 2 \) under the transformation \( w = \frac{z + i}{z - 1} \).

Step 1: Rearrange for \( z \). You need to get \( z = \dots \) in terms of \( w \).
\( w(z - 1) = z + i \)
\( wz - w = z + i \)
\( wz - z = w + i \)
\( z(w - 1) = w + i \)
\( z = \frac{w + i}{w - 1} \)

Step 2: Substitute into the given locus. We were told \( |z| = 2 \), so:
\( \left| \frac{w + i}{w - 1} \right| = 2 \)

Step 3: Simplify. Use the rule that \( \left| \frac{A}{B} \right| = \frac{|A|}{|B|} \):
\( \frac{|w + i|}{|w - 1|} = 2 \)
\( |w + i| = 2|w - 1| \)

Step 4: Identify the new locus. Look at the result. Does it look familiar? Yes! It’s a Circle of Apollonius in the \( w \)-plane. You can now use the "Complete the Square" method from Section 1 to draw it.

Don't worry if this seems tricky at first! The algebra can get messy, but the steps are always the same: Isolate \( z \), substitute, and simplify.

Key Takeaway: Transformations are just a game of "substitution." Rearrange the equation so \( z \) is the subject, then plug it into your original circle or line equation.

Summary Checklist

- Can I identify \( |z - a| = k|z - b| \) as a circle?
- Do I remember that \( \arg(\dots) = \beta \) is just an arc?
- Am I comfortable sketching regions with solid and dashed lines?
- Can I rearrange \( w = f(z) \) to make \( z \) the subject for transformations?

Did you know? Möbius transformations are used in modern physics to study how light bends around black holes! The math you are doing here is the foundation for understanding the shape of the universe.