Complex Numbers – Part 2

Complex Numbers – Part 2

Modulus, the complex plane and graphical representation

📐 The Complex Plane (Gauss Plane)

Every complex number \(z = a + bi\) can be represented as a point in the plane:

  • X-axis = real axis – represents a
  • Y-axis = imaginary axis – represents b
  • The number \(z = a + bi\) is represented by the point \((a, b)\)
Re Im 1 2 3 -1 -2 1 2 -1 -2 3+2i -2+i 1-2i -1-i O

📏 Modulus (Absolute Value)

The modulus of \(z = a + bi\) is the distance from the origin:

\(|z| = \sqrt{a^2 + b^2}\)

💡 This is exactly the Pythagorean theorem!

Re Im z = a+bi a b |z|

Examples:

\(|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

\(|1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\)

\(|-5| = \sqrt{(-5)^2 + 0^2} = 5\) (a real number)

\(|3i| = \sqrt{0^2 + 3^2} = 3\) (purely imaginary)

⭐ Connection to the conjugate:

\(|z|^2 = z \cdot \bar{z} = a^2 + b^2\)

📐 Properties of the Modulus

1. Always non-negative:

\(|z| \geq 0\)

\(|z| = 0 \iff z = 0\)

2. Multiplication:

\(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)

3. Division:

\(\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}\)   (when \(z_2 \neq 0\))

4. Conjugate:

\(|\bar{z}| = |z|\)

5. Triangle inequality:

\(|z_1 + z_2| \leq |z_1| + |z_2|\)

🪞 The Conjugate in the Complex Plane

The conjugate \(\bar{z}\) is the reflection of z across the real axis.

Re Im z = a+bi z̄ = a-bi |z| |z̄|

💡 Note:

  • z and \(\bar{z}\) are equidistant from the origin (\(|z| = |\bar{z}|\))
  • z and \(\bar{z}\) are symmetric with respect to the real axis
  • If z is on the real axis, then \(z = \bar{z}\)

📍 Distance Between Two Complex Numbers

The distance between \(z_1 = a + bi\) and \(z_2 = c + di\):

\(|z_1 - z_2| = \sqrt{(a-c)^2 + (b-d)^2}\)

Example: find the distance between \(z_1 = 3 + 2i\) and \(z_2 = -1 + 5i\)

\(z_1 - z_2 = (3 + 2i) - (-1 + 5i) = 4 - 3i\)

\(|z_1 - z_2| = |4 - 3i| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\)

⭕ Circle in the Complex Plane

The equation \(|z - z_0| = r\) describes a circle:

  • Centre: \(z_0\)
  • Radius: \(r\)

Examples:

\(|z| = 3\)

Circle centred at the origin with radius 3

\(|z - 2| = 4\)

Circle centred at \(z_0 = 2\) (i.e. the point (2,0)) with radius 4

\(|z - (1 + 2i)| = 5\)

Circle centred at (1,2) with radius 5

\(|z + 3i| = 2\), i.e. \(|z - (-3i)| = 2\)

Circle centred at (0,−3) with radius 2

📊 Inequalities in the Complex Plane

\(|z - z_0| < r\)

Interior of the circle (boundary not included)

\(|z - z_0| \leq r\)

Interior of the circle including the boundary (closed disk)

\(|z - z_0| > r\)

Exterior of the circle (boundary not included)

Example: describe the set \(|z - 1| \leq 2\)

All points inside and on the circle centred at (1,0) with radius 2.

📐 Properties of the Conjugate

1. \(\overline{\overline{z}} = z\)   (conjugate of conjugate = original)

2. \(\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}\)

3. \(\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}\)

4. \(\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}\)

5. \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}\)

6. \(z + \bar{z} = 2\text{Re}(z) = 2a\)   (always real!)

7. \(z - \bar{z} = 2\text{Im}(z) \cdot i = 2bi\)   (always purely imaginary!)

📋 Summary Table – Part 2

Concept Formula / Definition
Modulus \(|a+bi| = \sqrt{a^2 + b^2}\)
Distance between z₁ and z₂ \(|z_1 - z_2|\)
Circle \(|z - z_0| = r\)
Connection to conjugate \(|z|^2 = z \cdot \bar{z}\)
Conjugate graphically Reflection across the real axis

💡 Tips for the Exam

1️⃣ Modulus

It's Pythagoras! \(\sqrt{a^2 + b^2}\)

2️⃣ Circle

\(|z - z_0| = r\) is a circle with centre \(z_0\)

3️⃣ Useful relation

\(z \cdot \bar{z} = |z|^2\)

4️⃣ Multiplication

\(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)

📝 Summary – Part 2

\(|z| = \sqrt{a^2 + b^2}\)

\(|z - z_0| = r\) → circle

In the next part: quadratic equations and solving in complex numbers