IB Mathematics AA Complex numbers Study Notes
IB Mathematics AA Complex numbers Study Notes
IB Mathematics AA Complex numbers Study Notes Offer a clear explanation of Complex numbers , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Complex numbers.
Introduction to Complex Numbers
Introduction to Complex Numbers
Definition:
The imaginary unit \( i \) is defined as:
\( i^2 = -1 \)
Using this definition, we build the system of complex numbers.
Complex Number Format:
A complex number is written in the form:
\( z = a + bi \)
- \( a \) is the real part
- \( b \) is the imaginary part
- \( i \) is the imaginary unit, where \( i^2 = -1 \)
Key Powers of 𝑖 i:
| Power | Value |
|---|---|
| \( i^1 \) | \( i \) |
| \( i^2 \) | \( -1 \) |
| \( i^3 \) | \( -i \) |
| \( i^4 \) | \( 1 \) |
Then it repeats every 4 powers: \( i^5 = i \), \( i^6 = -1 \), etc.
Examples
- Simplify: \( i^7 \)
- Express \( \sqrt{-36} \) in terms of \( i \)
- Add \( (3 + 4i) + (2 – 6i) \)
▶️ Answer/Explanation
- \( i^7 \):
\( i^7 = i^{4 \cdot 1 + 3} = i^3 = -i \) - \( \sqrt{-36} \):
\( \sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i \) - \( (3 + 4i) + (2 – 6i) = (3 + 2) + (4i – 6i) = 5 – 2i \)
Complex Numbers in Cartesian Form
Complex Numbers in Cartesian Form
Standard Form:
\( z = a + bi \)
- \( a \) is the real part of the complex number
- \( b \) is the imaginary part
- \( i \) is the imaginary unit, where \( i^2 = -1 \)
Important Terms:
| Real part | \( \operatorname{Re}(z) = a \) |
| Imaginary part | \( \operatorname{Im}(z) = b \) |
| Conjugate | \( \bar{z} = a – bi \) |
| Modulus | \( |z| = \sqrt{a^2 + b^2} \) |
| Argument (arg) | \( \arg(z) = \theta = \tan^{-1}\left(\frac{b}{a}\right) \) |
Note: The argument is the angle the line connecting the origin to the point \( (a, b) \) makes with the positive real axis in the complex plane.
Examples
- Given \( z = 3 + 4i \), find the real part, imaginary part, conjugate, modulus, and argument.
- Given \( z = -5 – 12i \), compute the modulus .
▶️ Answer/Explanation
1. For \( z = 3 + 4i \):
- Real part: \( \operatorname{Re}(z) = 3 \)
- Imaginary part: \( \operatorname{Im}(z) = 4 \)
- Conjugate: \( \bar{z} = 3 – 4i \)
- Modulus: \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
- Argument: \( \arg(z) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \) or \( 0.93 \) radians
2. For \( z = -5 – 12i \):
- Modulus: \( |z| = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
The Complex Plane
The Complex Plane (Argand Diagram)
The complex plane is a way of visualizing complex numbers. It is also called the Argand diagram.
In this plane:
- The horizontal axis (x-axis) represents the real part of the complex number.
- The vertical axis (y-axis) represents the imaginary part.
Each complex number \( z = a + bi \) can be represented as a point \( (a, b) \) or as a vector from the origin to that point.
Features of the Argand Diagram
- Real Axis: Where \( b = 0 \); purely real numbers lie here.
- Imaginary Axis: Where \( a = 0 \); purely imaginary numbers lie here.
- Modulus: The distance from the origin to the point, \( |z| = \sqrt{a^2 + b^2} \).
- Argument: The angle \( \theta \) the vector makes with the positive real axis, measured counter-clockwise.
Examples
- Plot \( z = 4 + 3i \) on the complex plane.
- Plot \( z = -2 – 5i \) on the Argand diagram and find its modulus and argument.
▶️ Answer/Explanation
1. \( z = 4 + 3i \):
- Real part: 4
- Imaginary part: 3
- Plot the point at (4, 3)
- This is in the first quadrant
2. \( z = -2 – 5i \):
- Real part: -2
- Imaginary part: -5
- Point is in the third quadrant at (-2, -5)
- Modulus: \( |z| = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385 \)
- Argument: \( \arg(z) = \tan^{-1}\left(\frac{-5}{-2}\right) = \tan^{-1}(2.5) \approx 1.19 \) radians (but it’s in quadrant III, so \( \theta \approx -2.0 \) radians)
