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.
Complex Numbers
Complex numbers extend the concept of real numbers by including the imaginary unit \( i \), where \( i^2 = -1 \). They are essential in advanced mathematics and applications such as engineering, physics, and signal processing.
Key Concepts
- Definition:
- A complex number is expressed as \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with \( i^2 = -1 \).
- \( a \): Real part of \( z \). \( b \): Imaginary part of \( z \).
- Conjugate:
- The conjugate of \( z = a + bi \) is \( \bar{z} = a – bi \).
- Properties: \( z \cdot \bar{z} = a^2 + b^2 \) (modulus squared).
- Modulus:
- The modulus (magnitude) of \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \).
- Argument:
- The argument of \( z = a + bi \) is the angle \( \theta \) measured counterclockwise from the positive real axis in the Argand diagram.
- Expressed as \( \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \) for \( a > 0 \).
Guidance, Clarifications, and Syllabus Links
- Argand Diagram:
- The complex plane, also known as the Argand diagram, represents \( z = a + bi \) as a point \( (a, b) \).
- The real part (\( a \)) is plotted on the x-axis, and the imaginary part (\( b \)) on the y-axis.
- Operations in the Complex Plane:
- Addition: \( (a + bi) + (c + di) = (a + c) + (b + d)i \).
- Subtraction: \( (a + bi) – (c + di) = (a – c) + (b – d)i \).
- Multiplication: \( (a + bi)(c + di) = (ac – bd) + (ad + bc)i \).
- Applications:
- Complex numbers are used in solving quadratic equations, representing oscillations, and in electrical engineering.
Examples
Example 1: Modulus and Argument
Find the modulus and argument of \( z = 3 + 4i \):
- Modulus: \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \).
- Argument: \( \text{arg}(z) = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \).
Example 2: Conjugate and Multiplication
For \( z = 2 + 3i \) and \( w = 1 – 4i \):
- Conjugate of \( z \): \( \bar{z} = 2 – 3i \).
- Multiplication: \( z \cdot w = (2 + 3i)(1 – 4i) = 2 – 8i + 3i – 12i^2 = 14 – 5i \).
Visual Aids
The following graph demonstrates the representation of complex numbers \( z = a + bi \) and their modulus and argument on the Argand diagram:
IB Mathematics AA SL Complex numbers Exam Style Worked Out Questions
Question:
Consider the complex numbers z1 = 1 + bi and z2 = (1 – b2) – 2bi, where b ∈ R, b ≠ 0.
(a) Find an expression for z1z2 in terms of b.
▶️Answer/Explanation
Ans: z1z2 = (1 + bi) ((1-b2) – (2b)i)
= (1-b2 – 2i2b2) + i (-2b + b – b3)
= (1 + b2) + i (-b – b3)
Note: Award A1 for 1+ b2 and A1 for − bi – b3i .
(b) Hence, given that arg (z1 z2) = \(\frac{\pi }{4}\) , find the value of b.
▶️Answer/Explanation
Ans: arg (z1z2) = arctan \(\left ( \frac{-b-b^{3}}{1 + b^{2}} \right )= \frac{\pi }{4}\)
EITHER
arctan (-b) = \(\frac{\pi }{4}\) (since 1+b2 ≠, 0 for b ∈ R)
OR
-b – b3 = 1 + b2 (or equivalent)
THEN
b =−1
Question
Consider the complex numbers Z1 = cos \(\frac{11\pi }{12}+sin\frac{11\pi }{12}\) and Z2 = cos \(\frac{\pi }{6}+isin\frac{\pi }{6}\)
(a) (i) Find \(\frac{Z_1}{Z_2}\)
▶️Answer/Explanation
Ans: \(\frac{z_{1}}{z_{2}}= cos(\frac{11\pi }{12}-\frac{\pi }{6})+i sin (\frac{11\pi }{12}-\frac{\pi }{6}) = cos \frac{3\pi }{4}+i sin \frac{3\pi }{4}\)
(ii) Find \(\frac{Z_2}{Z_1}\) [3]
▶️Answer/Explanation
Ans: \(\frac{z_{2}}{z_{1}} = cos\frac{3\pi }{4}-isin \frac{3\pi }{4}\)
(b) \(\frac{Z_1}{Z_2}\) and \(\frac{Z_2}{Z_1}\) bare represented by three points O, A and B respectively on an Argand diagram. Determine the area of the triangle OAB. [2]