1. Addition and Subtraction of Complex Numbers
Complex numbers can be added or subtracted by combining their real parts and imaginary parts separately.
If \(z_1 = a + ib\) and \(z_2 = c + id\), then:
- Addition: \(z_1 + z_2 = (a + c) + i(b + d)\)
- Subtraction: \(z_1 - z_2 = (a - c) + i(b - d)\)
Examples:
- \((3 + 4i) + (2 + 5i) = 5 + 9i\)
- \((7 - 3i) - (4 + 2i) = 3 - 5i\)
2. Multiplication of Complex Numbers
To multiply two complex numbers, expand normally using algebra and replace \(i^2\) with \(-1\).
If \(z_1 = a + ib\) and \(z_2 = c + id\), then:
\(z_1 z_2 = (ac - bd) + i(ad + bc)\)
Example:
\((3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i\)
3. Conjugate of a Complex Number and Its Properties
The conjugate of a complex number changes the sign of its imaginary part.
If \(z = a + ib\), then its conjugate is:
\(\overline{z} = a - ib\)
Key Properties:
- \(z + \overline{z} = 2a\)
- \(z - \overline{z} = 2ib\)
- \(z \cdot \overline{z} = a^2 + b^2 = |z|^2\)
- \(\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}\)
Example: If \(z = 5 - 3i\), then \(\overline{z} = 5 + 3i\), and \(z \cdot \overline{z} = 25 + 9 = 34\).
4. Division of Complex Numbers and Rationalisation
To divide one complex number by another, multiply numerator and denominator by the conjugate of the denominator.
If \(z_1 = a + ib\) and \(z_2 = c + id\), then:
\(\dfrac{z_1}{z_2} = \dfrac{a + ib}{c + id} \cdot \dfrac{c - id}{c - id}\)
This removes \(i\) from the denominator.
Example:
\(\dfrac{3 + 2i}{1 - i} = \dfrac{(3 + 2i)(1 + i)}{1^2 - (-1)} = \dfrac{3 + 3i + 2i + 2i^2}{2} = \dfrac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i\)
5. Powers of i and Simplifying Expressions
Using the fundamental fact \(i^2 = -1\), all higher powers of \(i\) repeat in a cycle of 4.
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\)
Simplification rule: For any power \(i^n\), divide \(n\) by 4 and use the remainder.
Examples:
- \(i^{23} = i^{(23 \mod 4 = 3)} = i^3 = -i\)
- \(i^{50} = i^{(50 \mod 4 = 2)} = i^2 = -1\)
6. Solving Simple Linear Equations in z
Sometimes you need to find the value of a complex number \(z\) satisfying an equation.
Example: Solve \((2 + 3i)z = 7 - i\).
Solution:
\(z = \dfrac{7 - i}{2 + 3i}\)
Rationalise by multiplying by the conjugate:
\(z = \dfrac{(7 - i)(2 - 3i)}{(2 + 3i)(2 - 3i)}\)
Compute numerator:
\(7(2) - 7(3i) - 2i + 3i^2 = 14 - 21i - 2i - 3 = 11 - 23i\)
Denominator:
\(2^2 + 3^2 = 13\)
Final answer:
\(z = \dfrac{11}{13} - \dfrac{23}{13}i\)