Definitions [4]
z = x + iy, x, y∈ R and \[i=\sqrt{-1}\] is called a complex number.
x ⇒ Real Part Re(z)
iy ⇒ Imaginary Part Im(z)
If Re(z) = x = 0, then the complex number z is purely imaginary.
If Im(z) =y = 0, then complex number z is purely real.
Integral powers of iota (i):
\[\mathrm{i}^2=-1\]
\[\mathrm{i}^3=-\mathrm{i}\]
\[\mathrm{i}^{4}=1\]
In general,
\[1^{4n}=1\], \[\mathrm{i^{4n+1}=i}\], \[\mathrm{i^{4n+2}=-1}\], \[\mathrm{i^{4n+3}=-i}\] ...where n ∈ N
The polar form of a complex number z = x + iy is
z = r(cos θ + i sin θ), where x = r cos θ, y = r sin θ and r = \[r=\sqrt{x^{2}+y^{2}}\].
∴ z = a + ib = r(cos θ + i sin θ) = reiθ,
where r = |z| and θ = arg z is called an exponential form of a complex number.
The cube roots of unity are the solutions of the equation
x³ = 1
They are: 1, \[\frac{-1+i\sqrt{3}}{2}\], \[\frac{-1-i\sqrt{3}}{2}\]
They are denoted by 1, ω, ω²
Theorems and Laws [1]
- Quadratic Equation
ax² + bx + c = 0, where a ≠ 0 - Roots Formula
\[x=\frac{-\mathrm{b}+\sqrt{\mathrm{b}^{2}-4\mathrm{ac}}}{2\mathrm{a}}\]
Discriminant
D = b² − 4ac
- If D < 0 → roots are complex
- If D = 0 → roots are real and equal
Conjugate Roots
- If p + iq is a root, then p − iq is also a root
⇒ Complex roots occur in conjugate pairs
Key Points
| Operation | z₁ = a + ib, z₂ = c + id | Result |
|---|---|---|
| Addition | (a + ib) + (c + id) | (a + c) + i(b + d) |
| Subtraction | (a + ib) − (c + id) | (a − c) + i(b − d) |
| Multiplication | (a + ib)(c + id) | (ac − bd) + i(ad + bc) |
| Division |
\[\frac{\mathrm{a+ib}}{\mathrm{c+id}}\] |
\[\frac{\mathrm{ac+bd}}{\mathrm{c^{2}+d^{2}}}+\mathrm{i}\frac{\mathrm{bc-ad}}{\mathrm{c^{2}+d^{2}}}\] |
Let √(a + ib) = x + iy
- Square both sides
(x + iy)² = a + ib - Expand
x² − y² + 2ixy = a + ib - Equate real and imaginary parts
x² − y² = a
2xy = b - Solve these equations to find x and y
- Then, √(a + ib) = ±(x + iy)
1. Representation
- z = a + ib → point (a, b)
- X-axis → Real part (Re)
- Y-axis → Imaginary part (Im)
2. Modulus
- |z| = distance from origin
- |z| = √(a² + b²)
3. Argument (θ)
- Angle made with +X-axis (anticlockwise)
- θ = tan⁻¹(b/a)
| z = a + ib | Quadrant / Axis | θ = arg z |
|---|---|---|
| a > 0, b = 0 | On the positive real (X) axis | θ = 0 |
| a > 0, b > 0 | Quadrant I | \[\Theta=\tan^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)\] |
| a = 0, b > 0 | On the positive imaginary (Y) axis | θ = π/2 |
| a < 0, b > 0 | Quadrant II | \[\theta=\pi-\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] |
| a < 0, b = 0 | On the negative real (X) axis | θ = π |
| a < 0, b < 0 | Quadrant III | \[\Theta=\pi+\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] \[\theta=\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|-\pi\] |
| a = 0, b < 0 | On the negative imaginary (Y) axis | θ = 3π/2 |
| a > 0, b < 0 | Quadrant IV | \[\Theta=2\pi-\tan^{-1}\left|\frac{\mathrm{b}}{\mathrm{a}}\right|\] |
- ω³ = 1
- 1 + ω + ω² = 0
- ω² = 1/ω
- ω̄ = ω² and \[\left(\overline{\omega}\right)^2=\omega\]
- ω³ⁿ = 1
ω³ⁿ⁺¹ = ω
ω³ⁿ⁺² = ω² - ω + ω² = −1
- ωω² = 1
- arg(ω) = \[\frac{2\pi}{3}\]
arg(ω²) = \[\frac{4\pi}{3}\]
Set of Points in Complex Plane:
Let z = x + iy (variable point) and z₁ = x₁ + iy₁ (fixed point)
| Condition | Geometrical Meaning |
|---|---|
| |z − z₁| | Distance between point z and fixed point z₁ |
| |z − z₁| = r | Circle with centre z₁ and radius r |
| |z − z₁| = |z − z₂| | Perpendicular bisector of the line joining z₁ and z₂ |
