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Complex Numbers

Subject: Mathematics
Topic: 6
Cambridge Code: 0580

Introduction to Complex Numbers​

Complex number - Number of form a+bia + bi

Where:

  • a = real part
  • b = imaginary part
  • i = imaginary unit, i2=−1i^2 = -1

Imaginary Unit​

i=−1i = \sqrt{-1} i2=−1i^2 = -1 i3=−ii^3 = -i i4=1i^4 = 1

Pattern repeats every 4 powers

Complex Plane (Argand Diagram)​

Real part: Horizontal axis Imaginary part: Vertical axis Complex number z = a + bi: Point (a, b)

Operations with Complex Numbers​

Addition and Subtraction​

(a+bi)+(c+di)=(a+c)+(b+d)i(a + bi) + (c + di) = (a + c) + (b + d)i (a+bi)−(c+di)=(a−c)+(b−d)i(a + bi) - (c + di) = (a - c) + (b - d)i

Example:

  • (3+2i)+(1−i)=4+i(3 + 2i) + (1 - i) = 4 + i
  • (3+2i)−(1−i)=2+3i(3 + 2i) - (1 - i) = 2 + 3i

Multiplication​

(a+bi)(c+di)=ac+adi+bci+bdi2(a + bi)(c + di) = ac + adi + bci + bdi^2 =(ac−bd)+(ad+bc)i= (ac - bd) + (ad + bc)i

Example:

  • (2+i)(3−2i)=6−4i+3i−2i2(2 + i)(3 - 2i) = 6 - 4i + 3i - 2i^2
  • =6−4i+3i+2=8−i= 6 - 4i + 3i + 2 = 8 - i

Division​

a+bic+di=(a+bi)(c−di)(c+di)(c−di)\frac{a + bi}{c + di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}

Example: 2+i1+i=(2+i)(1−i)(1+i)(1−i)=2−2i+i−i21−i2\frac{2 + i}{1 + i} = \frac{(2+i)(1-i)}{(1+i)(1-i)} = \frac{2 - 2i + i - i^2}{1 - i^2} =2−i+11+1=3−i2=32−12i= \frac{2 - i + 1}{1 + 1} = \frac{3 - i}{2} = \frac{3}{2} - \frac{1}{2}i

Complex Conjugate​

Conjugate of z=a+biz = a + bi is zˉ=a−bi\bar{z} = a - bi

Properties:

  • z⋅zˉ=a2+b2z \cdot \bar{z} = a^2 + b^2 (always real, always positive)
  • z1+z2‾=z1ˉ+z2ˉ\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}
  • z1⋅z2‾=z1ˉ⋅z2ˉ\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}

Modulus and Argument​

Modulus (Absolute Value)​

Modulus of z=a+biz = a + bi: ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

  • Distance from origin in Argand diagram
  • Always non-negative real number

Properties:

  • ∣z1⋅z2∣=∣z1∣⋅∣z2∣|z_1 \cdot z_2| = |z_1| \cdot |z_2|
  • ∣z1z2∣=∣z1∣∣z2∣\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}
  • ∣z+w∣≤∣z∣+∣w∣|z + w| ≤ |z| + |w| (triangle inequality)

Argument​

Argument of z=a+biz = a + bi: arg⁡(z)=θ\arg(z) = \theta

Where tan⁡θ=ba\tan \theta = \frac{b}{a} and −π<θ≤π-π < \theta ≤ π

  • Angle from positive real axis (counterclockwise)
  • Usually given in radians or degrees

Example:

  • z=1+iz = 1 + i: arg⁡(z)=π4\arg(z) = \frac{\pi}{4} or 45°
  • z=−1+iz = -1 + i: arg⁡(z)=3π4\arg(z) = \frac{3\pi}{4} or 135°

Polar Form​

Polar form - Expressing complex number using r and θ

z=r(cos⁡θ+isin⁡θ)=r cis θz = r(\cos \theta + i \sin \theta) = r \text{ cis } \theta

Or using Euler's formula: z=reiθz = re^{i\theta}

Conversion​

From rectangular to polar:

  • r=∣z∣=a2+b2r = |z| = \sqrt{a^2 + b^2}
  • θ=arg⁡(z)\theta = \arg(z)

From polar to rectangular:

  • a=rcos⁡θa = r\cos \theta
  • b=rsin⁡θb = r\sin \theta

Operations in Polar Form​

Multiplication: z1z2=r1r2 cis(θ1+θ2)z_1 z_2 = r_1 r_2 \text{ cis}(\theta_1 + \theta_2)

Division: z1z2=r1r2 cis(θ1−θ2)\frac{z_1}{z_2} = \frac{r_1}{r_2} \text{ cis}(\theta_1 - \theta_2)

Powers (De Moivre's Theorem): zn=rn cis(nθ)z^n = r^n \text{ cis}(n\theta)

De Moivre's Theorem​

(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)

Or in exponential form: (eiθ)n=einθ(e^{i\theta})^n = e^{in\theta}

Finding Roots​

nth roots of z: zn=rn cis(θ+2πkn)\sqrt[n]{z} = \sqrt[n]{r} \text{ cis}\left(\frac{\theta + 2\pi k}{n}\right)

Where k = 0, 1, 2, ..., n-1

Example: Find cube roots of 8

  • z=8z = 8, so r=8r = 8, θ=0\theta = 0
  • 83=2 cis(0+2πk3)\sqrt[3]{8} = 2 \text{ cis}\left(\frac{0 + 2\pi k}{3}\right) for k = 0, 1, 2
  • Roots: 22, 2 cis(120°)2\text{ cis}(120°), 2 cis(240°)2\text{ cis}(240°)

Roots of Unity​

nth roots of unity - Solutions to zn=1z^n = 1

z=ei2πk/n=cos⁡(2πkn)+isin⁡(2πkn)z = e^{i2\pi k/n} = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right)

For k = 0, 1, 2, ..., n-1

Properties:

  • n distinct roots
  • Equally spaced on unit circle
  • Form regular n-gon

Sum of nth roots of unity = 0

Equations with Complex Numbers​

Quadratic Equations​

ax2+bx+c=0ax^2 + bx + c = 0

x=−b±b2−4ac2ax = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}

If discriminant < 0, roots are complex conjugates

Example: x2−2x+5=0x^2 - 2x + 5 = 0 x=2±4−202=2±−162=1±2ix = \frac{2 ± \sqrt{4-20}}{2} = \frac{2 ± \sqrt{-16}}{2} = 1 ± 2i

Polynomial Equations​

Fundamental Theorem of Algebra:

  • Polynomial of degree n has exactly n roots (counting multiplicity)
  • Complex roots occur in conjugate pairs (for real coefficients)

Applications​

Electrical Engineering​

AC circuits: Uses complex numbers to represent impedance

Quantum Mechanics​

Wave functions: Often complex-valued

Fluid Dynamics​

Potential flow: Complex analytic functions

Key Points​

  1. i2=−1i^2 = -1 defines imaginary unit
  2. Complex number = real part + imaginary part
  3. Operations follow usual algebra rules
  4. Conjugate used for division
  5. Modulus is distance from origin
  6. Argument is angle from real axis
  7. Polar form: z=r cisθz = r\text{ cis}\theta
  8. De Moivre's theorem for powers/roots
  9. Roots occur in conjugate pairs (for real polynomials)
  10. nth roots of unity equally distributed

Practice Questions​

  1. Perform operations with complex numbers
  2. Find modulus and argument
  3. Convert between forms
  4. Apply De Moivre's theorem
  5. Find roots of complex numbers
  6. Solve complex equations
  7. Prove identities
  8. Apply to practical problems
  9. Work with roots of unity
  10. Solve polynomial equations

Revision Tips​

  • Visualize in Argand diagram
  • Memorize De Moivre's theorem
  • Know conjugate properties
  • Practice conversions between forms
  • Understand argument calculation
  • Work with Euler's formula
  • Practice finding roots
  • Connect to real applications
  • Verify answers using conjugates