Complex numbers in polar form are a cornerstone of higher mathematics, offering a unique perspective on arithmetic operations. This comprehensive guide delves into the details of multiplication and division in polar form, highlighting the significance of modulus and argument in these processes. Understanding these concepts is crucial for students aiming to master complex number manipulations.

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Multiplication of Complex Numbers in Polar Form

• Complex Number in Polar Form: Represented as $z = r e^{i\theta}$, where:
  • $r$ = Modulus (distance from origin)
  • $\theta$ = Argument (angle from positive x-axis)
• Multiplying Two Complex Numbers: For $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$:
  • Procedure: Multiply their moduli (r-values) and add their arguments (θ-values).
  • Result: $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$
• Geometric Interpretation:
  • On Argand Plane, multiplication:
    • Stretches by the modulus of one number.
    • Rotates by the angle of the other number.
  • Helps visualize the effect on position and size of complex numbers.

Division of Complex Numbers in Polar Form

• Complex Number in Polar Form: Represented as $z = r e^{i\theta}$, where:
  • $r$ = Modulus (distance from origin)
  • $\theta$ = Argument (angle from positive x-axis)
• Dividing Two Complex Numbers: For $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$:
  • Procedure: Divide their moduli (r-values) and subtract their arguments (θ-values).
  • Result: $\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$
• Geometric Interpretation:
  • On Argand Plane, division:
    • Shrinks by the modulus of the divisor.
    • Rotates by the negative of the divisor's angle.
  • Helps visualize the effect on position and size of complex numbers.

Example Questions

Problem 1: Multiplying Complex Numbers

• Given: $4 e^{i\pi/3}$ and $4 e^{i\pi/3}$
• Step 1: Multiply Moduli
  • $4 \times 2 = 8$
• Step 2: Add Arguments
  • Convert to common denominator: $\frac{\pi}{3} = \frac{4\pi}{12}, \frac{\pi}{4} = \frac{3\pi}{12}$
  • Add: $\frac{4\pi}{12} + \frac{3\pi}{12} = \frac{7\pi}{12}$
• Result: $8 e^{i(7\pi/12)}$

Problem 2: Dividing Complex Numbers

• Given: $6 e^{i\pi/2}$ and $3 e^{i\pi/6}$
• Step 1: Divide Moduli
  • $6 \div 3 = 2$
• Step 2: Subtract Arguments
  • Convert to common denominator: $\frac{\pi}{2} = \frac{3\pi}{6}, \frac{\pi}{6} = \frac{1\pi}{6}$
  • Subtract: $\frac{3\pi}{6} - \frac{1\pi}{6} = \frac{2\pi}{6}$, simplify to $\frac{\pi}{3}$
• Result: $2 e^{i(\pi/3)}$