Write Z1 And Z2 In Polar Form

To write z1 and z2 in polar form, you must transform each complex number from its rectangular (Cartesian) representation into a magnitude‑angle representation. This conversion relies on the modulus (distance from the origin) and the argument (angle with the positive real axis). The resulting expressions take the shape r ∠ θ or r e^{iθ}, where r is the modulus and θ is the argument in radians or degrees. Understanding this process not only simplifies algebraic manipulations but also provides insight into the geometric interpretation of complex numbers.

Introduction

Complex numbers are typically introduced as a + bi, where a is the real part and b is the imaginary part. While this rectangular form is intuitive for addition and subtraction, many mathematical operations—especially multiplication, division, and exponentiation—are more naturally performed in polar form. In polar representation, a complex number is described by its distance from the origin (r) and the angle (θ) it makes with the positive real axis. Converting z1 and z2 from a + bi to r ∠ θ therefore involves two key calculations: the modulus and the argument. This article walks you through each step, explains the underlying science, and answers the most frequently asked questions.

Steps

The procedure to write z1 and z2 in polar form can be broken down into a clear sequence of actions. Follow these steps for each complex number:

1. Calculate the modulus (r).
   The modulus is the Euclidean distance from the origin to the point (a, b) in the complex plane. It is given by
   [ r = \sqrt{a^{2} + b^{2}} ]
   This value is always non‑negative.

2. Determine the argument (θ).
   The argument is the angle formed with the positive real axis. It can be found using the arctangent function:
   [ θ = \arctan!\left(\frac{b}{a}\right) ]
   However, because the arctangent alone does not indicate the correct quadrant, you must adjust θ based on the signs of a and b. Many calculators and software packages provide an atan2 function that automatically returns the correct angle.

3. Express the polar coordinates.
   Combine the modulus and argument into the polar notation:
   [ z = r \angle θ ]
   If you prefer exponential notation, you can write z = r e^{iθ}.

4. Apply the same procedure to z2.
   Repeat steps 1‑3 for the second complex number, using its own real and imaginary components.

Example Calculation

Suppose z1 = 3 + 4i and z2 = -2 + 2i.

• For z1:

  • Modulus: r₁ = √(3² + 4²) = 5
  • Argument: θ₁ = arctan(4/3) ≈ 53.13° (first quadrant)
  • Polar form: z1 = 5 ∠ 53.13°

• For z2:

  • Modulus: r₂ = √((-2)² + 2²) = √8 ≈ 2.83
  • Argument: θ₂ = arctan(2/(-2)) = arctan(-1) = -45°, but since the point lies in the second quadrant, add 180° → θ₂ ≈ 135°
  • Polar form: z2 = 2.83 ∠ 135°

Scientific Explanation

The conversion from rectangular to polar coordinates is rooted in basic trigonometry and vector geometry. The modulus r corresponds to the length of the vector representing the complex number, while the argument θ represents its direction. These concepts are formalized through the following relationships:

• Modulus calculation: The expression √(a² + b²) originates from the Pythagorean theorem, treating the real and imaginary parts as orthogonal components of a right‑angled triangle.
• Argument calculation: The tangent of the angle θ equals the ratio of the opposite side (b) to the adjacent side (a). Hence, θ = arctan(b/a). The need for quadrant correction stems from the periodic nature of the tangent function, which repeats every

The periodic nature of the tangent function, which repeats every π radians, necessitates careful quadrant adjustment to ensure the argument θ accurately reflects the complex number’s position. Here’s how to resolve ambiguities:

• Quadrant I (a > 0, b ≥ 0): θ = arctan(b/a) (no adjustment needed).
• Quadrant II (a < 0, b ≥ 0): θ = arctan(b/a) + π (or