Complex Numbers: Polar and Rectangular Forms

Converting Complex Numbers Between Forms

Complex numbers can be represented in two main forms: rectangular form (a + bi) and polar form (r(cos θ + i sin θ) or reiθ). Converting between these forms is a fundamental skill in trigonometry and complex analysis.

• Rectangular Form: , where a is the real part and b is the imaginary part.

• Polar Form: or , where r is the modulus (magnitude) and θ is the argument (angle).

Key Formulas for Conversion

• From Rectangular to Polar:

  • Magnitude:

  • Argument: (adjust for quadrant)

• From Polar to Rectangular:

  • Real part:

  • Imaginary part:

  • So, becomes

Example 1: Convert to Rectangular Form

Problem: Convert the complex number to rectangular form.

• Recall:

• So,

• Therefore,

Example 2: Convert to Rectangular Form

Problem: Convert the complex number to rectangular form.

• Therefore,

Summary Table: Conversion Between Forms

Additional info: The problems provided are typical exercises in converting complex numbers from polar (cis) form to rectangular form, a key skill in trigonometry and complex analysis.