This section provides step-by-step examples on how to perform operations with complex numbers in both rectangular (standard) and polar forms. You will learn how to handle the imaginary unit ii, perform arithmetic on complex numbers, find roots using De Moivre's theorem, and interpret them geometrically on the Argand plane.

Example

Case Study 1: Simplifying Powers of i Simplify the following powers of the imaginary unit ii:

  1. i15i^{15}
  2. i42i^{42}
  3. i100i^{100}
  4. i−3i^{-3}

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Example

Case Study 2: Solving Equations with Complex Roots Solve the quadratic equation x2+25=0x^2 + 25 = 0.

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Example

Example 1: Addition and Subtraction (Basic) Evaluate (3+4i)−(5−2i)+(1+i)(3 + 4i) - (5 - 2i) + (1 + i).

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Example

Example 2: Multiplication (Intermediate) Multiply the complex numbers: (2−3i)(4+5i)(2 - 3i)(4 + 5i).

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Example 3: Division and Complex Conjugates (Advanced) Divide the complex numbers and write the answer in standard form: 3+2i1−4i\frac{3 + 2i}{1 - 4i}.

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Example 1: Converting to Polar Form (Intermediate) Convert the complex number z=−1+i3z = -1 + i\sqrt{3} to polar form.

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Example 2: Applying De Moivre's Theorem (Advanced) Calculate (−1+i3)4(-1 + i\sqrt{3})^4 and write the result in standard rectangular form a+bia + bi.

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