Complex Numbers: Worked Examples

These problems progress from rectangular arithmetic to quadrant-aware polar form and De Moivre's theorem. Each root set is checked against the original power equation.

Imaginary-Unit Cycle with Positive and Negative Powers

Simplify i15i^{15}, i42i^{42}, i100i^{100}, and i−3i^{-3}.

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Quadratic with Purely Imaginary Roots

Solve x2+25=0x^2+25=0 over the complex numbers.

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Rectangular Addition and Subtraction

Evaluate (3+4i)−(5−2i)+(1+i)(3+4i)-(5-2i)+(1+i).

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Multiply Two Rectangular Complex Numbers

Compute (2−3i)(4+5i)(2-3i)(4+5i).

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Division by a Complex Conjugate

Simplify 3+2i1−4i\dfrac{3+2i}{1-4i} in rectangular form.

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Conjugate, Modulus, and Argument in Quadrant IV

For z=3−4iz=3-4i, determine z‾\overline z, ∣z∣|z|, and a principal argument.

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Rectangular-to-Polar Conversion in Quadrant II

Write z=−1+i3z=-1+i\sqrt3 in polar and exponential forms.

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Multiply by i as a Quarter-Turn Rotation

Find i(1+i)i(1+i) and describe its effect in the Argand plane.

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Fourth Power Using De Moivre's Theorem

Calculate (−1+i3)4(-1+i\sqrt3)^4 in rectangular form.

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All Fourth Roots of Unity with Exact Indexing

Find every distinct solution of w4=1w^4=1.

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Three Cube Roots of a Positive Real Number

Find every complex ww satisfying w3=8w^3=8.

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