Mathswell MATHSWELL

Complex Numbers

Impossible numbers that create endless possibilities

The Imaginary Unit i

We define i as something whose square is -1:

This "impossible" idea unlocks a whole new mathematical universe!

Complex Numbers: a + bi

Every complex number combines real and imaginary parts:

They follow all arithmetic rules, always simplifying to this form.

Factoring

See how complex numbers let us factor any quadratic, even when c is positive!

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Solving Any Quadratic

When the discriminant is negative, we can now find solutions using complex numbers!

Try your own quadratic:

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Key Insight

Accepting the impossible: By defining i such that i² = -1, we can solve previously unsolvable equations.

Algebraic Explorer

Master complex number arithmetic step by step

Worked Examples

Let's see how complex arithmetic works:

Example: Multiplication

Powers of i on the Unit Circle

Watch how powers of i rotate around the circle, repeating every 4 steps:

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Create Your Own Problem

Choose an operation type and I'll create numbers for you to work with:

Key Insights

1. Algebra stays the same: All the algebraic rules you already know still apply. The only new rule is that i² = -1.

2. Powers reveal geometry: The powers of i rotate around the unit circle, connecting algebra to geometry in a beautiful way.

Two Forms of Complex Operations

Multiplication in Rectangular Form

Expand and simplify with i² = -1

Multiplication in Polar Form

Multiply magnitudes, add angles - revealing rotation and scaling!

See Operations in Action

Watch how complex multiplication and division transform the unit square!

Select an operation to see its effect

Key Insights

Multiplication: Scales by |z₁||z₂| and rotates by θ₁ + θ₂

Division: Scales by |z₁|/|z₂| and rotates by θ₁ - θ₂

Complex arithmetic is geometric transformation!

Which Form to Use?

Problem:

Think about which approach would be more convenient: rectangular (a + bi) or polar r(cos θ + i sin θ)?

Rectangular Approach

Polar Approach

From Powers to De Moivre's Theorem

You've seen how polar form simplifies calculating high powers like zn. This is based on the geometric nature of multiplication.

To compute zn, we just multiply z by itself n times. For a complex number in polar form, this means we:

Multiply the moduli n times: The new modulus is rn.

Add the angles n times: The new angle is nθ.

This gives us the celebrated De Moivre's Theorem:

This theorem not only simplifies powers but also paves the way for the next logical step: finding the roots of complex numbers.

Finding Roots:

Find all solutions when z raised to a power equals a complex number

Key Insight

Use Rectangular Form (a+bi) for addition and subtraction. It's simple and direct.

Use Polar Form r(cos θ + i sin θ) for multiplication, division, powers, and roots. It reveals the geometric action of scaling and rotation.

The equation is fundamentally about repeated multiplication (z · z · ... · z). This is why the polar approach works so well:

  • The modulus of z must be the n-th root of the modulus of c.
  • n times the angle of z must be the angle of c (plus full circle rotations).