Complex Numbers and Quadratic Equations

The real number system is insufficient to solve equations like x² + 1 = 0. By introducing the imaginary unit i, where i² = -1, we extend real numbers to complex numbers, unlocking solutions to all polynomial equations.

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Complex Numbers

Key formulas

A complex number is z = a + ib, where a, b ∈ R.

a = Re(z) is the real part.

b = Im(z) is the imaginary part.

When b = 0, z is real. When a = 0, z is purely imaginary.

Equality: a + ib = c + id if and only if a = c and b = d.

Powers of i:
i¹ = i, i² = −1, i³ = −i, i⁴ = 1 (cycle of period 4)

To find iⁿ: divide n by 4; use the remainder.

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Operations

Addition: (a + ib) + (c + id) = (a+c) + i(b+d)
Subtraction: (a + ib) − (c + id) = (a−c) + i(b−d)
Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc)
Division: (a+ib)/(c+id) = [(a+ib)(c−id)] / (c²+d²)

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Modulus and Conjugate

Key formulas

Conjugate of z = a + ib is z-bar = a − ib.

Modulus of z is |z| = √(a² + b²).

z · z-bar = |z|² (used in division).

Key properties: |z1 · z2| = |z1| · |z2|; |z1/z2| = |z1|/|z2|.

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Argand Plane

A complex number a + ib is represented by the point (a, b) on the Argand plane (complex plane). The argument (or angle) θ satisfies tan θ = b/a.

Polar form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z).

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Quadratic Equations

For a quadratic ax² + bx + c = 0 (a ≠ 0), the discriminant is D = b² − 4ac.
D > 0: Two distinct real roots.
D = 0: Two equal real roots (x = −b/2a).
D < 0: Two complex conjugate roots x = (−b ± i·√(|D|)) / 2a.

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Worked Examples

Example 1

Compute i³⁵.
35 = 4×8 + 3, so i³⁵ = i³ = −i.

Example 2

Express (3 + 2i)(1 − i) in a + ib form.
= 3 − 3i + 2i − 2i² = 3 − i + 2 = 5 − i.

Example 3

Find |3 − 4i|.
|3 − 4i| = √(9 + 16) = √(25) = 5.

Example 4

Write 1/(2 + 3i) in a + ib form.
Multiply numerator and denominator by the conjugate (2 − 3i): (2−3i)/(4+9) = 2/13 − 3i/13.

Example 5

Find the conjugate of (2 + 3i)/(1 − i).
First simplify: (2+3i)(1+i)/((1)²+(1)²) = (2+2i+3i+3i²)/2 = (2+5i−3)/2 = (−1+5i)/2. Conjugate = (−1−5i)/2.

Example 6

Solve x² + 4 = 0 over C.
x² = −4, so x = ±2i.

Example 7

Solve x² − 4x + 13 = 0.
D = 16 − 52 = −36. x = (4 ± √(−36))/2 = (4 ± 6i)/2 = 2 ± 3i.

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Common mistakes

Forgetting i² = −1 during multiplication (writing i² = 1 is a common error).
Not multiplying by the conjugate correctly when simplifying a/z.
Confusing the modulus |z| with the imaginary part Im(z).

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Summary

Complex numbers extend real numbers by introducing i. All standard algebraic operations apply. Every quadratic equation has solutions in C. The Argand plane gives a geometric interpretation.