📝 Complex Numbers

Part 1 of 7 — Imaginary Numbers

i = √(-1), so i² = -1.

√(-a) = i√a for a > 0.

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Key Insight: Complex number: a + bi where a is real part, b is imaginary part.

SAT Tip: Pure imaginary: bi (when a = 0); real: a (when b = 0).

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Complex Number Operations

Part 2 of 7 — Complex Number Operations

Add/subtract: combine real parts and imaginary parts separately.

(a + bi) + (c + di) = (a+c) + (b+d)i.

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Key Insight: Multiply using FOIL: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.

SAT Tip: Remember: i² = -1, so replace i² with -1.

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Complex Number Applications

Part 3 of 7 — Complex Number Applications

Complex solutions come in conjugate pairs: a + bi and a - bi.

If the discriminant b² - 4ac < 0, the quadratic has complex roots.

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Key Insight: x = (-b ± i√(4ac - b²)) / (2a).

SAT Tip: Every polynomial of degree n has exactly n roots (counting complex and repeated).

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Powers of i

Part 4 of 7 — Powers of i

i¹ = i, i² = -1, i³ = -i, i⁴ = 1, then the pattern repeats.

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

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Key Insight: Remainder 0 → 1, remainder 1 → i, remainder 2 → -1, remainder 3 → -i.

SAT Tip: Example: i²³ → 23 ÷ 4 = 5 remainder 3 → i²³ = -i.

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Complex Conjugates

Part 5 of 7 — Complex Conjugates

Conjugate of a + bi is a - bi.

(a + bi)(a - bi) = a² + b² (always real and positive).

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Key Insight: Use conjugates to divide complex numbers: multiply numerator and denominator by the conjugate of the denominator.

SAT Tip: Rationalizing: (3 + 2i)/(1 - i) × (1 + i)/(1 + i).

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Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Conjugate of a + bi is a - bi.

(a + bi)(a - bi) = a² + b² (always real and positive).

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Key Insight: Use conjugates to divide complex numbers: multiply numerator and denominator by the conjugate of the denominator.

SAT Tip: Rationalizing: (3 + 2i)/(1 - i) × (1 + i)/(1 + i).

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Review & Applications

Part 7 of 7 — Review & Applications

Conjugate of a + bi is a - bi.

(a + bi)(a - bi) = a² + b² (always real and positive).

Check Your Understanding 🎯

Key Insight: Use conjugates to divide complex numbers: multiply numerator and denominator by the conjugate of the denominator.

SAT Tip: Rationalizing: (3 + 2i)/(1 - i) × (1 + i)/(1 + i).

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Match the Concepts 🔍