Unit 4: Complex Numbers & Quadratic Equations
Contents
1. Introduction to Complex Numbers
A Complex Number is a number that can be expressed in the form a + ib, where a and b are real numbers, and i (iota) is the imaginary unit.
- Real Part: Re(z) = a
- Imaginary Part: Im(z) = b
- Need for Complex Numbers: To solve equations like x2 + 1 = 0, where no real number exists as a solution.
i2 = -1, \quad i3 = -i, \quad i4 = 1
2. Algebra of Complex Numbers
Arithmetic operations on complex numbers follow standard algebraic rules, treating i as a variable and replacing i2 with -1.
- Addition/Subtraction: (a + ib) ± (c + id) = (a ± c) + i(b ± d)
- Multiplication: (a + ib)(c + id) = (ac - bd) + i(ad + bc)
- Conjugate: The conjugate of z = a + ib is z̄ = a - ib.
- Modulus: The distance from origin, |z| = √(a2 + b2).
3. Argand Plane and Polar Form
The Argand Plane (Complex Plane) is a geometric representation where the x-axis is the Real axis and the y-axis is the Imaginary axis.
Polar Representation
A complex number can be represented using its modulus (r) and the angle (θ) it makes with the positive real axis.
Where r = |z| and θ = arg(z) (Argument/Amplitude)
4. Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has exactly n complex roots (counting multiplicity).
Key Takeaway: An equation of degree 2 (quadratic) will always have exactly 2 roots in the complex number system.
5. Complex Roots of Quadratic Equations
For a quadratic equation ax2 + bx + c = 0, we check the discriminant D = b2 - 4ac.
- If D < 0, the roots are imaginary and occur in conjugate pairs.
6. Exam Focus Enhancements
- Powers of i: Simplify high powers of i by dividing the exponent by 4 and looking at the remainder. i102 = i2 = -1.
- Conjugate Pair: If one root of a quadratic is 2 + 3i, the other must be 2 - 3i (provided coefficients are real).
- Argument: Be careful with the quadrant when calculating θ = tan-1(b/a). Check the signs of a and b first.
- √(-a) · √(-b): Students often write √(ab). Correct result is i√(a) · i√(b) = i2√(ab) = -√(ab).
- Modulus: Forgetting that |z| is always a non-negative real number. |3 - 4i| = √(32 + (-4)2) = 5, not 5i.
- Polar Form: Forgetting that θ (the principal argument) should be in the range (-π, π].
Q: What is a purely imaginary number?
A: A complex number where the real part is zero (a=0), such as 5i.
Q: Why is the conjugate useful?
A: It is used to rationalize the denominator in complex division, turning it into a real number (z · z̄ = |z|2).