Revision Guide: Complex Numbers and Quadratic Equations

A complex number is in the form a + ib, with a and b as real numbers.

In z = a + ib, 'a' is the real part (Re z) and 'b' is the imaginary part (Im z).

Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.

z1 + z2 = (a + c) + i(b + d); closure, commutative, and associative properties hold.

z1 × z2 = (ac – bd) + i(ad + bc); maintains closure and commutativity.

The conjugate of z = a + ib is denoted as z̅ = a - ib; mirrors across the real axis.

The modulus |z| is √(a² + b²), the distance from the origin in the Argand plane.

i is defined such that i² = -1. Powers of i repeat every four: i, -1, -i, 1.

For ax² + bx + c = 0, use the discriminant D = b² – 4ac to determine roots.

If D < 0, roots are complex; if D = 0, roots are real and equal; if D > 0, distinct real roots.

Solutions are given by x = [-b ± √D] / (2a). Use this formula for finding roots.

The solutions of zⁿ = 1 are called roots of unity; evenly spaced points on the unit circle.

Complex numbers can be depicted as points in the Argand plane, with x-axis as Re and y-axis as Im.

Complex number operations (+, -, *, /) follow similar laws as real numbers.

Complex numbers represent vectors; their angle and magnitude can be analyzed geometrically.

Identities like (z1 + z2)² = z1² + z2² + 2z1z2 hold for complex numbers.

The square root of a negative number a is expressed as √(-a) = i√(a), introducing complex roots.

For z = a + ib, the multiplicative inverse is z⁻¹ = (a - ib)/(a² + b²).

Complex numbers model phenomena in physics and engineering, like electrical circuits.

Many confuse the imaginary unit 'i' with real numbers; understand its unique properties.

A complex number is of the form a + bi, where a and b are real. E.g., 3 + 4i.

For z = a + bi, Re(z) = a, Im(z) = b. They represent its real and imaginary parts.

Two complex numbers z1 = a + bi and z2 = c + di are equal if a = c and b = d.

The complex conjugate of z = a + bi is denoted as z = a - bi. It reflects across the real axis.

The modulus |z| of a + bi is |z| = √(a² + b²), representing the distance from the origin in the Argand plane.

For z1 = a + bi and z2 = c + di, the sum is z1 + z2 = (a + c) + i(b + d).

z1 - z2 = z1 + (-z2). Example: (3 + 2i) - (1 + i) = 2 + i.

z1 * z2 = (ac - bd) + i(ad + bc). E.g., (1 + i)(2 + 2i) = 0 + 4i.

The division z1/z2 = (z1 * conjugate(z2)) / |z2|². Example: (2 + 3i)/(1 + i).

Standard form: ax² + bx + c = 0. Roots can be complex if D < 0.

D = b² - 4ac determines the nature of roots: D > 0 (real), D = 0 (repeated), D < 0 (complex).

Roots are given by x = (-b ± √D)/(2a). Complex roots occur when D < 0.

For any integer k, i^4k = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i.

The square root of -a (a > 0) is expressed as i√a. Example: √(-4) = 2i.

e.g., (z1 + z2)² = z1² + z2² + 2z1z2. Useful for expanding expressions.

Complex numbers are represented in a 2D plane, where x-axis is real and y-axis is imaginary.

The graph of a quadratic function is a parabola, which can open upwards or downwards.

Complex numbers are used in electrical engineering, fluid dynamics, and quantum physics.

Confusing real and imaginary parts. Remember, a + bi is distinct from a and bi alone.

The complex numbers satisfying z^n = 1 are called roots of unity. They're spaced evenly on the unit circle.