Complex Numbers

Complex Numbers

Numbers expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Imaginary Unit

Denoted as i, it satisfies the equation i² = -1.

Real Part

The component 'a' in a complex number a + bi, represented on the x-axis in a two-dimensional plane.

Imaginary Part

The component 'b' in a complex number a + bi, represented on the y-axis in a two-dimensional plane.

Notation

A complex number is denoted as z = a + bi.

Conjugate

The conjugate of a complex number z = a + bi is denoted as \bar{z} = a - bi.

Addition of Complex Numbers

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

Subtraction of Complex Numbers

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

Multiplication of Complex Numbers

(a + bi)(c + di) = ac - bd + (ad + bc)i.

Division of Complex Numbers

\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}.

Polar Form

Representation of complex numbers as z = r(cosθ + i sinθ), where r is the magnitude and θ is the angle.

Magnitude

The magnitude of a complex number, denoted as r, is calculated as r = √(a² + b²).

Angle

The angle θ in polar form is calculated as θ = tan⁻¹(b/a).

Euler's Formula

A key relationship expressed as e^(iθ) = cosθ + i sinθ.

De Moivre's Theorem

A theorem stating (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ).

Complex Conjugate

The complex conjugate of z = a + bi is \bar{z} = a - bi, used in various calculations.