Complex Numbers

A complex number is a number of the form z = a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined by i² = -1. 'a' is called the real part of z, denoted Re(z), and 'b' is called the imaginary part of z, denoted Im(z). Complex numbers extend the real number system, allowing us to find roots of negative numbers. Operations such as addition, subtraction, multiplication, and division are performed by treating 'i' as a variable, remembering that i² = -1. For division, we multiply the numerator and denominator by the complex conjugate of the denominator.

An Argand diagram is a graphical representation of complex numbers. The complex number z = a + bi is represented as a point (a, b) in a Cartesian coordinate system, where the horizontal axis represents the real part (Real axis) and the vertical axis represents the imaginary part (Imaginary axis). This plane is often referred to as the complex plane.

The modulus of a complex number z = a + bi, denoted |z| or r, is the distance from the origin to the point (a, b) on the Argand diagram. It is always a non-negative real number. The argument of a complex number z, denoted arg(z) or θ, is the angle (in radians) that the line segment from the origin to the point (a, b) makes with the positive real axis, measured anti-clockwise. The principal argument is typically restricted to the interval (-π, π]. To find θ, we use tan θ = b/a and adjust for the quadrant in which the complex number lies.

A complex number z = a + bi can also be expressed in polar form (or modulus-argument form) using its modulus r and argument θ. From the Argand diagram, we can see that a = r cos θ and b = r sin θ. Substituting these into z = a + bi gives the polar form. This form is particularly useful for multiplication, division, powers, and roots of complex numbers.

De Moivre's Theorem provides a formula for finding powers of complex numbers in polar form. For any integer n, the nth power of a complex number z = r(cos θ + i sin θ) is given by raising the modulus to the power of n and multiplying the argument by n. This theorem is also fundamental for finding the nth roots of complex numbers.

The nth roots of unity are the solutions to the equation zⁿ = 1. These roots can be found using De Moivre's Theorem by expressing 1 in polar form (1 = 1(cos(0) + i sin(0))). There are always n distinct nth roots of unity. Geometrically, these roots are equally spaced points on the unit circle in the Argand diagram, forming the vertices of a regular n-sided polygon. The sum of the nth roots of unity is always 0 (for n > 1), and their product is (-1)^(n-1).