Properties of Complex Numbers

• Complex numbers are written in the form x + iy where x and y are real numbers and i is imaginary unit, defined by the property that i² = -1.
• The real part (Re) of the complex number, represented by x, forms the horizontal axis on the complex plane, and the imaginary part (Im), represented by y, forms the vertical axis.
• Complex numbers can be added, subtracted, multiplied, and divided just like real numbers. Please note that when multiplying and dividing, care should be taken with the imaginary unit i (e.g., i^2 = -1).

Modulus and Argument of Complex Numbers

• The modulus or absolute value of a complex number x + iy is given by the distance from the origin to the point representing the complex number in the complex plane, calculated using Pythagorean theorem as √(x² + y²).
• The argument of a complex number is the angle formed by the line from the origin to the point and the positive real axis on the complex plane, counted in an anti-clockly direction. It has value between 0 and 2π.
• The complex number can be represented in polar form r(cos θ + isin θ) where r is the modulus and θ is the argument.

Complex Conjugates

• The complex conjugate of a function represented as x + iy is identified as x - iy.
• Multiplying a complex number and its complex conjugate yields a real number, which is the square of the magnitude of the original complex number.

Functions on Complex Numbers

• Just like real numbers, we can build functions on complex numbers. The most common functions include power functions, exponential functions, and trigonometric functions.
• The fundamental theorem of algebra states that every non-constant polynomial equation has at least one complex root. This fact underpins the usage of complex numbers in finding solutions for a wide range of functions in mathematics.

Complex-Valued Functions

• Real functions produce a real number for each real input. Similarly, a function which produces a complex number for each complex input is called a complex-valued function.
• Functions of a complex variable carry many similar properties to their counterparts with real variables although care should be taken as the behaviour in the complex plane can often be counter-intuitive.

Loci in the Complex Plane

• The locus of a complex number z is the set of all points on the complex plane that satisfy a certain condition.
• Loci are useful in visually representing properties or conditions of complex numbers.
• Important loci include circles, lines and rays in the complex plane.