Definition and Representation
A complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit, which satisfies the fundamental equation i^2 = -1. Every complex number z can be expressed in the form z = x + iy, where x and y are real numbers. Since no real number satisfies the equation i^2 = -1, i was historically termed an imaginary number by René Descartes.
For a complex number z = x + iy, x is designated as the real part, denoted \Re(z), and y is designated as the imaginary part, denoted \Im(z). The set of all complex numbers is represented by the symbol \mathbb{C}.
The representation of a complex number z as x + iy naturally leads to its identification with an ordered pair (\Re(z), \Im(z)) of real numbers. This pair can be interpreted as the coordinates of a point in a two-dimensional space.
The most common space for this representation is the Euclidean plane, referred to as the complex plane or Argand diagram. An alternative visualization involves projecting these coordinates onto the two-dimensional surface of a sphere, known as the Riemann sphere.
In the Cartesian complex plane, the definition of complex numbers using two arbitrary real values suggests the use of Cartesian coordinates. The horizontal axis is conventionally used to display the real part, with values increasing to the right, while the vertical axis represents the imaginary part, with values increasing upwards.
A complex number plotted in this plane can be viewed either as a point (x,y) or as a position vector originating from the origin (0,0) and terminating at this point. The coordinate values of a complex number z are therefore expressed in its Cartesian, rectangular, or algebraic form.
The operations of addition and multiplication acquire a distinct geometric character when complex numbers are viewed as position vectors. Addition corresponds to standard vector addition. Multiplication involves a combination of scaling magnitudes and summing angles relative to the real axis. Multiplying a complex number by i corresponds to rotating its position vector counterclockwise by a quarter turn (90^\circ or \pi/2 radians) about the origin. This can be shown algebraically:
(x+iy) \cdot i = ix + i^2y = ix - y = -y + ix
An alternative coordinate system for the complex plane is the polar coordinate system. This system utilizes the distance of the point z from the origin (O), denoted r, and the angle \theta subtended between the positive real axis and the line segment Oz, measured in a counterclockwise direction.
This leads to the polar form of a complex number:
z = r(\cos \theta + i\sin \theta )
The term r represents the absolute value (or modulus or magnitude) of the complex number z = x + iy, and is calculated as:
r = |z| = \sqrt{x^2 + y^2}
If z is a real number (i.e., y = 0), then r = \sqrt{x^2} = |x|. Thus, the absolute value of a real number is consistent with its absolute value as a complex number. Geometrically the absolute value of a complex number is the distance from the origin to the point representing the complex number in the complex plane.
The argument of z (referred to as the “phase” and denoted \theta or \varphi) is the angle of the radius Oz with the positive real axis. It is written as \arg(z).
The argument can be determined from the rectangular form x + iy with a formula using a half-angle identity covering the range (-\pi, \pi] for the \arg-function:
\theta = \arg(x+iy) = \begin{cases} 2\arctan\left(\dfrac{y}{\sqrt{x^2+y^2}+x}\right) & \text{if } y \neq 0 \text{ or } x > 0, \\ \pi & \text{if } x < 0 \text{ and } y = 0, \\ \text{undefined} & \text{if } x = 0 \text{ and } y = 0. \end{cases}
Together, r and \theta provide another way to represent complex numbers, the polar form, as the modulus and argument uniquely specify a point’s position in the plane. The original rectangular coordinates (x,y) can be recovered from the polar form (r,\theta) using the trigonometric form:
z = r(\cos \theta + i\sin \theta) Utilizing Euler’s formula, e^{i\theta} = \cos \theta + i\sin \theta, this can also be written as:
z = re^{i\theta}
Complex numbers adhere to specific relations and operations. Two complex numbers z_1 = x_1 + iy_1 and z_2 = x_2 + iy_2 are equal if and only if their real and imaginary parts are respectively equal, i.e., x_1 = x_2 and y_1 = y_2.
Nonzero complex numbers in polar form are equal if and only if they possess the same magnitude and their arguments differ by an integer multiple of 2\pi.
The complex conjugate of a complex number z = x + iy is defined as \bar{z} = x - iy. This unary operation cannot be expressed solely through the basic arithmetic operations of addition, subtraction, multiplication, and division.
Geometrically, \bar{z} is the reflection of z about the real axis. Applying the conjugation operation twice returns the original complex number, \bar{\bar{z}} = z, making it an involution. This reflection preserves both the real part and the magnitude of z:
\begin{aligned} & \Re(\bar{z}) = \Re(z) \\ & |\bar{z}| = |z| \end{aligned}
However, the imaginary part and the argument of a complex number z change their sign under conjugation:
\begin{aligned} & \Im(\bar{z}) = -\Im(z) \\ & \arg(\bar{z}) \equiv -\arg(z) \pmod{2\pi} \end{aligned}
The product of a complex number z = x + iy and its conjugate \bar{z} is its absolute square, a non-negative real number equal to the square of its magnitude:
z \cdot \bar{z} = (x+iy)(x-iy) = x^2 - (iy)^2 = x^2 - i^2y^2 = x^2 + y^2 = |z|^2 = |\bar{z}|^2
This property is used in converting a fraction with a complex denominator into an equivalent fraction with a real denominator.
This process, often termed rationalization of the denominator, involves multiplying both the numerator and the denominator by the conjugate of the original denominator.
The real and imaginary parts of a complex number z can be extracted using its conjugate:
\begin{aligned} & \Re(z) = \frac{z + \bar{z}}{2} \\ & \Im(z) = \frac{z - \bar{z}}{2i} \end{aligned}
Furthermore, a complex number is real if and only if it is equal to its own conjugate (z = \bar{z}). Conjugation also distributes over the basic complex arithmetic operations:
\begin{aligned} & \overline{z \pm w} = \bar{z} \pm \bar{w} \\ & \overline{z \cdot w} = \bar{z} \cdot \bar{w} \\ & \overline{\left(\frac{z}{w}\right)} = \frac{\bar{z}}{\bar{w}}, \quad w \neq 0 \end{aligned}
Addition and subtraction of two complex numbers z_1 = x_1 + iy_1 and z_2 = x_2 + iy_2 are performed by separately adding or subtracting their real and imaginary parts:
\begin{aligned} & z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2) \\ & z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2) \end{aligned}
Multiplication of a complex number z = x+iy by a real number k involves multiplying both the real and imaginary parts of z by k:
kz = k(x+iy) = kx + iky
Subtraction can also be viewed as adding the negation of the subtrahend:
z_1 - z_2 = z_1 + (-1)z_2
When visualized in the complex plane, addition has a geometric interpretation: the sum of two complex numbers z_1 and z_2 (viewed as points or vectors) is the point obtained by constructing a parallelogram from the origin O and the points corresponding to z_1 and z_2. If these points are A (for z_1) and B (for z_2), their sum z_1+z_2 corresponds to the fourth vertex X of the parallelogram OAXB.
The multiplication of complex numbers adheres to the distributive property, commutative properties (for both addition and multiplication), and the defining property i^2 = -1. For z_1 = x_1 + iy_1 and z_2 = x_2 + iy_2:
\begin{aligned} z_1 z_2 & = (x_1 + iy_1)(x_2 + iy_2) \\ & = x_1x_2 + ix_1y_2 + iy_1x_2 + i^2y_1y_2 \\ & = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1) \end{aligned}
In particular, the square of a complex number z = x+iy is:
z^2 = (x+iy)^2 = x^2 + 2ixy + (iy)^2 = x^2 - y^2 + 2ixy
The reciprocal of a non-zero complex number z = x + iy can be found using its conjugate:
\frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} = \frac{x - iy}{x^2 + y^2} = \frac{x}{x^2 + y^2} - i\frac{y}{x^2 + y^2}
This is valid since z \neq 0 implies x^2 + y^2 > 0. Division of a complex number z_2 by a non-zero complex number z_1 can then be expressed as z_2 \cdot \frac{1}{z_1}:
\frac{z_2}{z_1} = \frac{(x_1x_2 + y_1y_2) + i(y_1x_2 - x_2y_1)}{x_1^2 + y_1^2}
Operations are often simpler in polar form. Given two complex numbers z_1 = r_1(\cos\theta_1 + i\sin\theta_1) and z_2 = r_2(\cos\theta_2 + i\sin\theta_2), and using the trigonometric sum identities:
\begin{aligned} & \cos(a+b) = \cos a \cos b - \sin a \sin b \\ & \sin(a+b) = \sin a \cos b + \cos a \sin b \end{aligned}
their product is:
\begin{aligned} z_1 z_2 = & r_1 r_2 (\cos\theta_1 \cos\theta_2 - \sin\theta_1 \sin\theta_2) \\ & + i r_1 r_2 (\sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2) \\ = & r_1 r_2 (\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)) \end{aligned}
In terms of Euler’s form, if z_1 = r_1 e^{i\theta_1} and z_2 = r_2 e^{i\theta_2}, then z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}, when multiplying complex numbers, their absolute values are multiplied, and their arguments are added.
For example, multiplication by i (which has r=1, \theta=\pi/2) corresponds to a counter-clockwise rotation by \pi/2 radians. This is consistent with i^2 = -1, as two such rotations result in a rotation by \pi radians (a point (x,y) becomes (-x,-y)).
Similarly, division in polar form is given by:
\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))
Or, in Euler’s form:
\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}
When dividing complex numbers, their absolute values are divided, and the argument of the denominator is subtracted from the argument of the numerator.