Complex Numbers Solver

Definition of a Complex Number | Use de-moivres-theorem

A complex number is a number that can be expressed in the form $z = a + ib$, where:

$a$ is the real part of the complex numbers, denoted as $\text{Re}(z)$.
$b$ is the imaginary part of the complex numbers, denoted as $\text{Im}(z)$.
$i$ is the imaginary unit, defined by its property $i^2 = -1$. Thus, $i = \sqrt{-1}$.

The set of complex numbers is denoted by $\mathbb{C}$. Examples include $3 + 2i$, $5 - i$, $7i$ (where $a=0$), and $4$ (a real number, where $b=0$).

Key Properties

Conjugate: The conjugate of a complex number $z = a + ib$ is denoted as $\bar{z}$ or $z^*$ and is given by $\bar{z} = a - ib$. It is obtained by changing the sign of the imaginary part.

If $z = a+ib$, then $\bar{z} = a-ib$

Modulus: The modulus (or magnitude, absolute value) of $z = a+ib$ is denoted by $|z|$ and is calculated as $|z| = \sqrt{a^2 + b^2}$. Geometrically, it represents the distance from the origin $(0,0)$ to the point $(a,b)$ representing $z$ in the complex plane (Argand diagram).

If $z = a+ib$, then $|z| = \sqrt{a^2 + b^2}$

Quadratic Equation Solver

Solve $ax^2 + bx + c = 0$ (Roots can be complex)

Coefficient $a$ (cannot be 0):

Coefficient $b$:

Coefficient $c$:

Roots will appear here...

Argand Diagram Visualizer

Plot $z = a + ib$ and see its properties

Enter Complex Number $z = a+ib$:

+ i

Modulus $|z|$: , Argument $\arg(z)$: ° ( rad)

Cartesian Complex Number Operations

Perform $z_1 \text{ op } z_2$ where $z_1 = a_1+ib_1, z_2 = a_2+ib_2$

Complex Number $z_1$:

+ i

Complex Number $z_2$:

+ i

Operation:

Result will appear here...

Polar & Exponential Forms

A complex number $z = a+ib$ can also be represented in polar form as $z = r(\cos\theta + i\sin\theta)$, where:

$r = |z| = \sqrt{a^2+b^2}$ is the modulus.
$\theta = \arg(z)$ is the argument (angle in radians or degrees), satisfying $\cos\theta = a/r$ and $\sin\theta = b/r$. The principal argument is usually taken in the interval $(-\pi, \pi]$ or $(-180^\circ, 180^\circ]$.

Using Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, this leads to the exponential form:

z = re^{i\theta} \quad (\text{where } \theta \text{ is in radians})

Multiplication and division are simpler in polar/exponential form:
$z_1 z_2 = r_1 r_2 (\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)) = r_1 r_2 e^{i(\theta_1+\theta_2)}$
$\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)) = \frac{r_1}{r_2} e^{i(\theta_1-\theta_2)}$

Cartesian $\leftrightarrow$ Polar/Exponential Conversion

Conversion Type:

Enter Cartesian $z = a+ib$:

+ i

Conversion result will appear here...

Polar Form Operations ($z_1 \text{ op } z_2$)

Enter $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$. Angles in degrees.

$z_1$: ($r_1, \theta_1$)

$z_2$: ($r_2, \theta_2$)

Operation:

Result will appear here...

Solve Simple Complex Equations

Solve for Complex Variable $X$

Type 1: $z_1 + X = z_2 \implies X = z_2 - z_1$

$z_1 = a_1 + ib_1$

$z_2 = a_2 + ib_2$

Result for $X$ will appear here...

Type 2: $z_1 \cdot X = z_2 \implies X = z_2 / z_1$

$z_1 = a_1 + ib_1$ ($z_1 \neq 0$)

$z_2 = a_2 + ib_2$

Result for $X$ will appear here...

Applications of Complex Numbers

Complex numbers are not just abstract mathematical constructs; they are essential tools in various fields of science and engineering:

Electrical Engineering: Fundamental for analyzing AC (alternating current) circuits. Impedance (resistance to AC) and phasors (representing sinusoidal voltages/currents) are expressed as complex numbers, greatly simplifying circuit analysis.
Signal Processing: The Fourier Transform, which decomposes signals (like sound or images) into their constituent frequencies, extensively uses complex numbers. This is vital in audio engineering, image compression (JPEG), and telecommunications.
Quantum Mechanics: The state of a quantum system is described by a complex-valued wave function. Complex numbers are an indispensable part of the mathematical framework of quantum theory.
Control Systems: Used to analyze the stability and response of dynamic systems (e.g., in robotics, aerospace). The locations of poles and zeros (often complex) in the s-plane determine system behavior.
Fluid Dynamics: Complex analysis techniques, particularly conformal mapping, can be used to model two-dimensional potential flow around objects.
Fractal Geometry: Many intricate and beautiful fractals, such as the Mandelbrot set and Julia sets, are generated through iterative calculations involving complex numbers.
Geometry and Transformations: Complex numbers provide a concise way to represent points in a 2D plane and to describe geometric transformations like rotations, translations, and scaling.