Complex Number Calculator

About Complex Number Calculator

Welcome to our Complex Number Calculator, a comprehensive tool designed to perform various operations on complex numbers with detailed step-by-step solutions and visualizations. This calculator is perfect for students, engineers, and anyone dealing with complex numbers in mathematics or engineering fields.

Features of the Complex Number Calculator

• Arithmetic Operations: Addition, Subtraction, Multiplication, and Division of complex numbers.
• Conversions: Convert between rectangular and polar forms.
• Complex Functions: Compute modulus, argument, conjugate, powers, and roots of complex numbers.
• Step-by-Step Solutions: Understand each step involved in the calculations.
• Visualizations: Plot complex numbers on the complex plane.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

Rectangular Form

In rectangular form, a complex number is represented as \( z = a + bi \).

Polar Form

In polar form, a complex number is represented as \( z = r(\cos \theta + i \sin \theta) \) or \( z = re^{i\theta} \), where:

• \( r = |z| \) is the modulus of \( z \)
• \( \theta = \arg(z) \) is the argument of \( z \)

Operations Explained

Below are the operations you can perform with complex numbers using this calculator, along with their respective formulas:

Addition

To add two complex numbers in rectangular form:

\[ (a + bi) + (c + di) = (a + c) + (b + d)i \]

Subtraction

To subtract one complex number from another in rectangular form:

\[ (a + bi) - (c + di) = (a - c) + (b - d)i \]

Multiplication

To multiply two complex numbers in rectangular form:

\[ (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i \]

Alternatively, in polar form:

\[ re^{i\theta} \times se^{i\phi} = (rs)e^{i(\theta + \phi)} \]

Division

To divide one complex number by another in rectangular form:

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

In polar form:

\[ \frac{re^{i\theta}}{se^{i\phi}} = \left(\frac{r}{s}\right)e^{i(\theta - \phi)} \]

Modulus

The modulus of a complex number \( z = a + bi \) is calculated as:

\[ |z| = \sqrt{a^2 + b^2} \]

Argument

The argument of a complex number \( z = a + bi \) is the angle \( \theta \) it makes with the positive real axis, calculated as:

\[ \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \]

Conjugate

The conjugate of a complex number \( z = a + bi \) is:

\[ \overline{z} = a - bi \]

Rectangular to Polar Conversion

To convert a complex number from rectangular to polar form:

\[ z = a + bi \Rightarrow r = \sqrt{a^2 + b^2}, \quad \theta = \tan^{-1}\left(\frac{b}{a}\right) \] \[ z = re^{i\theta} \]

Polar to Rectangular Conversion

To convert a complex number from polar to rectangular form:

\[ z = re^{i\theta} \Rightarrow a = r\cos\theta, \quad b = r\sin\theta \] \[ z = a + bi \]

Power

To raise a complex number \( z \) to an integer power \( n \) in polar form:

\[ z^n = \left(re^{i\theta}\right)^n = r^n e^{in\theta} \]

In rectangular form, use the binomial expansion:

\[ (a + bi)^n \]

Root

To find the \( n \)-th roots of a complex number \( z = re^{i\theta} \) in polar form:

\[ \sqrt[n]{z} = \sqrt[n]{r} e^{i\left(\frac{\theta + 2k\pi}{n}\right)}, \quad k = 0, 1, 2, \dots, n-1 \]

How to Use the Complex Number Calculator

• Enter the first complex number in the desired format (rectangular or polar).
• Choose the operation you want to perform.
• If required, enter the second complex number.
• Specify the input and output forms.
• For operations like Power or Root, provide the necessary exponent.
• Click on "Compute" to process your inputs.
• View the result along with step-by-step solutions and graphs.

Applications of Complex Numbers

Complex numbers are widely used in various fields such as:

• Electrical Engineering: Analyzing AC circuits.
• Quantum Physics: Describing quantum states.
• Signal Processing: Fourier transforms and filters.
• Control Systems: Stability analysis.
• Mathematics: Solving polynomial equations.

Additional Resources

For more information on complex numbers and their applications, check out the following resources:

• Complex Number - Wikipedia
• Intro to complex numbers - Khan Academy

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