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Complex Number Calculator

Perform operations with complex numbers and get detailed step-by-step solutions!

Complex Number Calculator
Compute Complex Number Operations:

Try the following examples:

[Addition] [Subtraction] [Multiplication] [Division] [Modulus] [Argument] [Conjugate] [Rectangular to Polar] [Polar to Rectangular] [Power] [Roots]
First Complex Number:
Second Complex Number:
Operation:
Exponent (for Power or Root):

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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

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:

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

Applications of Complex Numbers

Complex numbers are widely used in various fields such as:

Additional Resources

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

Reference this content, page, or tool as:

"Complex Number Calculator" at https://MiniWebtool.com/complex-number-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Nov 27, 2024

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