Complex Number Calculator
Perform operations with complex numbers and get detailed step-by-step solutions!
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:
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
You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.
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