Complex Number Calculator - All Operations
Free calculator for complex number operations. Add, subtract, multiply, divide, and convert to polar form instantly
Complex Number Calculator
Result
Common Examples
What is a Complex Number Calculator?
A complex number calculator is a mathematical tool that performs operations on complex numbers in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i² = -1. This calculator handles addition, subtraction, multiplication, division, and conversion to polar form.
Complex numbers are fundamental in advanced mathematics, physics, engineering, and signal processing. They solve equations that have no real solutions, describe alternating current circuits in electrical engineering, represent quantum mechanical wave functions in physics, and model oscillations and waves in various applications.
This calculator performs all basic operations with complex numbers and converts results to both rectangular form a + bi and polar form r∠θ. It computes magnitude (modulus) and argument (angle) automatically, providing complete information about the result in multiple representations for different applications.
For solving quadratic equations that yield complex solutions, try our quadratic formula calculator. The system of equations calculator solves linear systems. For basic arithmetic, use our fraction calculator.
How the Complex Number Calculator Works
For addition and subtraction, the calculator combines real and imaginary parts separately. Addition: (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i. These operations treat real and imaginary components independently.
For multiplication, the calculator uses the FOIL method: (a + bi)(c + di) = ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i. Division multiplies numerator and denominator by the conjugate of the denominator to eliminate imaginary parts from the denominator.
The calculator computes polar form using magnitude r = √(a² + b²) and argument θ = arctan(b/a), adjusting for quadrant. Polar form r∠θ or r(cos θ + i sin θ) is especially useful for multiplication and division where magnitudes multiply and arguments add.
Key Concepts Explained
Imaginary Unit
The number i satisfies i² = -1. It allows solutions to equations like x² + 1 = 0. Combined with real numbers, it forms the complex number system.
Rectangular Form
Complex numbers written as a + bi where a is real part and b is imaginary coefficient. Standard form for addition and subtraction operations.
Polar Form
Represents complex number as r∠θ where r is magnitude and θ is argument. Useful for multiplication, division, and powers of complex numbers.
Complex Conjugate
Conjugate of a + bi is a - bi. Multiplying by conjugate produces real number a² + b². Essential for dividing complex numbers.
How to Use This Calculator
Select Operation
Choose the operation you want to perform: addition, subtraction, multiplication, or division of complex numbers
Enter First Number
Input real part a and imaginary part b for the first complex number a + bi
Enter Second Number
Input real part c and imaginary part d for the second complex number c + di
View Result
Click Calculate to see result in standard form, plus magnitude, argument, and polar form
Benefits of Using This Calculator
Using this complex number calculator eliminates errors in multi-step calculations involving imaginary numbers. It handles all operations and conversions instantly, saving significant time on complex arithmetic.
Instant Results: Perform complex operations immediately instead of working through multiple calculation steps
Multiple Formats: See results in both rectangular and polar forms simultaneously for different uses
Avoid Errors: Eliminate mistakes in sign handling and imaginary unit calculations
Verify Homework: Check your manual complex number calculations for accuracy
Learn Patterns: Understand how operations affect magnitude and argument in polar form
Factors That Affect Your Results
The values you enter and operation selected determine the result form. Understanding how operations affect complex numbers helps predict outcomes and choose appropriate forms.
Operation Type
Addition and subtraction are simplest in rectangular form. Multiplication and division are easier in polar form where magnitudes multiply/divide and arguments add/subtract.
Magnitude & Argument
Magnitude represents distance from origin. Argument represents angle from positive real axis. These values change differently for each operation type.
Real-Imaginary Balance
When imaginary part is zero, result is real. When real part is zero, result is purely imaginary. Balance affects quadrant and argument calculation.
Frequently Asked Questions (FAQ)
Q: What are complex numbers?
A: Complex numbers are numbers of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). They extend the real number system to solve equations like x² + 1 = 0.
Q: How do you add complex numbers?
A: Add complex numbers by adding their real parts and imaginary parts separately. For (a + bi) + (c + di) = (a + c) + (b + d)i. For example, (3 + 2i) + (1 + 4i) = 4 + 6i.
Q: What is polar form of complex numbers?
A: Polar form expresses complex numbers as r(cos θ + i sin θ) or r∠θ, where r is the magnitude (or modulus) and θ is the argument (or angle). Magnitude r = √(a² + b²) and argument θ = arctan(b/a).
Q: How do you multiply complex numbers?
A: Multiply using FOIL method: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i. Remember i² = -1. For example, (2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i² = -4 + 7i.
Q: What is the complex conjugate?
A: The complex conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate gives a real number: (a + bi)(a - bi) = a² + b². This is useful for dividing complex numbers.
Q: How do you divide complex numbers?
A: Divide by multiplying numerator and denominator by the conjugate of the denominator. For (a + bi)/(c + di), multiply by (c - di)/(c - di) to get a real denominator, then simplify to get the result in standard form.