Complex Number Calculator
Add, subtract, multiply, divide complex numbers and convert to polar form.
Perform operations on complex numbers z₁ = a + bi and z₂ = c + di
z₁ = a + bi
z₂ = c + di
What Is the Complex Number Calculator?
The Complex Number Calculator performs arithmetic operations (addition, subtraction, multiplication, division) on two complex numbers and also computes the modulus (magnitude), argument (angle), conjugate, and polar form. Complex numbers have a real part (a) and an imaginary part (b) where i = √(−1).
Formula
How to Use
Enter the real and imaginary parts of two complex numbers: z₁ = a + bi and z₂ = c + di. Select the operation from the dropdown. Click Calculate to see the result with step-by-step working.
Example Calculation
z₁ = 3 + 4i, z₂ = 1 − 2i Multiplication: (3+4i)(1−2i) = 3×1 + 3×(−2i) + 4i×1 + 4i×(−2i) = 3 − 6i + 4i − 8i² = 3 − 2i − 8(−1) = 11 − 2i Division (3+4i)÷(1−2i): Multiply by conjugate (1+2i)/(1+2i): = (3+4i)(1+2i) / (1+4) = (3+6i+4i+8i²)/5 = (3+10i−8)/5 = −1 + 2i
Understanding Complex Number
Complex numbers extend the real number line into a 2D plane (the complex plane or Argand diagram). Every complex number corresponds to a unique point, and the four arithmetic operations have elegant geometric interpretations: addition is vector addition; multiplication scales and rotates.
Euler's formula e^(iθ) = cos(θ) + i·sin(θ) is one of the most profound results in mathematics. It connects exponential functions, trigonometry, and complex numbers in a single equation, showing these seemingly different topics are deeply unified.
Applications of complex numbers are ubiquitous in physics and engineering: alternating current (AC) circuit analysis (impedance as a complex number), quantum mechanics (wave functions are complex-valued), signal processing (Fourier transforms), and fluid dynamics (conformal mapping) all rely on complex arithmetic.
Frequently Asked Questions
What is an imaginary number?
The imaginary unit i is defined as √(−1). While no real number squares to give a negative result, i satisfies i² = −1. A purely imaginary number is a real multiple of i, like 3i.
Why do complex numbers exist?
Complex numbers arise naturally as the roots of polynomials. The quadratic x²+1=0 has no real solutions, but has complex solutions x = ±i. Introducing complex numbers ensures every polynomial equation has solutions (Fundamental Theorem of Algebra).
What is the modulus of a complex number?
The modulus |z| = √(a²+b²) is the distance from the origin to the point (a,b) in the complex plane. It is always a non-negative real number.
What is the complex conjugate?
The conjugate of a+bi is a−bi. Multiplying z by its conjugate gives |z|²: (a+bi)(a−bi) = a²+b², a real number. This is used in division to eliminate i from the denominator.
What is Euler's formula?
Euler's formula states: e^(iθ) = cos(θ) + i·sin(θ). Setting θ=π gives Euler's identity: e^(iπ) + 1 = 0, often called the most beautiful equation in mathematics.