Complex Number Calculator — a+bi
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 — a+bi?
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 — a+bi
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.
Related Tools
Quadratic Equation Solver | DigitHelm
Solve quadratic equations (ax² + bx + c = 0) and find both real and complex root…
Trigonometry Calculator — Sin, Cos, Tan
Calculate sine, cosine, tangent, and their inverses for any angle.…
Basic Calculator
Perform addition, subtraction, multiplication, and division with an easy-to-use …