Perform addition, subtraction, multiplication, division, modulus, conjugate, polar form, and more for complex numbers a+bi and c+di with step-by-step explanations.
This calculator performs arithmetic and analytical operations on complex numbers of the form a + bi and c + di. It supports addition, subtraction, multiplication, division, modulus, conjugate, and polar form conversion — with clear step-by-step explanations.
Perform operations between two complex numbers:
The calculator automatically applies the correct formula and shows the result in standard form.
Additional complex number operations include:
This helps students understand geometric interpretation of complex numbers.
Every calculation includes detailed steps:
This makes it ideal for homework practice and exam preparation.
Results update automatically when values change:
1️⃣ Enter the real and imaginary parts of the first complex number z₁.
2️⃣ Enter the real and imaginary parts of the second complex number z₂.
3️⃣ Select the desired operation from the dropdown menu (Addition, Subtraction, Multiplication, Division, Modulus, Conjugate, or Polar Form).
4️⃣ The calculator will automatically compute the result and display it instantly.
5️⃣ For most operations, a step-by-step explanation is provided to help you understand how the real and imaginary parts are calculated.
This interactive feedback helps students and users understand both the algebraic and geometric interpretation of complex numbers without needing to memorize formulas.
Complex numbers are numbers in the form z = a + bi, where a is the real part and b is the imaginary part. Operations with complex numbers follow specific algebraic and geometric formulas. Understanding these formulas helps in arithmetic calculations, solving equations, and visualizing numbers in the complex plane.
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Add the real parts together and the imaginary parts together. Example: (2 + 3i) + (1 − 4i) = (2+1) + (3−4)i = 3 − 1i
Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i
Subtract the real parts and the imaginary parts separately. Example: (5 + 2i) − (3 + 6i) = (5−3) + (2−6)i = 2 − 4i
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Multiply using distributive property and combine like terms. Example: (2 + 3i)(1 − 4i) = 2*1 − 3*−4 + (2*−4 + 3*1)i = 14 − 5i
(a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
Multiply numerator and denominator by the conjugate of the denominator and simplify. Example: (3 + 2i) ÷ (1 − i) = [(3*1 + 2*−1) + (2*1 − 3*−1)i] / (1² + (−1)²) = (1 + 5i)/2 = 0.5 + 2.5i
Modulus: |a + bi| = √(a² + b²)
The modulus represents the distance of the complex number from the origin in the complex plane. Example: |3 + 4i| = √(3² + 4²) = 5
Conjugate: z* = a − bi
The conjugate flips the sign of the imaginary part, useful in division and polar form. Example: (5 + 2i)* = 5 − 2i
Polar Form: z = r(cosθ + i sinθ), where r = √(a² + b²), θ = arctan(b/a)
Converts a complex number to magnitude-angle form for geometric interpretation. Example: 3 + 3i → r = √(3² + 3²) = 4.24, θ = arctan(3/3) = 0.785 rad → z = 4.24(cos0.785 + i sin0.785)
Using these formulas, you can perform any arithmetic or geometric operation on complex numbers easily. Combining step-by-step calculation with visual understanding of real and imaginary parts ensures a deeper grasp of complex number algebra.
Suppose:
z₁ = 2 + 3i and z₂ = 1 − 4i
For multiplication:
(2+3i)(1−4i) = 14 − 5i
The calculator shows each intermediate step so you can clearly see how the real and imaginary parts are computed.
Everything you need to know about performing operations on complex numbers (a + bi and c + di) accurately and step-by-step.
This calculator performs arithmetic and advanced operations on two complex numbers (z₁ = a₁ + b₁i and z₂ = a₂ + b₂i). It supports addition, subtraction, multiplication, division, modulus, conjugate, and polar form conversion with step-by-step explanations.
You can perform: Addition (z₁ + z₂), Subtraction (z₁ − z₂), Multiplication, Division, Modulus (|z|), Conjugate (z*), and Polar Form conversion. Simply select the desired operation from the dropdown menu.
Yes. The calculator requires both z₁ and z₂ to be entered. For operations like modulus, conjugate, and polar form, it calculates results for both numbers separately.
Division is performed using the standard formula: (a + bi) / (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²). If the denominator (c² + d²) equals zero, the calculator displays an error because division by zero is undefined.
Multiplication follows the distributive property: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The calculator shows each intermediate step so you can understand how the final real and imaginary parts are computed.
The modulus (or magnitude) of a complex number a + bi is calculated as √(a² + b²). The calculator computes |z₁| and |z₂| separately and displays both values.
Polar form is expressed as r(cosφ + i sinφ), where r = √(a² + b²) and φ = arctan(b/a). The calculator computes the modulus and angle (in radians) for each complex number and displays them clearly.
All numerical results are rounded to two decimal places for clarity and readability. This ensures accurate yet clean output for most academic and practical purposes.
If any input field is empty or contains invalid numbers, the calculator will not display results. In case of division by zero, it shows a clear error message to prevent undefined calculations.
Yes. Use the “Clear All” button to remove all inputs instantly, or click “Reload Calculator” to fully reset the tool. The calculator is also fully responsive and works smoothly on mobile and desktop devices.
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