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 = a₁ + b₁i.
2️⃣ Enter the real and imaginary parts of the second complex number z₂ = a₂ + b₂i.
3️⃣ Select the desired operation from the dropdown menu.
4️⃣ The calculator applies the appropriate mathematical formula:
Addition: (a₁+a₂) + (b₁+b₂)i
Multiplication: (a₁a₂ − b₁b₂) + (a₁b₂ + a₂b₁)i
Division: [(a₁a₂ + b₁b₂) / (a₂² + b₂²)] + [(b₁a₂ − a₁b₂) / (a₂² + b₂²)]i
Modulus: √(a² + b²)
Polar Form: r = √(a² + b²), θ = atan(b/a)
5️⃣ The result and step-by-step explanation are displayed instantly.
This interactive feedback helps students understand both algebraic and geometric interpretations of complex numbers.
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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