Complex Number Calculator
Perform arithmetic on complex numbers (a + bi). Add, subtract, multiply, and divide imaginary numbers with step-by-step simplification and polar form conversion.
Complex numbers combine real and imaginary parts. Use this tool to perform operations as follows:
Step 1: Enter the Real and Imaginary parts for the first number (e.g., Real = 3, Imaginary = 4 for 3 + 4i).
Step 2: Enter the Real and Imaginary parts for the second number.
Step 3: Select the operation: Addition (+), Subtraction (-), Multiplication (×), or Division (÷).
Step 4: Click "Calculate."
Step 5: Review the results. The calculator will provide: - The result in standard form (a + bi). - The absolute value (magnitude or modulus). - The argument (phase angle) in degrees or radians. - The polar representation (r∠θ).
Complex number operations follow standard algebraic rules, with the critical definition that i² = −1.
1. Addition: (a + bi) + (c + di) = (a+c) + (b+d)i 2. Subtraction: (a + bi) − (c + di) = (a−c) + (b−d)i 3. Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i 4. Division: Multiply the numerator and denominator by the conjugate of the denominator (c − di) to remove the imaginary part from the bottom.
Modulus (Magnitude): |z| = √(a² + b²) Conjugate: z* = a − bi
The Complex Number Calculator is a specialized tool for advanced mathematics, physics, and electrical engineering. In the physical world, complex numbers are used to model alternating current (AC) circuits, signal processing, and fluid dynamics. For example, impedance in an electrical circuit is represented as a complex number where the real part is resistance and the imaginary part is reactance. This tool simplifies the often-tedious process of manual multiplication and division of imaginary units, allowing students and engineers to focus on the high-level analysis of their systems.
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