Instantly perform arithmetic operations on complex numbers with our free online calculator.
Calculate addition, subtraction, multiplication, division, magnitude, and other complex number properties.
Perfect for students, engineers, and anyone working with complex mathematics.
Enter complex numbers to calculate
Complex numbers have both real and imaginary parts. Example: 3 + 4i
Our calculator makes working with complex numbers simple and efficient
Enter the real and imaginary parts for the first complex number (a + bi format)
Enter the real and imaginary parts for the second complex number (c + di format)
View instant results for all basic operations (addition, subtraction, multiplication, division)
See additional analysis including magnitude, phase, and conjugate of the first number
Toggle 'Show detailed calculation breakdown' to see step-by-step formulas
Use 'Round results to 2 decimal places' for cleaner output if needed
Calculate complex impedance in AC circuits where Z = R + jX (resistance + reactance). Example: (5 + 3i) + (2 + 4i) = 7 + 7i
Analyze complex frequency responses in digital signal processing. Example: (1 + 2i) * (3 - i) = 5 + 5i
Find complex solutions to quadratic equations. Example: Dividing complex numbers to simplify expressions like (4 + 2i) / (1 + i)
Work with wave functions containing complex coefficients. Example: Calculating probability amplitudes with complex conjugates
Calculate AC circuit parameters, impedance analysis, and phasor calculations where complex numbers represent magnitude and phase of electrical quantities
Analyze digital filters, Fourier transforms, and frequency domain representations where signals are expressed as complex values
Determine system stability using complex roots of characteristic equations and analyze transfer functions in the complex plane
Work with wave functions, probability amplitudes, and operators that involve complex number calculations
Learn and verify complex number operations, visualize concepts, and check homework problems involving complex mathematics
Complex numbers are mathematical entities that combine real and imaginary parts, expressed in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit with the property that i² = -1.
The real part of a complex number can be any real number (positive, negative, or zero), while the imaginary part is also a real number that multiplies the imaginary unit. Complex numbers extend the concept of one-dimensional number line to a two-dimensional complex plane, where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis.
Basic operations with complex numbers follow specific rules: Addition and subtraction involve adding/subtracting the real parts and imaginary parts separately. Multiplication uses the distributive property along with the fact that i² = -1. Division involves multiplying both numerator and denominator by the complex conjugate of the denominator.
The magnitude (or modulus) of a complex number a + bi is calculated as √(a² + b²), which represents the distance from the origin to the point (a, b) in the complex plane. The phase (or argument) is the angle the line makes with the positive real axis, calculated as arctan(b/a) with proper quadrant consideration.
The complex conjugate of a + bi is a - bi, which is useful in division operations and has important properties in various mathematical and engineering applications.
Forgetting that i² = -1 when multiplying complex numbers
Always remember to replace i² with -1 during multiplication operations to avoid incorrect results
Incorrectly dividing complex numbers without using the conjugate
To divide (a + bi) ÷ (c + di), multiply both numerator and denominator by (c - di) to make the denominator real
Confusing the magnitude and phase calculations
Magnitude is √(a² + b²) and phase is arctan(b/a) with correct quadrant adjustment using atan2 function
Misidentifying the real and imaginary parts in complicated expressions
Always rearrange complex expressions to standard form a + bi before inputting values, ensuring proper signs for both parts
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1. The real number 'a' is called the real part, and 'b' is called the imaginary part.
To add complex numbers, add the real parts together and add the imaginary parts together. For example, (a + bi) + (c + di) = (a + c) + (b + d)i. Our calculator performs this operation automatically when you input the values.
To multiply complex numbers, use the distributive property: (a + bi) × (c + di) = ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i. Our calculator shows this process in the detailed breakdown.
The complex conjugate of a complex number a + bi is a - bi. The conjugate has the same real part but an imaginary part with the opposite sign. It's useful in division operations and finding the modulus of a complex number.
To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator. This eliminates the imaginary part in the denominator, allowing for a standard division. For example, (a + bi) ÷ (c + di) = [(a + bi)(c - di)] ÷ [(c + di)(c - di)].
The magnitude (or modulus) of a complex number a + bi is √(a² + b²), representing its distance from the origin in the complex plane. The phase (or argument) is the angle it makes with the positive real axis, typically calculated using the arctan function with proper quadrant consideration.
Yes, our complex number calculator is completely free to use with no registration required. Simply enter your values and get instant results for all complex number operations.