Complex Number Calculator
Perform arithmetic operations with complex numbers
A complex number is a number that combines a real number and an imaginary number in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). Complex numbers extend the concept of the one-dimensional number line to a two-dimensional complex plane, allowing us to solve equations that have no real solutions.
Addition and Subtraction
Add or subtract the real and imaginary parts separately: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
Multiplication
Use the FOIL method and remember that i² = -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division
Multiply numerator and denominator by the complex conjugate of the denominator: (a + bi)/(c + di) = ((ac + bd)/(c² + d²)) + ((bc - ad)/(c² + d²))i
- Modulus (|z|): The distance from the origin to the point in the complex plane, calculated as √(a² + b²)
- Argument (arg z): The angle between the positive real axis and the line from the origin to the point, calculated as tan⁻¹(b/a)
- Complex conjugate: For z = a + bi, its conjugate is z̄ = a - bi
- Polar form: z = r(cos θ + i sin θ), where r is the modulus and θ is the argument
Complex numbers are used in many fields:
- Electrical engineering and circuit analysis
- Quantum mechanics and wave functions
- Signal processing and control systems
- Computer graphics and geometric transformations
- Fluid dynamics and aerodynamics
- Advanced algebra and number theory
Example 1: Addition
- (3 + 2i) + (1 - 4i)
- = (3 + 1) + (2 - 4)i
- = 4 - 2i
Example 2: Multiplication
- (2 + i)(1 + 3i)
- = 2(1) + 2(3i) + i(1) + i(3i)
- = 2 + 6i + i - 3
- = -1 + 7i
- Keep track of i² = -1 when multiplying complex numbers
- Use complex conjugates to rationalize denominators in division
- Remember that complex numbers can be represented in both rectangular (a + bi) and polar (r∠θ) forms
- Practice visualizing complex numbers on the complex plane
- Pay attention to signs when performing operations