Complex Number
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = −1. The real part a and imaginary part b together extend the real number line into a two-dimensional complex plane, enabling solutions to equations like x² + 1 = 0 that have no real solutions. Complex numbers are fundamental in electrical engineering, quantum mechanics, signal processing, and control theory.
Key Formula
z = a + bi, |z| = sqrt(a² + b²)
LaTeX: z = a + bi,\quad |z| = \sqrt{a^2 + b^2}
| Symbol | Meaning | Unit |
|---|---|---|
| z | complex number | dimensionless |
| a | real part of the complex number | dimensionless |
| b | imaginary part of the complex number | dimensionless |
| i | imaginary unit (i² = −1) | dimensionless |
| |z| | modulus (absolute value) of z | dimensionless |
Worked Example
Problem
Multiply the complex numbers z₁ = 3 + 2i and z₂ = 1 − 4i, then find the modulus of the result.
Solution
Step 1: Expand using FOIL: (3 + 2i)(1 − 4i) = 3·1 + 3·(−4i) + 2i·1 + 2i·(−4i) = 3 − 12i + 2i − 8i² Step 2: Replace i² = −1: = 3 − 12i + 2i − 8(−1) = 3 + 8 − 10i = 11 − 10i Step 3: Modulus: |11 − 10i| = √(11² + (−10)²) = √(121 + 100) = √221 ≈ 14.87
Answer
Product = 11 − 10i; |product| ≈ 14.87
Properties and Forms of Complex Numbers
| Form | Expression | Real Part | Imaginary Part | Modulus |
|---|---|---|---|---|
| Rectangular | a + bi | a | b | √(a²+b²) |
| Polar | r(cosθ + i sinθ) | r cosθ | r sinθ | r |
| Euler's | re^(iθ) | r cosθ | r sinθ | r |
| Pure Real | a + 0i | a | 0 | |a| |
| Pure Imaginary | 0 + bi | 0 | b | |b| |
| Conjugate of z | a − bi | a | −b | √(a²+b²) |
Interactive Tools
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Related Terms
Imaginary Number
An imaginary number is a number whose square is a non-positive real number, defined as a real multiple of the imaginary unit i, where i = √(−1). Pure imaginary numbers take the form bi (with b ≠ 0), and they arise naturally as solutions to equations such as x² = −4. Despite their name, imaginary numbers are indispensable in physics, electrical engineering, and signal processing, where they model oscillations and phase differences.
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point in the plane is specified by a radial distance r from a fixed origin (pole) and an angle θ measured from a fixed reference direction (polar axis), written as the ordered pair (r, θ). Unlike Cartesian coordinates that use perpendicular axes, polar coordinates are natural for describing curves with rotational symmetry such as circles, spirals, roses, and limaçons. They are widely used in physics (orbital mechanics, wave interference), engineering (antenna patterns), and complex number representation.
The term "complex number" was introduced by Carl Friedrich Gauss in 1831, from the Latin "complexus" meaning "embracing" or "comprising," reflecting how these numbers comprise both real and imaginary parts. The imaginary unit i was introduced by Leonhard Euler in the 18th century.