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.
Key Formula
i = sqrt(-1), i² = -1, i³ = -i, i⁴ = 1
LaTeX: i = \sqrt{-1},\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1
| Symbol | Meaning | Unit |
|---|---|---|
| i | imaginary unit | dimensionless |
| bi | pure imaginary number (b is real) | dimensionless |
Worked Example
Problem
Simplify i²³ and express it in terms of i or a real number.
Solution
Step 1: Recall the cycle of powers of i: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then repeats. Step 2: Divide the exponent by 4 to find the remainder: 23 ÷ 4 = 5 remainder 3. Step 3: Therefore i²³ = i³ = −i.
Answer
i²³ = −i
Cyclic Powers of the Imaginary Unit i
| Power | Value | Pattern Remainder (mod 4) | Notes |
|---|---|---|---|
| i¹ | i | 1 | Base imaginary unit |
| i² | −1 | 2 | Defines imaginary unit |
| i³ | −i | 3 | Negative imaginary unit |
| i⁴ | 1 | 0 | Returns to real 1 |
| i⁵ | i | 1 | Cycle repeats |
| i⁰ | 1 | 0 | Any non-zero base to power 0 |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
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.
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.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
The term "imaginary" was coined by René Descartes in 1637 in his work "La Géométrie," initially used dismissively to describe roots he considered impossible. Leonhard Euler later formalized the notation i in 1777, and Gauss helped legitimize imaginary numbers by developing the complex plane in the early 19th century.