What Is a Complex Number?
a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = −1. The part a is called the real part, and b is the imaginary part.For example, in the complex number 3 + 4i, the real part is 3 and the imaginary part is 4.
Complex numbers were invented to solve equations that have no real solutions. The equation x² + 1 = 0 has no real solution because no real number squared gives −1. But if we allow x = i, then i² = −1 makes the equation work.
Extending the real numbers
Every real number is also a complex number (with imaginary part zero), so the complex numbers extend the real numbers rather than replacing them. The set of complex numbers is denoted ℂ, and it includes all real numbers plus all numbers with a nonzero imaginary part.
a + bi and c + di are equal if and only if a = c and b = d — real and imaginary parts must match independently.Despite sounding abstract, complex numbers are essential in electrical engineering, quantum mechanics, signal processing, and many areas of pure mathematics.
Powers of i
Since i² = −1, higher powers of i cycle through a short repeating pattern.
| Power | Value | Power | Value |
|---|---|---|---|
i¹ | i | i⁵ | i |
i² | −1 | i⁶ | −1 |
i³ = i² · i | −i | i⁷ | −i |
i⁴ = i² · i² | (−1)(−1) = 1 | i⁸ | 1 |
i, divide the exponent by 4 and take the remainder. i^n depends only on n mod 4: remainder 0 → 1, remainder 1 → i, remainder 2 → −1, remainder 3 → −i.Example: Evaluate i^25.
Since 25 = 4(6) + 1, the remainder is 1, so i^25 = i.
iExample: Evaluate i^102.
Since 102 = 4(25) + 2, the remainder is 2, so i^102 = −1.
−1This mod-4 cycling is one of the most useful facts about complex numbers. It lets you simplify expressions with arbitrarily high powers of i in seconds.
Adding, Subtracting, and Multiplying Complex Numbers
Addition and subtraction
Addition and subtraction: Combine real parts with real parts and imaginary parts with imaginary parts, just like combining like terms.
(3 + 4i) + (2 − 5i) = (3 + 2) + (4 − 5)i = 5 − i.
(7 + 2i) − (1 + 6i) = (7 − 1) + (2 − 6)i = 6 − 4i.
Multiplication
Multiplication: Use the distributive property (FOIL), then simplify using i² = −1.
Example: (3 + 4i)(2 − 5i).
FOIL: 3(2) + 3(−5i) + 4i(2) + 4i(−5i) = 6 − 15i + 8i − 20i².
Replace i² = −1: 6 − 15i + 8i − 20(−1) = 6 − 15i + 8i + 20 = 26 − 7i.
26 − 7iMultiplication by the imaginary unit: i(a + bi) = ai + bi² = ai − b = −b + ai. Multiplying by i rotates a complex number 90° counterclockwise in the complex plane — a fact that becomes central when you think geometrically.
Dividing Complex Numbers (Complex Conjugates)
a + bi is a − bi. Multiplying a complex number by its conjugate gives a real number: (a + bi)(a − bi) = a² − (bi)² = a² + b².This "rationalizing" trick eliminates the i from the denominator.
Example: Simplify (3 + 2i)/(1 + 4i).
Multiply numerator and denominator by the conjugate (1 − 4i): [(3 + 2i)(1 − 4i)]/[(1 + 4i)(1 − 4i)] = [3 − 12i + 2i − 8i²]/[1² + 4²] = [3 − 12i + 2i + 8]/17 = (11 − 10i)/17 = 11/17 − (10/17)i.
11/17 − 10/17 · iMore properties of the conjugate
The complex conjugate has two more important properties: (1) if a polynomial with real coefficients has a complex root a + bi, then a − bi is also a root; (2) the product of a complex number with its conjugate equals the square of its modulus (absolute value): |a + bi|² = a² + b².
The Complex Plane and Modulus
Every complex number a + bi corresponds to a point (a, b) in the complex plane — a 2D plane where the horizontal axis is the real axis and the vertical axis is the imaginary axis. Visualizing complex numbers as points or vectors makes many operations intuitive.
a + bi is |a + bi| = √(a² + b²). This is just the Pythagorean theorem applied to the complex plane.For example, |3 + 4i| = √(9 + 16) = 5.
5The argument (angle)
θ from the positive real axis to the vector a + bi is given by θ = arctan(b/a), with adjustments based on quadrant.For 3 + 4i, θ = arctan(4/3) ≈ 53.13°.
θ ≈ 53.13°Geometric interpretation of addition: Adding complex numbers follows the parallelogram rule for vectors. Geometric interpretation of multiplication: Multiplying two complex numbers multiplies their moduli and adds their arguments. This is why multiplying by i (modulus 1, argument 90°) rotates a number by 90°.
Polar Form and De Moivre's Theorem
z = r(cos θ + i sin θ), where r = |z| is the modulus and θ is the argument. This is often abbreviated as r · cis θ or, using Euler's formula, as r · e^(iθ).Converting from rectangular to polar
- Compute the modulus
Given
z = a + bi, computer = √(a² + b²). - Compute the argument
Compute
θ = arctan(b/a)(with quadrant adjustment).
Example: Convert z = 1 + i to polar form.
r = √2, θ = arctan(1/1) = π/4.
Polar form: z = √2 · (cos(π/4) + i sin(π/4)) or z = √2 · e^(iπ/4).
√2 · e^(iπ/4)[r(cos θ + i sin θ)]^n = r^n (cos(nθ) + i sin(nθ)). This lets you raise complex numbers to powers easily.Example: Compute (1 + i)^8.
From above, 1 + i = √2 · cis(π/4).
Raising to the 8th power: (√2)^8 · cis(8 · π/4) = 16 · cis(2π) = 16 · (1 + 0i) = 16.
16Without polar form, expanding (1 + i)^8 by FOIL would take 20+ lines. With De Moivre's theorem, it's two lines. This is why polar form is so powerful for higher powers and roots of complex numbers. For more complex problems or if you want to verify your work, scan them with Solver AI to see detailed step-by-step solutions.