Complex Number Definition

Define i as the number satisfying i² = −1, so i = √(−1). Powers of i follow a repeating cycle of period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, i⁵ = i, and so on. To find iⁿ for any positive integer n, compute the remainder r = n mod 4 and use the corresponding value from the cycle.

For a complex number z = a + bi, the real part is Re(z) = a and the imaginary part is Im(z) = b. Note that Im(z) is the real coefficient b, not bi itself. For example, if z = 3 − 5i, then Re(z) = 3 and Im(z) = −5. Two complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal simultaneously.

The complex numbers contain familiar subsets as special cases. When b = 0 the complex number a + 0i = a is simply a real number; every real number lives inside the complex numbers. When a = 0 the number 0 + bi = bi is called purely imaginary. The number 0 + 0i = 0 is both real and purely imaginary. This nesting means ℝ ⊂ ℂ (the reals are a subset of the complexes).

Formally, the set of complex numbers is defined as ℂ = {a + bi : a, b ∈ ℝ, i² = −1}. Every element of ℂ is an ordered pair (a, b) of real numbers, written in the algebraic notation a + bi. This two-component structure means complex numbers can be visualised as points in a plane (the Argand plane), which you will explore in the concept Argand Plane.

The term 'imaginary' is a historical accident — René Descartes used it dismissively in 1637. Imaginary numbers are no less real as mathematical objects than negative numbers or irrational numbers. They obey consistent algebraic rules, have geometric interpretations, and describe physical phenomena. Thinking of i as simply a new algebraic symbol that satisfies i² = −1 is more helpful than worrying about whether it 'exists'.

• z — Complex number (dimensionless): The complex number being represented; it is a single mathematical object with two real components.
• a — Real part, Re(z) (dimensionless): The real-number component of z. It sits on the horizontal axis of the Argand plane.
• b — Imaginary part, Im(z) (dimensionless): The real-number coefficient of i. It sits on the vertical axis of the Argand plane. Note: the imaginary part is the real number b, not the term bi.
• i — Imaginary unit (dimensionless): The fundamental imaginary unit defined by i² = −1. It is not a variable; it is a fixed mathematical constant analogous to π or e.

Problem. Given z₁ = 4 − 7i and z₂ = −4 + 7i, (a) state Re(z₁), Im(z₁), Re(z₂), and Im(z₂); (b) determine whether z₁ = z₂; (c) find the value of n ∈ {1, 2, 3, 4} such that iⁿ = −i.

Answer. (a) Re(z₁) = 4, Im(z₁) = −7, Re(z₂) = −4, Im(z₂) = 7. (b) z₁ ≠ z₂ because their real parts (and imaginary parts) differ. (c) n = 3.

• Algebra of Complex Numbers — Once you can write a complex number as a + bi, the natural next step is learning how to add, subtract, multiply, and divide such numbers — all of which build directly on the a + bi representation.
• Modulus and Conjugate — The modulus |z| = √(a² + b²) and the conjugate z̄ = a − bi are two of the most important quantities attached to a complex number; both are defined directly from its real and imaginary parts.
• Argand Plane — The Argand plane gives the a + bi form a geometric home: a is the horizontal coordinate and b is the vertical coordinate, turning complex numbers into points or vectors in a 2-D plane.
• Polar Form of Complex Numbers — Polar form r(cosθ + i sinθ) is an alternative representation of the same complex number a + bi, using distance r and angle θ instead of rectangular coordinates — essential for multiplication and powers.
• De Moivre's Theorem and Roots — De Moivre's theorem uses the complex number structure to compute powers and nth roots of complex numbers efficiently, representing the deepest algebraic application of the a + bi definition introduced here.