The standard way of writing a complex number, as a sum of a real part a and an imaginary part b scaled by the imaginary unit i. Also called rectangular form or standard form. Every complex number has a unique algebraic form, and every pair of real numbers (a,b) produces exactly one complex number this way.
Expresses a complex number using its distance r from the origin and its angle θ from the positive real axis. Also called polar form. Particularly useful for multiplication, division, and powers, where the geometric meaning becomes transparent.
Compact representation derived from Euler’s formula eiθ=cosθ+isinθ. Encodes modulus and argument in a single exponential expression. Multiplication, division, and powers reduce to elementary exponent rules.
For z=a+bi, the modulus measures the straight-line distance from the origin to z in the complex plane. Direct application of the Pythagorean theorem to the right triangle with legs ∣a∣ and ∣b∣.
The argument is the angle from the positive real axis to the line segment from the origin to z, measured counterclockwise. Determined only up to multiples of 2π — the principal argument Arg(z)∈(−π,π] provides the canonical choice.
Recovers the rectangular components a and b from polar parameters r and θ. Direct trigonometric projection: the horizontal leg of the right triangle is rcosθ, the vertical leg is rsinθ.
Extraction functions returning the real components of a complex number. Despite its name, Im(z) is itself a real number — it is the coefficient of i, not the term bi.
Two complex numbers are equal if and only if both their real parts and their imaginary parts match. A single complex equation thus splits into two independent real equations — a powerful technique for solving complex equations by separation into real and imaginary parts.
Add real parts to real parts and imaginary parts to imaginary parts. The two components are independent — no interaction occurs between them. Geometrically, addition follows the parallelogram rule: place the two complex numbers as vectors from the origin and the sum is the diagonal.
Subtract real parts and imaginary parts independently. Equivalently, z1−z2=z1+(−z2) where −z2 is the additive inverse. Geometrically, z1−z2 is the vector from z2 to z1, and its modulus ∣z1−z2∣ is the distance between the two points in the complex plane.
Multiply numerator and denominator by the conjugate of the denominator. The denominator becomes the real number ∣z2∣2=c2+d2, converting the quotient to standard algebraic form. This technique is sometimes called rationalizing the denominator.
Multiply the moduli and add the arguments. Geometrically, multiplication by z2 scales by ∣z2∣ and rotates counterclockwise by arg(z2). Dramatically simpler than rectangular multiplication when both factors are in polar form.
Divide the moduli and subtract the arguments. Geometrically, division by z2 shrinks by 1/∣z2∣ and rotates clockwise by arg(z2). The inverse of polar multiplication.
Reflects z across the real axis in the complex plane. The real part stays unchanged; the sign of the imaginary part flips. Conjugation is the most important auxiliary operation in complex arithmetic — it underlies division, modulus computation, and classification of real and pure imaginary numbers.
Conjugation is an involution: applying it twice returns the original number. Geometrically, reflecting twice across the same axis brings every point back to its starting position.
Conjugation distributes over addition. The conjugate of a sum equals the sum of the conjugates. Allows term-by-term conjugation of expressions involving complex sums.
Conjugation distributes over multiplication. Allows conjugating each factor independently before multiplying — often simpler than conjugating the full product.
Conjugation passes through integer powers. Follows from repeated application of the product rule for positive integers and from the quotient rule for negative integers.
The product of a complex number with its conjugate is always the real, non-negative quantity ∣z∣2. The cornerstone identity of complex arithmetic — it makes division by complex numbers possible by converting complex denominators into real ones.
A complex number is real if and only if it equals its own conjugate. Geometrically, real numbers lie on the real axis — the mirror line for conjugation — so they are fixed under reflection.
A complex number is pure imaginary if and only if its conjugate equals its negative. Pure imaginaries lie on the imaginary axis, perpendicular to the mirror, so reflection sends them to their opposites.
The square of the modulus equals the sum of squares of the real and imaginary parts. Avoids the square root in ∣z∣ — useful in proofs and algebraic manipulations where the squared form is cleaner.
Conjugation preserves modulus. Reflection across the real axis does not change distance from the origin, so z and z lie on the same circle centered at the origin.
The modulus of a product equals the product of the moduli. Reduces modulus calculations on complicated products to multiplication of individual magnitudes.
Raising a complex number to the n-th power raises its modulus to the n-th power. Combined with De Moivre’s theorem, this makes computing moduli of powers trivial.
The modulus of a sum never exceeds the sum of moduli. Geometrically, one side of a triangle (the direct path from 0 to z1+z2) cannot exceed the sum of the other two sides (the detour through z1 or z2). Foundation for estimation throughout complex analysis.
Raising a unit-modulus complex number to the n-th power multiplies its argument by n. Reduces high powers of complex numbers to elementary angle arithmetic — no binomial expansion, no tracking powers of i.
Exponential-form statement of De Moivre’s theorem. Raises modulus to the n-th power and multiplies argument by n. Direct consequence of the exponent rule (eiθ)n=einθ.
Every nonzero complex number w=Reiϕ has exactly n distinct n-th roots. They share the modulus R1/n and are spaced uniformly around the origin at angular intervals of 2π/n, forming the vertices of a regular n-gon.
The n solutions to zn=1. All lie on the unit circle, equally spaced at angles 2π/n apart, forming the vertices of a regular n-gon with one vertex at 1.
The sum of all n-th roots of unity is zero whenever n≥2. Geometrically, the roots form a regular polygon centered at the origin — placed tip-to-tail as vectors, they return to the starting point. Algebraically, this is a finite geometric series with ratio ω=1.
Connects exponential and trigonometric functions through the imaginary unit. Foundation of the exponential form of complex numbers and of every operation in polar coordinates.
Special case of Euler’s formula at θ=π. Connects five fundamental constants — e, i, π, 1, and 0 — in a single equation. Often cited as the most elegant equation in mathematics.
Multiplying complex numbers adds their arguments. Geometrically, multiplication by z2 rotates by arg(z2). Equality holds modulo 2π since arguments are determined only up to multiples of 2π.
The modulus of the inverse is the reciprocal of the modulus. Numbers far from the origin have inverses close to the origin, and vice versa. Numbers on the unit circle have inverses also on the unit circle.
Euclidean distance between two points in the complex plane. The modulus of the difference equals the straight-line distance — direct application of the Pythagorean theorem to the right triangle with legs ∣a1−a2∣ and ∣b1−b2∣.
Every non-constant polynomial with complex coefficients factors completely into linear terms over C. A polynomial of degree n has exactly n roots, counted with multiplicity. This is what makes C algebraically closed — no further extension of the number system is needed to solve polynomial equations.
For a polynomial with real coefficients, complex roots come in conjugate pairs. If z0 is a root, so is z0. Consequence: every real polynomial of odd degree has at least one real root.
The quadratic formula extends to complex coefficients without modification. Every quadratic equation az2+bz+c=0 has exactly two roots in C (counting multiplicity), regardless of whether the discriminant is real or complex.
Relates the roots of a quadratic az2+bz+c=0 to its coefficients without solving the equation. Same form as in the real case — Vieta’s formulas extend to complex roots unchanged.
For a polynomial of degree n, the elementary symmetric sums of its roots equal the coefficients (up to sign and division by an). The sum of roots and the product of roots are the simplest cases; intermediate symmetric sums correspond to coefficients in between.