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Basics

Complex Numbers

The imaginary unit, arithmetic operations, modulus, and complex conjugate.

A complex number has the form a + ib, where a and b are real and i is the imaginary unit. Treating i as an algebraic symbol with i2 = -1 lets arithmetic proceed like ordinary polynomial arithmetic, with the modulus and conjugate adding extra structure. Geometrically, every complex number is a point in the Argand plane: real part on the horizontal axis, imaginary part on the vertical.

z = a + ibReImab
A complex number z = a + ib plotted on the Argand plane; the real and imaginary parts are the horizontal and vertical projections from the origin.

The imaginary unit is defined as a square root of negative one. This single symbol extends the reals into an algebraically closed field in which every non-constant polynomial has a root (the fundamental theorem of algebra). The choice of i over -i is conventional; both satisfy the defining relation below.

i=−1i=\sqrt{-1}

Squaring the imaginary unit returns the defining relation. This rule is what distinguishes complex arithmetic from polynomial arithmetic in i: every i2 is replaced by -1. Higher powers of i cycle through i, -1, -i, 1 with period four.

i2=−1i^{2}=-1

Square roots of negative real numbers factor out i. The non-negative a under the inner radical keeps that operation real-valued; the i outside carries the sign. The rule blocks the common false step √(-a)·√(-b) = √(ab) when both operands are negative.

−a=ia   (if a≥0)\sqrt{-a}=i\sqrt{a}\text{ }\text{ }\text{ }\left(if\text{ }a\geq 0\right)

Addition combines real and imaginary parts component-wise, like vector addition in R2. On the Argand diagram, two complex numbers add as the parallelogram-rule sum of their position vectors.

(a+ib)+(c+id)=(a+c)+(b+d)i\left(a+ib\right)+\left(c+id\right)=\left(a+c\right)+\left(b+d\right)i

Subtraction is also component-wise. Reversing the sign of the second operand reflects it through the origin before the vector addition.

(a+ib)−(c+id)=(a−c)+(b−d)i\left(a+ib\right)-\left(c+id\right)=\left(a-c\right)+\left(b-d\right)i

Multiplication uses the distributive law with i2 = -1. The cross terms iad and ibc sit in the imaginary part; i2bd = -bd moves into the real part. Geometrically, multiplication rotates and scales: moduli multiply and arguments add.

(a+ib)(c+id)=(ac+bd)+(ab+bc)i\left(a+ib\right)\left(c+id\right)=\left(ac+bd\right)+\left(ab+bc\right)i

Multiplying a complex number by its conjugate gives a real result, which is how complex denominators are rationalised. The cross terms iab and -iab cancel, leaving the sum of squares a2 + b2, the squared distance from the origin in the Argand plane.

(a+ib)(c−ib)=a2+b2\left(a+ib\right)\left(c-ib\right)=a^{2}+b^{2}

The modulus is the distance from the origin in the complex plane. By the Pythagorean theorem applied to the horizontal and vertical projections in the diagram, the length of the vector from 0 to a + ib is √(a2 + b2).

∣a+ib∣=a2+b4   (Complex Modulus)\left|a+ib\right|=\sqrt{a^{2}+b^{4}}\text{ }\text{ }\text{ }\left(Complex\text{ }Modulus\right)

The conjugate flips the sign of the imaginary part, reflecting the number across the real axis. Conjugation is an involution: applying it twice recovers the original. It also commutes with addition and multiplication, making it a field automorphism of the complex numbers.

a+ib‾=a+ib   (Complex Conjugate)\overline{a+ib}=a+ib\text{ }\text{ }\text{ }\left(Complex\text{ }Conjugate\right)

A number times its conjugate equals the modulus squared. This expresses |z|2 without taking a square root, which is the cleaner quantity to work with whenever the modulus appears under a power.

(a+ib‾)(a+ib)=∣a+ib∣2\left(\overline{a+ib}\right)\left(a+ib\right)=\left|a+ib\right|^{2}