A complex number is a number expressible as (z=a+bi), where (a) and (b) are real numbers and (i) is the imaginary unit, defined by (i^2=-1). The numbers (a) and (b) are respectively the real and imaginary parts of (z); the imaginary part is (b), not (bi). The set of complex numbers is denoted (\mathbb C). It contains the real numbers as the special case (b=0), while allowing equations such as (z^2+1=0) to have solutions. (ocw.mit.edu)
Arithmetic and algebraic structure
Two complex numbers are equal precisely when their corresponding real and imaginary parts are equal. Addition and subtraction operate componentwise, while multiplication follows ordinary distributive rules together with (i^2=-1): [ (a+bi)+(c+di)=(a+c)+(b+d)i, ] [ (a+bi)(c+di)=(ac-bd)+(ad+bc)i. ] For example, ((2+3i)(1-i)=5+i). These operations preserve the familiar associative, commutative, and distributive laws of arithmetic. (ocw.mit.edu)
The complex conjugate of (z=a+bi) is (\overline z=a-bi). Because [ z\overline z=a^2+b^2, ] every nonzero complex number has the reciprocal [ \frac1z=\frac{\overline z}{a^2+b^2}. ] Division can therefore be performed by multiplying numerator and denominator by the denominator’s conjugate. Division by zero remains undefined. (ocw.mit.edu)
With these operations, (\mathbb C) is a field. It can also be constructed rigorously from ordered pairs of real numbers, with multiplication defined by [ (a,b)(c,d)=(ac-bd,ad+bc). ] In this construction, (i=(0,1)). This shows that complex numbers do not depend on treating the square root of a negative number as an unexplained symbol. (math.mit.edu)
Geometric representation
The complex plane represents (a+bi) by the point ((a,b)), with a horizontal real axis and a vertical imaginary axis. Addition corresponds to vector addition, while conjugation reflects a point across the real axis. This connects complex arithmetic with geometry and makes (\mathbb C) a two-dimensional real vector space. (ocw.mit.edu)
The modulus, or absolute value, is [ |z|=\sqrt{a^2+b^2}. ] It measures distance from the origin; similarly, (|z-w|) is the Euclidean distance between two complex points. For (z\ne0), an argument (\theta) specifies its angle from the positive real axis. Arguments differing by an integer multiple of (2\pi) represent the same direction. Zero has modulus zero but no defined argument. (ocw.mit.edu)
Using trigonometry, a nonzero complex number has polar form [ z=r(\cos\theta+i\sin\theta),\qquad r=|z|. ] Euler’s formula, [ e^{i\theta}=\cos\theta+i\sin\theta, ] gives the equivalent exponential form (z=re^{i\theta}). Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments. In particular, multiplication by (i) rotates the plane counterclockwise through (90^\circ). (ocw.mit.edu)
Powers, roots, and polynomial equations
Polar form simplifies powers through De Moivre’s formula: [ [r(\cos\theta+i\sin\theta)]^n =r^n(\cos n\theta+i\sin n\theta). ] For a positive integer (n), every nonzero (z=re^{i\theta}) has exactly (n) distinct (n)th roots: [ r^{1/n}e^{i(\theta+2\pi k)/n}, \qquad k=0,\ldots,n-1. ] They are evenly spaced around a circle. The roots of unity are the special case (z=1); the fourth roots are (1,i,-1,-i). (ocw.mit.edu)
The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root. Consequently, a degree-(n) polynomial factors into (n) linear factors over (\mathbb C), counting repeated roots according to multiplicity. This property is called algebraic closure. For polynomials with real coefficients, nonreal roots occur in conjugate pairs, since conjugating the equation preserves its coefficients. (math.wustl.edu)
Complex functions and analysis
Complex analysis studies functions of a complex variable. Its central notion is complex differentiability: the derivative [ f'(z)=\lim_{h\to0}\frac{f(z+h)-f(z)}h ] must have the same value regardless of the direction from which (h) approaches zero in the plane. A function differentiable throughout an open region is called holomorphic there. (math.wustl.edu)
For (f=u+iv), complex differentiability imposes the Cauchy–Riemann equations, (u_x=v_y) and (u_y=-v_x). Holomorphic functions have strong additional properties: they possess derivatives of every order and can be represented locally by convergent power series. These features distinguish complex differentiation from ordinary real differentiation. (math.berkeley.edu)
Historical development
Complex numbers emerged from sixteenth-century work on cubic equations. Gerolamo Cardano’s Ars Magna (1545) included calculations involving square roots of negative numbers. Rafael Bombelli’s L’Algebra (1572) developed rules for manipulating them and demonstrated how complex intermediate expressions could yield real solutions. Later geometric representations, notably those associated with Wessel, Argand, and Carl Friedrich Gauss, helped establish their interpretation as points in a plane. (math.mit.edu)
Applications
In electrical engineering, complex impedance and phasors encode both the magnitude and phase of sinusoidal voltages and currents. The notation (j) often replaces (i) to avoid confusion with electric current. Complex exponentials also underpin the Fourier transform, which represents signals through frequency components, and simplify the treatment of oscillatory differential equations. (ocw.mit.edu)
In quantum mechanics, a wave function generally takes complex values. Its squared modulus supplies the position probability density in the position representation, while relative phases affect interference. Thus complex quantities encode information that their real parts or magnitudes alone cannot retain. (live.ocw.mit.edu)