Complex number ($\mathbb{C}$)

A complex number is $z=x+iy$, where $x,y$ are real numbers and the imaginary unit satisfies $i^2=-1$. The real part is $x$ and the imaginary part is $y$: for $3+4i$ they are $3$ and $4$, respectively. Equality requires both coordinates to agree. Thus the complex plane gives each complex number a concrete position, while ordinary addition and multiplication become geometric operations.

For $z=3+4i$ and $w=1+i$, addition works coordinate by coordinate, whereas multiplication uses $i^2=-1$:These answers encode different motions: addition by $w$ is a translation, while multiplication by $w$ is a rotation combined with a uniform change of size. Keeping these pictures distinct prevents a common beginner's confusion.

The complex numbers form a field: every nonzero $z$ has a multiplicative inverse, computable using its complex conjugate and modulus. They also supply the missing solutions of a quadratic equation such as $u^2+1=0$, namely $u=\pm i$. The surprising payoff is that extending arithmetic also makes Euclidean geometry easier to calculate. For a systematic introduction to these definitions, see Jeremy Orloff, MIT 18.04, topic 1, sections 1.3–1.5.

A nonreal complex number is a complex number with nonzero imaginary part. Being nonreal is a property of one number; forming a complex conjugate pair additionally specifies the relation between two numbers.

Multiplication is defined by $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$. In polar form, it multiplies magnitudes and adds arguments. This arithmetic operation is distinct from elliptic-curve complex multiplication.

Arithmetic operations on complex numbers. Writing two inputs as $a+bi$ and $c+di$ turns multiplication into $(ac-bd)+(ad+bc)i$ and addition into $(a+c)+(b+d)i$. This is distinct from the specialized theory of complex multiplication of elliptic curves.

A nonzero complex number has polar formwhere $r$ is its modulus and $\theta$ is an argument. The equality uses Euler's formula; sine and cosine recover the imaginary part and real part from polar coordinates. For $z=3+4i$, take $r=5$ and $\theta =\arctan (4/3)\approx 0.927295$ radians, so $\cos \theta =3/5$ and $\sin \theta =4/5$.

If $w=s{e}^{i\varphi }\ne 0$, thenThus multiplication multiplies moduli and adds arguments modulo $2\pi$; division divides moduli and subtracts arguments modulo $2\pi$. Since $1+i=\sqrt{2}{e}^{i\pi /4}$, multiplying $3+4i$ by $1+i$ gives $-1+7i$, of modulus $5\sqrt{2}$ and argument $\theta +\pi /4\approx 1.712693$ radians. One formula simultaneously describes its new size and direction.

Repeated multiplication now becomes especially simple: $(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }$ for a positive integer $n$. This explains the evenly spaced arguments of the roots of a complex number. For example, $(3+4i{)}^2=-7+24i$ has modulus $25$ and principal argument $2\theta \approx 1.854590$ radians. The Cartesian calculation and the polar form provide independent checks on both its location and its size. Choose polar form when the task concerns multiplication, division, or repeated turns; use real parts and imaginary parts for addition.

For $z=r{e}^{i\theta }\ne 0$ and a positive integer $n$, all solutions of $w^n=z$ are $r^{1/n}{e}^{i(\theta +2\pi k)/n}$ for $k=0,\dots ,n-1$. They differ by multiplication by roots of unity and are distinct. If $z=0$, the sole root is zero.

For a nonzero complex number $z=r{e}^{i\theta }$ with $r>0$, its two square roots are $\pm \sqrt{r}{e}^{i\theta /2}$. They are distinct and differ by a sign. At $z=0$ the only square root is zero. A principal value convention chooses $\theta =\mathrm{Arg}z\in (-\pi ,\pi ]$ and the plus sign; continuity across the negative real axis then fails. A holomorphic square root is a different, domain-dependent notion: it requires a holomorphic function whose square is $z$ on the specified domain. It cannot exist on every punctured neighborhood surrounding zero.

The complex modulus of $z=x+iy$ is $|z|=\sqrt{x^2+y^2}=\sqrt{z\overline{z}}$. It is the Euclidean norm of $(x,y)$ in the complex plane, and satisfies $|zw|=|z||w|$ and the triangle inequality.

The imaginary unit is the complex number $i$ satisfying $i^2=-1$. Its integer powers repeat with period four.

For a complex number $z=x+iy$, its real part is $\mathrm{Re}z=x$.

For a complex number $z=x+iy$, its imaginary part is $\mathrm{Im}z=y$.

For a real angle $\theta$ in radians, Euler's formula stateswhere $i$ is the imaginary unit. It connects the complex exponential to the sine and cosine coordinates of the unit circle. In particular, $|e^{i\theta }|=1$, and multiplying by $e^{i\theta }$ rotates a complex number through angle $\theta$ without changing its modulus. At $\theta =\pi /3$ it gives $e^{i\pi /3}=1/2+i\sqrt{3}/2$; at $\theta =\pi$, it gives $e^{i\pi }=-1$.

The identity makes angle addition algebraic: $e^{i\alpha }{e}^{i\beta }=e^{i(\alpha +\beta )}$. Expanding the product and comparing real parts and imaginary parts recovers the sine addition formula and cosine addition formula. Differentiating $e^{i\omega t}$ gives $i\omega e^{i\omega t}$, and differentiating again gives $-{\omega }^2{e}^{i\omega t}$. Its real and imaginary coordinates therefore solve the harmonic oscillator equation. This is why complex exponentials are useful building blocks in Fourier analysis: differentiation acts on each frequency by simple multiplication.

A short justification separates the even and odd powers in the absolutely convergent power series for $e^{i\theta }$. The even powers give ${\sum }_{n\ge 0}(-1{)}^n{\theta }^{2n}/(2n)!=\cos \theta$, and the odd powers give $i{\sum }_{n\ge 0}(-1{)}^n{\theta }^{2n+1}/(2n+1)!=i\sin \theta$. The absolute convergence permits this regrouping. The formula thus links an analytic definition of the complex exponential to a geometric motion rather than merely introducing a convenient notation.

Leonhard Euler presents the relation between imaginary exponentials and sines and cosines in his 1748 Introductio in analysin infinitorum, volume 1, chapter VIII, section 138. His derivation uses limiting powers after connecting trigonometric addition to algebra. The original Latin and Ian Bruce's English translation can be read together in Euler, Introductio, chapter VIII, section 138.

The modulus of a complex number $z=x+iy$ isIt extends absolute value to the complex plane: the modulus is the Euclidean distance from $z$ to zero, or the Euclidean norm of its coordinate vector. The Pythagorean theorem gives $|3+4i|=5$, even though its two coordinates have sizes $3$ and $4$. The modulus is always nonnegative, and is zero exactly at zero.

Two essential rules areThe second is the triangle inequality. For $z=3+4i$ and $w=1+i$, the product has modulus $\sqrt{50}=5\sqrt{2}\approx 7.071068$, while the sum has modulus $\sqrt{41}\approx 6.403124<5+\sqrt{2}\approx 6.414214$. The strict inequality records that the two displacement vectors point in different directions.

The Euclidean distance between two complex numbers $z,w$ is $|z-w|$. Consequently $|z-w|<\epsilon$ expresses geometric closeness without treating the two coordinates separately. This makes the modulus a natural measure of error in complex analysis; the triangle inequality bounds accumulated error, while the product rule tracks how multiplication changes its size. Equal moduli do not imply equal complex numbers: $5$, $-5$, and $3+4i$ all lie on the same circle around zero. The modulus records size and deliberately forgets direction; the argument supplies the missing information.

The complex conjugate of $z=x+iy$ is $\overline{z}=x-iy$. It keeps the real part and negates the imaginary part. In the complex plane, this is a reflection across the horizontal axis. Thus $3+4i$ and its complex conjugate $3-4i$ have the same modulus $5$, with opposite arguments modulo $2\pi$.

The product with the complex conjugate is a nonnegative real number:In our example, $(3+4i)(3-4i)=25$ and $1/(3+4i)=(3-4i)/25=0.12-0.16i$. Its modulus is $1/5$. Geometrically, taking the multiplicative inverse reverses the argument and reciprocates the modulus; the reflected direction alone is only part of the operation.

Applying the complex conjugate twice returns the starting point. Moreover, a complex number equals its own complex conjugate exactly when its imaginary part is zero. The horizontal axis is therefore characterized algebraically by $z=\overline{z}$.

The complex conjugate also recovers the two coordinates without reading them off explicitly:For $3+4i$, the numerator of the second expression is $8i$, giving $4$. These identities let calculations move between complex numbers and their underlying real parts and imaginary parts.

Complex conjugation sends a complex number $z=a+ib$ to its complex conjugate $\overline{z}=a-ib$. It is an involution and a field automorphism of $\mathbb{C}$, with fixed field $\mathbb{R}$. Applied componentwise to a complex vector, it produces the conjugated vector appearing in a Choi state for a rank-one Kraus operator.

This operation reflects a holomorphic function across the real axis while preserving holomorphic dependence on the new argument. A transform of real data with real kernel coefficients satisfies $F^{♯}=F$. In a polygonal global relation, complex rotation factors are conjugated as well; an explicit prefactor $i$ changes sign.

An argument of a nonzero complex number $z$ is any angle $\theta$ for which $z=|z|e^{i\theta }$. All arguments are $\theta +2\pi k$, with $k$ an integer. Zero has no argument: it has no distinguished direction. A commonly chosen representative, the principal argument $\mathrm{Arg}z$, lies in $(-\pi ,\pi ]$. Choosing representatives is part of specifying a branch of a multivalued function. These conventions are described in Jeremy Orloff, MIT 18.04, topic 1, sections 1.5 and 1.14.

For $3+4i$, the principal argument is $\arctan (4/3)\approx 0.927295$ radians, computed with the inverse tangent. But the coordinate ratio alone loses the quadrant: $-3-4i$ has the same ratio $4/3$, yet its principal argument is $\arctan (4/3)-\pi \approx -2.214297$ radians.

For nonzero $z,w$, the correct rule isOrdinary equality can fail. With $z=w=e^{3\pi i/4}$, their principal arguments sum to $3\pi /2$, but $zw=-i$ has principal argument $-\pi /2$. They differ by one full turn. As a point crosses the negative horizontal axis, the principal argument jumps by $2\pi$. A continuous choice on a suitable region therefore requires a branch cut; the jump belongs to the choice of representative, although the point moves smoothly.

The principal argument selects a single complex argument of a nonzero complex number, conventionally in $(-\pi ,\pi ]$. It is the imaginary part of the principal logarithm. The alternative interval $[-\pi ,\pi )$ changes the convention on the negative real axis. Neither convention makes the selected complex argument continuous across that axis.

The complex plane identifies the complex number $x+iy$ with the point $(x,y)$ in Cartesian coordinates. The horizontal coordinate is its real part, the vertical coordinate its imaginary part. Arrows from zero identify the same points with vectors. Adding $1+i$ moves $3+4i$ to $4+5i$: every point undergoes the same translation.

Multiplication has a richer geometry. Multiplying by the imaginary unit sends $x+iy$ to $-y+ix$, a counterclockwise quarter-turn rotation. In particular $i(3+4i)=-4+3i$. Multiplying by $1+i$ rotates through $\pi /4$ and scales every Euclidean distance by $\sqrt{2}$. More generally, multiplication by $a+ib$ acts through the real matrixFor $a+ib\ne 0$, this preserves oriented angles between nonzero displacement vectors: every direction turns equally. Its determinant is $a^2+b^2>0$, while its common scale is $\sqrt{a^2+b^2}$. This is the elementary geometric reason complex numbers are so useful for conformal maps.

The name Argand plane recalls Jean-Robert Argand's geometric interpretation. In his 1813 article he explicitly reports publishing his Essai in 1806; the original article is available in Argand, Annales de mathématiques, volume 4, page 133. A ^[ref] preserves the essay and its geometric drawings. Reading these alongside a modern complex plane reveals how directions and lengths give arithmetic a visible meaning.

The region between two rays with a common vertex, with no upper bound on the modulus. In the displayed convention the vertex and boundary rays are excluded. A holomorphic logarithm with $\alpha <\arg z<\beta$ defines powers $z^p=e^{p\mathrm{Log}z}$ there. For real $p>0$ with $p(\beta -\alpha )\le 2\pi$, the power is a conformal bijection onto the sector $p\alpha <\arg w<p\beta$: it multiplies arguments by $p$ and sends moduli bijectively by $r\mapsto r^p$.

Given a compact planar set $K$ and an open neighborhood $V$, a finite square-grid construction gives a compact polygonal neighborhood inside $V$ with finitely many complementary components. Remove thin polygonal corridors disjoint from $K$ joining holes that lie in the same component of $\mathbb{C}\setminus K$, and joining holes in its unbounded component to the exterior. Corridors can also include a prescribed point from each of finitely many specified bounded components. There remains a neighborhood whose bounded holes lie in distinct bounded components of $\mathbb{C}\setminus K$, each containing its chosen point. Standard small polygonal rounding gives a finite union of planar surfaces with boundary. The index map from its first integral homology group to one integer per bounded hole is an isomorphism. Triangulate a surrounding disc: a one-cycle bounds a two-chain whose coefficients are constant in each complementary component. Zero hole coefficients give a filling supported inside the neighborhood; the face chain of each hole supplies a dual cycle. The cycles can be sums of boundary curves in different components. This uses finite triangulation, not the general Jordan curve theorem.

Represent Cartesian coordinates $(x,y)$ in the plane by the complex number $z=x+iy$, whose real part and imaginary part recover them. Multiplication by $e^{i\theta }$ rotates the coordinates through angle $\theta$, so a rotating-frame calculation becomes a complex exponential change of variable.

For $c\ne 0$ and real $r$, this equation defines a straight line. Writing $z=x+iy$ and $c=u+iv$ gives $2ux+2vy+r=0$, so $c$ supplies a normal direction. Conversely every real affine line $Ax+By+C=0$ has this representation with $c=(A+iB)/2$ and $r=C$. The condition $c\ne 0$ excludes an empty set or the entire complex plane.

The imaginary axis is the set of complex numbers $iy$ with $y\in \mathbb{R}$. It is the vertical coordinate axis in the complex plane.

The complex unit circle is the subgroup $\{z\in \mathbb{C}:|z|=1\}$ under multiplication.

The real axis consists of the points of the complex plane whose imaginary part is zero. These points represent the real numbers.