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A representation is faithful when its kernel is trivial.

Representations $\rho :G\to \mathrm{GL}(V)$ and ${\rho }^{\prime }:G\to \mathrm{GL}(V^{\prime })$ are isomorphic when an invertible linear map $T:V\to V^{\prime }$ intertwines the actions: $T\rho (g)={\rho }^{\prime }(g)T$ for every $g\in G$.

The degree of a finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ is $\dim V$.

A finite-dimensional continuous representation of a topological group is a continuous homomorphism $\rho :G\to \mathrm{GL}(V)$, where $V$ is a finite-dimensional complex vector space.

If $x=c_1(\mathcal{O}(1))$ has degree two, thenReducing coefficients modulo two gives

The complex Hopf fibration is the principal circle bundleIt is the unit-sphere bundle of the tautological complex line bundle.

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 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.

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.

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.

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

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

If $G$ is the principal complex inverse sine, all values reached by analytic continuation areThis follows from the monodromy reflections about the branch values $-\pi /2$ and $\pi /2$.

A first-order theory is $\kappa$-categorical when it has a model of a first-order theory of cardinality $\kappa$ and any two such models of a first-order theory are isomorphic. Existence is required, so a missing model size does not make categoricity vacuously true.

For the adjacency matrix $A$ of $K_n$,Hence $I,A,A^2$ are linearly dependent for every $n\ge 2$.

Every compactly supported distribution is a finite sum of distributional derivatives of bounded continuous functions. One proof convolves it with a sufficiently high-order Bessel potential and then applies a power of $1-\Delta$.

A space is locally compact when every point has a base of neighbourhoods containing compact neighbourhoods. A compact Hausdorff space is locally compact because regular separation produces $x\in V\subseteq \overline{V}\subseteq U$.

Every compact Hausdorff space is normal: disjoint closed subsets have disjoint open neighbourhoods. Pointwise Hausdorff separation becomes uniform first over one compact closed set and then over the other by taking finite subcovers.

Every open cover of a compact metric space has a number $\delta >0$ such that every subset of diameter less than $\delta$ lies in one member of the cover.

Every quotient of a compact space is compact because the quotient projection is continuous and surjective.