nLab complex number




A complex number is a number of the form a+iba + \mathrm{i} b, where aa and bb are real numbers and i 2=1\mathrm{i}^2 = - 1 is an imaginary unit. The set of complex numbers (in fact a field and topological vector space) is denoted C\mathbf{C} or \mathbb{C}.

This can be thought of as:

We think of \mathbb{R} as a subset (in fact {\mathbb{R}}-vector subspace) of \mathbb{C} by identifying aa with a+i0a + \mathrm{i} 0. \mathbb{C} is equipped with a \mathbb{R}-linear involution , called complex conjugation, that maps i\mathrm{i} to i¯=i\bar{\mathrm{i}} = -\mathrm{i}. Concretely, a+ib¯=aib \overline{a + \mathrm{i} b} = a - \mathrm{i} b .

Complex conjugation is the nontrivial field automorphism of \mathbb{C} which leaves {\mathbb{R}} invariant. In other words, the Galois group Gal(/)Gal({\mathbb{C}}/\mathbb{R}) is cyclic of order two and generated by complex conjugation. \mathbb{C} also has an absolute value:

|a+ib|=a 2+b 2; |{a + \mathrm{i} b}| = \sqrt{a^2 + b^2} ;

notice that the absolute value of a complex number is a nonnegative real number, with

|z| 2=zz¯. |z|^2 = z \bar{z} .

Most concepts in analysis can be extended from \mathbb{R} to \mathbb{C}, as long as they do not rely on the order in \mathbb{R}. Sometimes \mathbb{C} even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in \mathbb{R} is important, often it is enough to order the absolute values of complex numbers. See ground field for some of the concepts whose precise definition may vary with the choice of \mathbb{R} or \mathbb{C} (or even other possibilities).



See also at normed division algebra – automorphism.

Over other subfields, the automorphism group may be considerably larger. Over the rational numbers, for instance, \mathbb{C} has transcendence degree equal to the cardinality of the continuum, i.e., there is an algebraic extension (X)\mathbb{Q}(X) \hookrightarrow \mathbb{C} with |X|=𝔠=2 0{|X|} = \mathfrak{c} = 2^{\aleph_0}. Any bijection XXX \to X induces a field automorphism (X)(X)\mathbb{Q}(X) \to \mathbb{Q}(X) which may be extended to an automorphism of \mathbb{C} over \mathbb{Q}. Therefore the number of automorphisms of \mathbb{C} is at least 2 𝔠2^\mathfrak{c} (and in fact at most this as well, since the number of functions \mathbb{C} \to \mathbb{C} is also 2 𝔠2^\mathfrak{c}).

See also at automorphism of the complex numbers.

Geometry of complex numbers

The complex numbers form a plane, the complex plane. Indeed, a map 2\mathbb{C} \to \mathbb{R}^2 given by sending x+iy\mathrm{x} + \mathrm{i}\mathrm{y} to the standard real-valued coordinates (x,y)(\mathrm{x},\mathrm{y}) on this plane is a bijection. Much of complex analysis can be understood through differential topology by identifying \mathbb{C} with 2\mathbb{R}^2, using either x\mathrm{x} and y\mathrm{y} or z\mathrm{z} and z¯\bar{\mathrm{z}}. (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)

It is often convenient to use the Alexandroff compactification of \mathbb{C}, the Riemann sphere P 1\mathbb{C}P^1. One may think of P 1\mathbb{C}P^1 as {}\mathbb{C} \cup \{\infty\}; functions valued in \mathbb{C} but containing ‘poles’ may be taken to be valued in ¯\overline{\mathbb{C}}, with f(ζ)=f(\zeta) = \infty whenever ζ\zeta is a pole of ff.

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string


On the history of the notion of complex numbers:

See also:

Last revised on April 22, 2023 at 16:59:12. See the history of this page for a list of all contributions to it.