nLab
complex number

Context

Complex geometry

Algebra

Contents

Definition

A complex number is an expression of the form a+iba + \mathrm{i} b, where aa and bb are real numbers and i 2=1\mathrm{i}^2 = - 1. 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).

Properties

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.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{O} the octonionsheterotic/type II string

References

Revised on July 17, 2014 05:03:05 by Colin Tan (182.55.187.4)