section_path: Characterizations, generalizations and related notions/Other number systems
section_name: Other number systems
paragraph_text: Other number systems
  
  The process of extending the field R {\displaystyle \mathbb {R} } of reals to C {\displaystyle \mathbb {C} } is an instance of the Cayley–Dickson construction. Applying this construction iteratively to C {\displaystyle \mathbb {C} } then yields the quaternions, the octonions and the sedenions. This construction turns out to diminish the structural properties of the involved number systems.
  
  Unlike the reals, C {\displaystyle \mathbb {C} } is not an ordered field, that is to say, it is not possible to define a relation z < z that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i = −1 precludes the existence of an ordering on C. {\displaystyle \mathbb {C}.} Passing from C {\displaystyle \mathbb {C} } to the quaternions H {\displaystyle \mathbb {H} } loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed division algebras over R {\displaystyle \mathbb {R} }. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.
  
  The Cayley–Dickson construction is closely related to the regular representation of C, {\displaystyle \mathbb {C},} thought of as an R {\displaystyle \mathbb {R} } - algebra (an R {\displaystyle \mathbb {R} } -vector space with a multiplication), with respect to the basis (1, i). This means the following: the R {\displaystyle \mathbb {R} } -linear map C → C z ↦ w z {\displaystyle {\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}} for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is (Re (w) − Im (w) Im (w) Re (w)), {\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},} that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C {\displaystyle \mathbb {C} } in the 2 × 2 real matrices, it is not the only one. Any matrix J = (p q r − p), p 2 + q r + 1 = 0 {\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0} has the property that its square is the negative of the identity matrix: J = − I. Then { z = a I + b J: a, b ∈ R } {\displaystyle \{z=aI+bJ:a,b\in \mathbb {R} \}} is also isomorphic to the field C, {\displaystyle \mathbb {C},} and gives an alternative complex structure on R 2. {\displaystyle \mathbb {R} ^{2}.} This is generalized by the notion of a linear complex structure.
  
  Hypercomplex numbers also generalize R, {\displaystyle \mathbb {R},} C, {\displaystyle \mathbb {C},} H, {\displaystyle \mathbb {H},} and O. {\displaystyle \mathbb {O}.} For example, this notion contains the split-complex numbers, which are elements of the ring R / (x 2 − 1) {\displaystyle \mathbb {R} /(x^{2}-1)} (as opposed to R / (x 2 + 1) {\displaystyle \mathbb {R} /(x^{2}+1)} for complex numbers). In this ring, the equation a = 1 has four solutions.
  
  The field R {\displaystyle \mathbb {R} } is the completion of Q, {\displaystyle \mathbb {Q},} the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q {\displaystyle \mathbb {Q} } lead to the fields Q p {\displaystyle \mathbb {Q} _{p}} of p -adic numbers (for any prime number p), which are thereby analogous to R {\displaystyle \mathbb {R} }. There are no other nontrivial ways of completing Q {\displaystyle \mathbb {Q} } than R {\displaystyle \mathbb {R} } and Q p, {\displaystyle \mathbb {Q} _{p},} by Ostrowski's theorem. The algebraic closures Q p ¯ {\displaystyle {\overline {\mathbb {Q} _{p}}}} of Q p {\displaystyle \mathbb {Q} _{p}} still carry a norm, but (unlike C {\displaystyle \mathbb {C} }) are not complete with respect to it. The completion C p {\displaystyle \mathbb {C} _{p}} of Q p ¯ {\displaystyle {\overline {\mathbb {Q} _{p}}}} turns out to be algebraically closed. By analogy, the field is called p -adic complex numbers.
  
  The fields R, {\displaystyle \mathbb {R},} Q p, {\displaystyle \mathbb {Q} _{p},} and their finite field extensions, including C, {\displaystyle \mathbb {C},} are called local fields.