Editing Cayley–Dickson construction (section)

=== Complex numbers as ordered pairs ===
{{Main|Complex number}}
The [[complex numbers]] can be written as [[ordered pair]]s {{math|(''a'', ''b'')}} of [[real number]]s {{mvar|a}} and {{mvar|b}}, with the addition operator being component-wise and with multiplication defined by

: <math>(a, b) (c, d) = (a c - b d, a d + b c).\,</math>

A complex number whose second component is zero is associated with a real number: the complex number {{math|(''a'', 0)}} is associated with the real number&nbsp;{{mvar|a}}.

The [[complex conjugate]] {{math|(''a'', ''b'')*}} of {{math|(''a'', ''b'')}} is given by

: <math>(a, b)^* = (a^*, -b) = (a, -b)</math>

since {{mvar|a}} is a real number and is its own conjugate.

The conjugate has the property that

: <math>(a, b)^* (a, b)  = (a a + b b, a b - b a) = \left(a^2 + b^2, 0\right),\,</math>

which is a non-negative real number.  In this way, conjugation defines a ''[[norm (mathematics)|norm]]'', making the complex numbers a [[normed vector space]] over the real numbers:  the norm of a complex number&nbsp;{{mvar|z}} is

: <math>|z| = \left(z^* z\right)^\frac12.\,</math>

Furthermore, for any non-zero complex number&nbsp;{{mvar|z}}, conjugation gives a [[inverse element|multiplicative inverse]],

: <math>z^{-1} = \frac{z^*}{|z|^2}.</math>

As a complex number consists of two independent real numbers, they form a two-dimensional [[vector space]] over the real numbers.

Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.