Editing Split-complex number (section)

===Conjugate, modulus, and bilinear form===
Just as for complex numbers, one can define the notion of a '''split-complex conjugate'''. If

<math display=block> z = x + jy ~,</math>

then the conjugate of {{mvar|z}} is defined as

<math display=block> z^* = x - jy ~.</math>

The conjugate is an [[involution (mathematics)|involution]] which satisfies similar properties to the [[complex conjugate]]. Namely,

<math display=block>\begin{align}
           (z + w)^* &= z^* + w^* \\
              (zw)^* &= z^* w^* \\
  \left(z^*\right)^* &= z.
\end{align}</math>

The squared '''modulus''' of a split-complex number <math>z=x+jy</math> is given by the [[isotropic quadratic form]]

<math display=block>\lVert z \rVert^2 = z z^* = z^* z = x^2 - y^2 ~.</math>

It has the [[composition algebra]] property:

<math display=block>\lVert z w \rVert = \lVert z \rVert \lVert w \rVert ~.</math>

However, this quadratic form is not [[definite bilinear form|positive-definite]] but rather has [[metric signature|signature]] {{math|(1, −1)}}, so the modulus is ''not'' a [[norm (mathematics)|norm]].

The associated [[bilinear form]] is given by

<math display=block>\langle z, w \rangle = \operatorname\mathrm{Re}\left(zw^*\right) = \operatorname\mathrm{Re} \left(z^* w\right) = xu - yv ~,</math>

where <math>z=x+jy</math> and <math>w=u+jv.</math> Here, the ''real part'' is defined by <math>\operatorname\mathrm{Re}(z) = \tfrac{1}{2}(z + z^*) = x</math>. Another expression for the squared modulus is then

<math display=block> \lVert z \rVert^2 = \langle z, z \rangle ~.</math>

Since it is not positive-definite, this bilinear form is not an [[inner product]]; nevertheless the bilinear form is frequently referred to as an ''indefinite inner product''. A similar abuse of language refers to the modulus as a norm.

A split-complex number is invertible [[if and only if]] its modulus is nonzero {{nowrap|(<math>\lVert z \rVert \ne 0</math>),}} thus numbers of the form {{math|''x'' ± ''j x''}} have no inverse. The [[multiplicative inverse]] of an invertible element is given by

<math display=block>z^{-1} = \frac{z^*}{ {\lVert z \rVert}^2} ~.</math>

Split-complex numbers which are not invertible are called [[null vector]]s. These are all of the form {{math|(''a'' ± ''j a'')}} for some real number {{mvar|a}}.