Editing Cayley–Dickson construction (section)

=== Quaternions ===
{{Main|Quaternion}}
[[File:Cayley_Q8_multiplication_graph.svg|thumb|link={{filepath:Cayley_Q8_multiplication_graph.svg}}|Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of ''i'' (red), ''j'' (green) and ''k'' (blue). In [{{filepath:Cayley_Q8_quaternion_multiplication_graph.svg}} the SVG file,] hover over or click a path to highlight it.]]

The next step in the construction is to generalize the multiplication and conjugation operations.

Form ordered pairs {{math|(''a'', ''b'')}} of complex numbers {{mvar|a}} and {{mvar|b}}, with multiplication defined by

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

Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.

The order of the factors seems odd now, but will be important in the next step.

Define the conjugate {{math|(''a'', ''b'')*}} of {{math|(''a'', ''b'')}} by

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

These operators are direct extensions of their complex analogs:  if {{mvar|a}} and {{mvar|b}} are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

The product of a nonzero element with its conjugate is a non-negative real number:

: <math>\begin{align} (a, b)^* (a, b) &= (a^*, -b) (a, b) \\ &= (a^* a + b^* b, b a^* - b a^*) \\ &= \left(|a|^2 + |b|^2, 0 \right).\, \end{align}</math>

As before, the conjugate thus yields a norm and an inverse for any such ordered pair.  So in the sense we explained above, these pairs constitute an algebra something like the real numbers.  They are the [[quaternion]]s, named by [[William Rowan Hamilton|Hamilton]] in 1843.

As a quaternion consists of two independent complex numbers, they form a four-dimensional vector space over the real numbers.

The multiplication of quaternions is not quite like the multiplication of real numbers, though; it is not [[commutative]] – that is, if {{mvar|p}} and {{mvar|q}} are quaternions, it is not always true that {{math|''pq'' {{=}} ''qp''}}.