Editing Quaternion (section)

==== As a union of complex planes ====
Each [[antipodal points|antipodal pair]] of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If {{nowrap|{{math|''q''<sup>2</sup> {{=}} −1}},}} then the copy is the [[image (mathematics)|image]] of the function

<math display=block>a + bi \mapsto a + b q.</math>

This is an [[injective function|injective]] [[ring homomorphism]] from <math>\mathbb C</math> to <math>\mathbb H,</math> which defines a field [[isomorphism]] from <math>\Complex</math> onto its [[image (mathematics)|image]]. The images of the embeddings corresponding to {{mvar|q}} and −{{mvar|q}} are identical.

Every non-real quaternion generates a [[subalgebra]] of the quaternions that is isomorphic to <math>\mathbb C,</math> and is thus a planar subspace of <math>\mathbb H\colon</math> write {{mvar|q}} as the sum of its scalar part and its vector part:

<math display=block>q = q_s + \vec{q}_v.</math>

Decompose the vector part further as the product of its norm and its [[versor]]:

<math display=block>q = q_s + \lVert\vec{q}_v\rVert\cdot\mathbf{U}\vec{q}_v=q_s+\|\vec q_v\|\,\frac{\vec q_v}{\|\vec q_v\|}.</math>

(This is not the same as <math>q_s + \lVert q\rVert\cdot\mathbf{U}q</math>.) The versor of the vector part of {{mvar|q}}, <math>\mathbf{U}\vec{q}_v</math>, is a right versor with –1 as its square. A straightforward verification shows that
<math display=block>a + bi \mapsto a + b\mathbf{U}\vec{q}_v</math>
defines an injective [[algebra homomorphism|homomorphism]] of [[normed algebra]]s from <math>\mathbb C</math> into the quaternions. Under this homomorphism, {{mvar|q}} is the image of the complex number <math>q_s + \lVert\vec{q}_v\rVert i</math>.

As <math>\mathbb H</math> is the [[union (set theory)#Arbitrary unions|union]] of the images of all these homomorphisms, one can view the quaternions as a [[pencil of planes]] intersecting on the [[real line]]. Each of these [[complex plane]]s contains exactly one pair of [[antipodal points]] of the sphere of square roots of minus one.