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Quaternion

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by $1$ (for Hamilton), or in blackboard bold by $$\mathbb H.$$ Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form

$$a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k,$$

where the coefficients $a$, $b$, $c$, $d$ are real numbers, and $i$, $j$ are the basis vectors or basis elements.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, magnetic resonance imaging and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern terms, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified as $$\operatorname{Cl}_{0,2}(\mathbb R)\cong \operatorname{Cl}_{3,0}^+(\mathbb R).$$ It was the first noncommutative division algebra to be discovered.

According to the Frobenius theorem, the algebra $$\mathbb H$$ is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra.

The unit quaternions give a group structure on the 3-sphere $k$ isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.

[[File:Quaternion 2.svg|thumb|right|Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of $1$ The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually $1$. • In blue: • $i$ (1/i plane)

• $j$ (i/k plane)

• In red: • $k$ (1/j plane)

• $i$ (j/k plane)]]

History
[[File:Inscription on Broom Bridge (Dublin) regarding the discovery of Quaternions multiplication by Sir William Rowan Hamilton.jpg|right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge]], Dublin, which reads:

]]

Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact, Ferdinand Georg Frobenius later proved in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras: $$\mathbb {R, C}$$ (complex numbers) and $$\mathbb H$$ (quaternions) which have dimension 1, 2, and 4 respectively.

The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,

$$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i\,j\,k} = -1$$

into the stone of Brougham Bridge as he paused on it. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science; Hamilton states: "And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth."

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long; it was edited by his son and published shortly after his death.

After Hamilton's death, the Scottish mathematical physicist Peter Tait became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described entirely in terms of quaternions. There was even a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems.

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.

However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles, they are not susceptible to "gimbal lock". For this reason, quaternions are used in computer graphics, computer vision, robotics, nuclear magnetic resonance image sampling, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.

Quaternions in physics
The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions (in the form of the famous Pauli spin matrices) furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "Plate trick"). , their use has not overtaken rotation groups.

Definition
A quaternion is an expression of the form

$$a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}\ ,$$

where $a$, $b$, $c$, $d$, are real numbers, and $i$, $−1$, $k$, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of $a$, $b$, $c$, $d$ is 0, the corresponding term is omitted; if $a$, $b$, $c$, $d$ are all zero, the quaternion is the zero quaternion, denoted 0; if one of $b$, $c$, $d$ equals 1, the corresponding term is written simply $−j$, or $j$.

Hamilton describes a quaternion $$q = a + b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}$$, as consisting of a scalar part and a vector part. The quaternion $$ b\,\mathbf{i} + c\,\mathbf{j} + d\,\mathbf{k}$$ is called the vector part (sometimes imaginary part) of $q$, and $a$ is the scalar part (sometimes real part) of $q$. A quaternion that equals its real part (that is, its vector part is zero) is called a scalar or real quaternion, and is identified with the corresponding real number. That is, the real numbers are embedded in the quaternions. (More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.) A quaternion that equals its vector part is called a vector quaternion.

The set of quaternions is a 4-dimensional vector space over the real numbers, with $$\left\{ 1, \mathbf i, \mathbf j, \mathbf k\right\}$$ as a basis, by the component-wise addition

$$\begin{align} &(a_1+b_1\,\mathbf i + c_1\,\mathbf j + d_1\,\mathbf k) + (a_2 + b_2\,\mathbf i + c_2\,\mathbf j + d_2\,\mathbf k) \\[3mu] &\qquad = (a_1 + a_2) + (b_1 + b_2)\,\mathbf i + (c_1 + c_2)\,\mathbf j + (d_1 + d_2)\,\mathbf k, \end{align}$$

and the component-wise scalar multiplication

$$\lambda(a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k) = \lambda a + (\lambda b)\,\mathbf i + (\lambda c)\,\mathbf j + (\lambda d)\,\mathbf k.$$

A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way: Thus the quaternions form a division algebra.
 * The real quaternion $j$ is the identity element.
 * The real quaternions commute with all other quaternions, that is $−k$ for every quaternion $q$ and every real quaternion $a$. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
 * The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the distributive property and the center property of the real quaternions. The Hamilton product is not commutative, but is associative, thus the quaternions form an associative algebra over the real numbers.
 * Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product: $$(a + b\,\mathbf i + c\,\mathbf j + d \,\mathbf k)^{-1} = \frac{1}{a^2 + b^2 + c^2 + d^2}\,(a - b\,\mathbf i - c\,\mathbf j- d\,\mathbf k).$$

Multiplication of basis elements
The multiplication with $−1$ of the basis elements $i$, and $k$ is defined by the fact that $k$ is a multiplicative identity, that is,

$$\mathbf i \, 1 = 1 \, \mathbf i = \mathbf i, \qquad \mathbf j \, 1 = 1 \, \mathbf j = \mathbf j, \qquad \mathbf k \, 1 = 1 \, \mathbf k= \mathbf k \,.$$

The products of other basis elements are

$$\begin{align} \mathbf i^2 &= \mathbf j^2 = \mathbf k^2 = -1, \\[5mu] \mathbf{i\,j} &= - \mathbf{j\,i} = \mathbf k, \qquad \mathbf{j\,k} = - \mathbf{k\,j} = \mathbf i, \qquad \mathbf{k\,i} = - \mathbf{i\,k} = \mathbf j. \end{align}$$

Combining these rules,

$$\begin{align} \mathbf{i\,j\,k}&=-1. \end{align}$$

Center
The center of a noncommutative ring is the subring of elements $c$ such that $j$ for every $x$. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if $−i$ belongs to the center, then

$$0 = \mathbf i\,q - q\,\mathbf i = 2c\,\mathbf{ij} + 2d\,\mathbf{ik} = 2c\,\mathbf k - 2d\,\mathbf j\,,$$

and $−1$. A similar computation with $H$ instead of $1, i, j$ shows that one has also $k$. Thus $S^{3}$ is a real quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a polynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation ${1, i, j, k}.$, has infinitely many quaternion solutions, which are the quaternions $i⋅j = −(j⋅i)$ such that $1⋅i=i$. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.

Hamilton product
For two elements $i⋅j=k$ and $1⋅j=j$, their product, called the Hamilton product ($j⋅i=−k$) ($i^{2} = j^{2} = k^{2} = ijk = −1$), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

$$\begin{alignat}{4} &a_1a_2 &&+ a_1b_2 \mathbf i   &&+ a_1c_2 \mathbf j   &&+ a_1d_2 \mathbf k\\ {}+{} &b_1a_2 \mathbf i &&+ b_1b_2 \mathbf i^2 &&+ b_1c_2 \mathbf{ij} &&+ b_1d_2 \mathbf{ik}\\ {}+{} &c_1a_2 \mathbf j &&+ c_1b_2 \mathbf{ji} &&+ c_1c_2 \mathbf j^2 &&+ c_1d_2 \mathbf{jk}\\ {}+{} &d_1a_2 \mathbf k &&+ d_1b_2 \mathbf{ki} &&+ d_1c_2 \mathbf{kj} &&+ d_1d_2 \mathbf k^2 \end{alignat}$$

Now the basis elements can be multiplied using the rules given above to get:

$$\begin{alignat}{4} &a_1a_2 &&- b_1b_2 &&- c_1c_2 &&- d_1d_2\\ {}+{} (&a_1b_2 &&+ b_1a_2 &&+ c_1d_2 &&- d_1c_2) \mathbf i\\ {}+{} (&a_1c_2 &&- b_1d_2 &&+ c_1a_2 &&+ d_1b_2) \mathbf j\\ {}+{} (&a_1d_2 &&+ b_1c_2 &&- c_1b_2 &&+ d_1a_2) \mathbf k \end{alignat}$$

Scalar and vector parts
A quaternion of the form $i$, where $a$ is a real number, is called scalar, and a quaternion of the form $j$, where $b$, $c$, and $d$ are real numbers, and at least one of $b$, $c$, or $d$ is nonzero, is called a vector quaternion. If $k$ is any quaternion, then $a$ is called its scalar part and $i, j$ is called its vector part. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space $$\mathbb R^3.$$

Hamilton also called vector quaternions right quaternions and real numbers (considered as quaternions with zero vector part) scalar quaternions.

If a quaternion is divided up into a scalar part and a vector part, that is,

$$ \mathbf q = (r,\ \vec{v}), \mathbf q \in \mathbb{H}, r \in \mathbb{R}, \vec{v}\in \mathbb{R}^3, $$

then the formulas for addition, multiplication, and multiplicative inverse are

$$\begin{align} (r_1,\ \vec{v}_1) + (r_2,\ \vec{v}_2) &= (r_1 + r_2,\ \vec{v}_1+\vec{v}_2)\,, \\[5mu] (r_1,\ \vec{v}_1) (r_2,\ \vec{v}_2) &= (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\ r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2)\,, \\[5mu] (r, \vec{v})^{-1} &= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v}}, \frac{-\vec{v}}{r^2 + \vec{v}\cdot\vec{v}}\right)\,, \end{align}$$

where "$$\cdot$$" and "$$\times$$" denote respectively the dot product and the cross product.

Conjugation, the norm, and reciprocal
Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let $$q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k $$ be a quaternion. The conjugate of $q$ is the quaternion $$ q^* = a - b\,\mathbf i - c\,\mathbf j - d\,\mathbf k $$. It is denoted by $k$, qt, $$\tilde q$$, or $\overline{q}$. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if $p$ and $q$ are quaternions, then $1$, not $aq = qa$.

The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

$$ q^* = - \frac{1}{2} (q + \,\mathbf i \,q \,\mathbf i + \,\mathbf j \,q \,\mathbf j + \,\mathbf k \,q \,\mathbf k). $$

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of $p$ is $1$, and the vector part of $p$ is $i$.

The square root of the product of a quaternion with its conjugate is called its norm and is denoted $j$ (Hamilton called this quantity the tensor of q, but this conflicts with the modern meaning of "tensor"). In formulas, this is expressed as follows:

$$\lVert q \rVert = \sqrt{\,qq^*~} = \sqrt{\,q^*q~} = \sqrt{\,a^2 + b^2 + c^2 + d^2~}$$

This is always a non-negative real number, and it is the same as the Euclidean norm on $$\mathbb H$$ considered as the vector space $$\mathbb R^4$$. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if $α$ is real, then

$$\lVert\alpha q\rVert = \left| \alpha\right|\,\lVert q\rVert~.$$

This is a special case of the fact that the norm is multiplicative, meaning that

$$\lVert pq \rVert = \lVert p \rVert\,\lVert q \rVert$$

for any two quaternions $p$ and $q$. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively it follows from the identity

$$ \det \begin{pmatrix} a + i b & i d + c \\ i d - c & a - i b \end{pmatrix} = a^2 + b^2 + c^2 + d^2,$$

(where $i$ denotes the usual imaginary unit) and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distance $k$ between $p$ and $q$ as the norm of their difference:

$$d(p, q) = \lVert p - q \rVert~.$$

This makes $$\mathbb H$$ a metric space. Addition and multiplication are continuous in regard to the associated metric topology. This follows with exactly the same proof as for the real numbers $$\mathbb R$$ from the fact that $$\mathbb H$$ is a normed algebra.

Unit quaternion
A unit quaternion is a quaternion of norm one. Dividing a nonzero quaternion $q$ by its norm produces a unit quaternion $1$ called the versor of $q$:

$$\mathbf{U}q = \frac{q}{\lVert q\rVert}.$$

Every nonzero quaternion has a unique polar decomposition $$ q = \lVert q \rVert \cdot \mathbf{U} q $$, while the zero quaternion can be formed from any unit quaternion.

Using conjugation and the norm makes it possible to define the reciprocal of a nonzero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of $$q$$ and $$q^* / \left \Vert q \right \| ^2$$ is 1 (for either order of multiplication). So the reciprocal of $q$ is defined to be

$$q^{-1} = \frac{q^*}{\lVert q\rVert^2}.$$

Since the multiplication is non-commutative, the quotient quantities $1$ or $i$ are different (except if $p$ and $q$ are scalar multiples of each other or if one is a scalar): the notation $j$ is ambiguous and should not be used.

Algebraic properties

The set $$\mathbb H$$ of all quaternions is a vector space over the real numbers with dimension 4. Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions $$\mathbb H$$ are a non-commutative, associative algebra over the real numbers. Even though $$\mathbb H$$ contains copies of the complex numbers, it is not an associative algebra over the complex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three: $$\mathbb R$$, $$\mathbb C$$, and $$\mathbb H$$. The norm makes the quaternions into a normed algebra, and normed division algebras over the real numbers are also very rare: Hurwitz's theorem says that there are only four: $$\mathbb R$$, $$\mathbb C$$, $$\mathbb H$$, and $$\mathbb O$$ (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra.

Because the product of any two basis vectors is plus or minus another basis vector, the set $k$ forms a group under multiplication. This non-abelian group is called the quaternion group and is denoted $i$. The real group ring of $i$ is a ring $$\mathbb R[\mathrm Q_8]$$ which is also an eight-dimensional vector space over $$\mathbb R.$$ It has one basis vector for each element of $$\mathrm Q_8.$$ The quaternions are isomorphic to the quotient ring of $$\mathbb R[\mathrm Q_8]$$ by the ideal generated by the elements $−1$, $k$, $−j$, and $j$. Here the first term in each of the differences is one of the basis elements $j$, and $−k$, and the second term is one of basis elements $−1$, and $i$, not the additive inverses of $k$, and $k$.

Quaternions and three-dimensional geometry
The vector part of a quaternion can be interpreted as a coordinate vector in $$\mathbb R^3;$$ therefore, the algebraic operations of the quaternions reflect the geometry of $$\mathbb R^3.$$ Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics.

For the remainder of this section, $j$, $−i$, and $−1$ will denote both the three imaginary basis vectors of $$\mathbb H$$ and a basis for $$\mathbb R^3.$$ Replacing $1$ by $i, j$, $k$ by $1$, and $cx = xc$ by $q = a + b i + c j + d k$ sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

For two vector quaternions $c = d = 0$ and $j$ their dot product, by analogy to vectors in $$\mathbb R^3,$$ is

$$p \cdot q = b_1 b_2 + c_1 c_2 + d_1 d_2~.$$

It can also be expressed in a component-free manner as

$$p \cdot q = \textstyle\frac{1}{2}(p^*q + q^*p) = \textstyle\frac{1}{2}(pq^* + qp^*).$$

This is equal to the scalar parts of the products $i$. Note that their vector parts are different.

The cross product of $p$ and $q$ relative to the orientation determined by the ordered basis $b = 0$, and $q = a$ is

$$p \times q = (c_1 d_2 - d_1 c_2)\mathbf i + (d_1 b_2 - b_1 d_2)\mathbf j + (b_1 c_2 - c_1 b_2)\mathbf k\,.$$

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product $z^{2} + 1 = 0$ (as quaternions), as well as the vector part of $z = b i + c j + d k$. It also has the formula

$$p \times q = \textstyle\tfrac{1}{2}(pq - qp).$$

For the commutator, $b^{2} + c^{2} + d^{2} = 1$, of two vector quaternions one obtains

$$[p,q]= 2p \times q.$$

In general, let $p$ and $q$ be quaternions and write

$$\begin{align} p &= p_\text{s} + p_\text{v}, \\[5mu] q &= q_\text{s} + q_\text{v}, \end{align}$$

where $a_{1} + b_{1}i + c_{1}j + d_{1}k$ and $a_{2} + b_{2}i + c_{2}j + d_{2}k$ are the scalar parts, and $a_{1} + b_{1}i + c_{1}j + d_{1}k$ and $a_{2} + b_{2}i + c_{2}j + d_{2}k$ are the vector parts of $p$ and $q$. Then we have the formula

$$pq = (pq)_\text{s} + (pq)_\text{v} = (p_\text{s}q_\text{s} - p_\text{v}\cdot q_\text{v}) + (p_\text{s} q_\text{v} + q_\text{s} p_\text{v} + p_\text{v} \times q_\text{v}).$$

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.

Unit quaternions can be identified with rotations in $$\mathbb R^3$$ and were called versors by Hamilton. Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.

See Hanson (2005) for visualization of quaternions.

Matrix representations
Just as complex numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related representations. These are injective homomorphisms from $$\mathbb H$$ to the matrix rings $a + 0 i + 0 j + 0 k$ and $0 + b i + c j + d k$, respectively.

The quaternion $a + b i + c j + d k$ can be represented using a 2 × 2 complex matrix as

 * $$ \left[ \begin{array}{rr} a+bi & c+di \\ -c + d i & a - b i \end{array} \right].$$

This representation has the following properties:
 * Constraining any two of $b$, $c$ and $d$ to zero produces a representation of complex numbers. For example, setting $b i + c j + d k$ produces a diagonal complex matrix representation of complex numbers, and setting $q^{∗}$ produces a real matrix representation.
 * The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
 * The scalar part of a quaternion is one half of the matrix trace.
 * The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
 * By restriction this representation yields an isomorphism between the subgroup of unit quaternions and their image SU(2). Topologically, the unit quaternions are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group $(pq)^{∗} = q^{∗}p^{∗}$ is important for describing spin in quantum mechanics; see Pauli matrices.
 * There is a strong relation between quaternion units and Pauli matrices. The 2 × 2 complex matrix above can be written as $$a I + b i \sigma_3 + c i \sigma_2 + d i \sigma_1$$, so in this representation the quaternion units $p^{∗}q^{∗}$ correspond to $1⁄2(p + p^{∗})$ = $1⁄2(p − p^{∗})$, so multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1. One obtains −1 via $\|q\|$; e.g. the last equality is $$\mathbf{i\;j\;k} = \sigma_1 \sigma_2 \sigma_3 \sigma_1 \sigma_2 \sigma_3 = -1.$$
 * The representation in $d(p, q)$ is not unique. A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose
 * $$1 \mapsto \mathbf{I}\,, \quad \mathbf{i} \mapsto - i \sigma_1 = - \sigma_2 \sigma_3\,, \quad \mathbf{j} \mapsto - i \sigma_2 = - \sigma_3 \sigma_1\,, \quad \mathbf{k} \mapsto - i \sigma_3 = - \sigma_1 \sigma_2\,,$$
 * This gives an alternative representation,
 * $Uq$  $$ \mapsto \left[ \begin{array}{rr}  a-di & -c-bi \\ c - b i & a + d i \end{array} \right].$$

Using 4 × 4 real matrices, that same quaternion can be written as

$$\begin{align} \left[ \begin{array}{rrrr} a & -b & -c & -d \\ b & a & -d & c \\ c & d & a & -b \\ d & -c & b & a \end{array} \right] &= a \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] + b \left[ \begin{array}{rrrr} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{array} \right] \\[10mu] &\qquad + c \left[ \begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{array} \right] + d \left[ \begin{array}{rrrr} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array} \right]. \end{align}$$

However, the representation of quaternions in $p q^{−1}$ is not unique. For example, the same quaternion can also be represented as

$$\begin{align} \left[ \begin{array}{rrrr} a & d & -b & -c \\ -d & a & c & -b \\ b & -c & a & -d \\ c & b & d & a \end{array} \right] &= a \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] + b \left[ \begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right] \\[10mu] &\qquad + c \left[ \begin{array}{rrrr} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array} \right] + d \left[ \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{array} \right]. \end{align}$$

There exist 48 distinct matrix representations of this form in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48 sets of quadruples of matrices with these symmetry constraints such that a function sending $q^{−1}p$, and $p⁄q$ to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices. In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting $Q_{8}$.

Each 4×4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table

which is isomorphic — through $$\{a \mapsto 1, b \mapsto i, c \mapsto j, d \mapsto k\}$$ — to

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing $i$ with $a$, $j$ with $b$, $i$ with $c$, and $j$ with $d$ and removing the row and column headers yields a matrix representation of $k$.

Lagrange's four-square theorem
Quaternions are also used in one of the proofs of Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.

Quaternions as pairs of complex numbers
Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

Let $$\mathbb C^2$$ be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements $\{±1, ±i, ±j, ±k\}$ and $Q_{8}$. A vector in $$\mathbb C^2$$ can be written in terms of the basis elements $Q_{8}$ and $1 + (−1)$ as

$$(a + b i)1 + (c + d i)\mathbf j\,. $$

If we define $i + (−i)$ and $j + (−j)$, then we can multiply two vectors using the distributive law. Using $k + (−k)$ as an abbreviated notation for the product $1, i, j$ leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion $k$. If we write the elements of $$\mathbb C^2$$ as ordered pairs and quaternions as quadruples, then the correspondence is

$$(a + bi,\ c + di) \leftrightarrow (a, b, c, d).$$

Square roots of −1
In the complex numbers, $$\mathbb C,$$ there are exactly two numbers, $i$ and $−1, −i, −j$, that give −1 when squared. In $$\mathbb H$$ there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in $$\mathbb R^3.$$ To see this, let $−k$ be a quaternion, and assume that its square is −1. In terms of $a$, $b$, $c$, and $d$, this means

$$\begin{align} a^2 - b^2 - c^2 - d^2 &= -1, \vphantom{x^|} \\[3mu] 2ab &= 0, \\[3mu] 2ac &= 0, \\[3mu] 2ad &= 0. \end{align}$$

To satisfy the last three equations, either $1, i, j$ or $b$, $c$, and $d$ are all 0. The latter is impossible because a is a real number and the first equation would imply that $k$. Therefore, $i$ and $j$. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere.

Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0).

As a union of complex planes
Each antipodal pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If $k$, then the copy is the image of the function

$$a + bi \mapsto a + b q\,.$$

This is an injective ring homomorphism from $$\mathbb C$$ to $$\mathbb H,$$ which defines a field isomorphism from $$\Complex$$ onto its image. The images of the embeddings corresponding to $q$ and −$q$ are identical.

Every non-real quaternion generates a subalgebra of the quaternions that is isomorphic to $$\mathbb C,$$ and is thus a planar subspace of $$\mathbb H\colon$$ write $q$ as the sum of its scalar part and its vector part:

$$q = q_s + \vec{q}_v.$$

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

$$q = q_s + \lVert\vec{q}_v\rVert\cdot\mathbf{U}\vec{q}_v=q_s+\|\vec q_v\|\,\frac{\vec q_v}{\|\vec q_v\|}.$$

(This is not the same as $$q_s + \lVert q\rVert\cdot\mathbf{U}q$$.) The versor of the vector part of $q$, $$\mathbf{U}\vec{q}_v$$, is a right versor with –1 as its square. A straightforward verification shows that $$a + bi \mapsto a + b\mathbf{U}\vec{q}_v$$ defines an injective homomorphism of normed algebras from $$\mathbb C$$ into the quaternions. Under this homomorphism, $q$ is the image of the complex number $$q_s + \lVert\vec{q}_v\rVert i$$.

As $$\mathbb H$$ is the 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 planes contains exactly one pair of antipodal points of the sphere of square roots of minus one.

Commutative subrings
The relationship of quaternions to each other within the complex subplanes of $$\mathbb H$$ can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions $p$ and $q$ commute (i.e., $i$) only if they lie in the same complex subplane of $$\mathbb H$$, the profile of $$\mathbb H$$ as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring.

Square roots of arbitrary quaternions
Any quaternion $$\mathbf q = (r,\, \vec{v})$$ (represented here in scalar–vector representation) has at least one square root $$\sqrt{\mathbf q} = (x,\, \vec{y})$$ which solves the equation $$\sqrt{\mathbf q}^2 = (x,\, \vec{y})^2 = \mathbf q$$. Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions

$$ \sqrt{\mathbf q} = \sqrt{(r,\, \vec{v}\,)} = \pm\left(\sqrt{\frac{\|\mathbf q\|+r}{2}},\ \frac{\vec{v}}{\|\vec{v}\|}\sqrt{\frac{\|\mathbf q\|-r}{2}}\right), $$

where $\|\vec{v}\| = \sqrt{\vec{v}\cdot\vec{v}}=\sqrt{-\vec v^2}$ is the norm of $$\vec{v}$$ and $\|\mathbf q\| = \sqrt{\mathbf q^*\mathbf q} = r^2 + \|\vec{v}\|^2$  is the norm of $$\mathbf q$$. For any scalar quaternion $$\mathbf q$$, this equation provides the correct square roots if $\frac{\vec{v}}{\|\vec{v}\|}$ is interpreted as an arbitrary unit vector.

Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on $$\{0\} \times S^2(\sqrt{-r})$$, i.e., where the scalar part is zero and the vector part is located on the 2-sphere with radius $$\sqrt{-r}$$.

Functions of a quaternion variable
Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4-dimensional space.

Exponential, logarithm, and power functions
Given a quaternion,

$$ q = a + b\mathbf i + c\mathbf j + d\mathbf k = a + \mathbf{v}, $$

the exponential is computed as

$$ \exp(q) = \sum_{n=0}^\infty \frac{q^n}{n!} = e^{a} \left(\cos \|\mathbf{v}\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \sin \|\mathbf{v}\|\right), $$

and the logarithm is

$$\ln(q) = \ln \|q\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \arccos \frac{a}{\|q\|}.$$

It follows that the polar decomposition of a quaternion may be written

$$q=\|q\|e^{\hat{n}\varphi} = \|q\| \left(\cos(\varphi) + \hat{n} \sin(\varphi)\right),$$

where the angle $$\varphi$$

$$a = \| q \| \cos( \varphi )$$

and the unit vector $$\hat{n}$$ is defined by:

$$\mathbf{v} = \hat{n} \|\mathbf{v}\|= \hat{n}\|q\|\sin(\varphi)\,.$$

Any unit quaternion may be expressed in polar form as:

$$q=\exp{(\hat{n}\varphi)}.$$

The power of a quaternion raised to an arbitrary (real) exponent $x y$ is given by:

$$q^x = \|q\|^x e^{\hat{n} x \varphi} = \|q\|^x \left(\cos(x\varphi) + \hat{n}\,\sin(x\varphi)\right)~.$$

Geodesic norm
The geodesic distance $−i$ between unit quaternions $x$ and $p$ is defined as:

$$d_\text{g}(p, q) = \lVert \ln(p^{-1} q) \rVert.$$

and amounts to the absolute value of half the angle subtended by $q$ and $p$ along a great arc of the $j$ sphere. This angle can also be computed from the quaternion dot product without the logarithm as:

$$\arccos(2(p \cdot q)^2 - 1).$$

Three-dimensional and four-dimensional rotation groups
The word "conjugation", besides the meaning given above, can also mean taking an element $q$ to $−j$ where $a$ is some nonzero quaternion. All elements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of $$\mathbb R^3$$ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part $k$ is a rotation by an angle $−k$, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

 * Avoiding gimbal lock, a problem with systems such as Euler angles.
 * Faster and more compact than matrices.
 * Nonsingular representation (compared with Euler angles for example).
 * Pairs of unit quaternions represent a rotation in 4D space (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations).

The set of all unit quaternions (versors) forms a 3-sphere $p = b_{1}i + c_{1}j + d_{1}k$ and a group (a Lie group) under multiplication, double covering the group $$\text{SO}(3,\mathbb{R})$$ of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See plate trick.

The image of a subgroup of versors is a point group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group.

The versors' group is isomorphic to $q = b_{2}i + c_{2}j + d_{2}k$, the group of complex unitary 2×2 matrices of determinant 1.

Let $r$ be the set of quaternions of the form $pq^{∗}, qp^{∗}, p^{∗}q, and q^{∗}p$ where $A$ and $a, b, c,$ are either all integers or all half-integers. The set $d$ is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a regular 24 cell with Schläfli symbol $i, j$ They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600 cell with Schläfli symbol $k$ can be taken as the unit icosians, corresponding to the double cover of the rotational symmetry group of the regular icosahedron. The double cover of the rotational symmetry group of the regular octahedron corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell.

Quaternion algebras
The Quaternions can be generalized into further algebras called quaternion algebras. Take $A$ to be any field with characteristic different from 2, and $F$ and $a$ to be elements of $b$; a four-dimensional unitary associative algebra can be defined over $F$ with basis $pq$ and $−q^{∗}p^{∗}$, where $[p, q] = pq − qp$, $p_{s}$ and $q_{s}$ (so $p_{v}$).

Quaternion algebras are isomorphic to the algebra of 2×2 matrices over $F$ or form division algebras over $F$, depending on the choice of $F$ and $a$.

Quaternions as the even part of $q_{v}$
The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part $$\operatorname{Cl}_{3,0}^+(\mathbb R)$$ of the Clifford algebra $$\operatorname{Cl}_{3,0}(\mathbb R).$$ This is an associative multivector algebra built up from fundamental basis elements $M(2,C)$ using the product rules

$$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1,$$ $$\sigma_i \sigma_j = - \sigma_j \sigma_i \qquad (j \neq i).$$

If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that the reflection of a vector $b$ in a plane perpendicular to a unit vector $r$ can be written:

$$r^{\prime} = - w\, r\, w.$$

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

$$r^{\prime\prime} = \sigma_2 \sigma_1 \, r \, \sigma_1 \sigma_2 $$

corresponds to a rotation of 180° in the plane containing σ1 and σ2. This is very similar to the corresponding quaternion formula,

$$r^{\prime\prime} = -\mathbf{k}\, r\, \mathbf{k}. $$

Indeed, the two structures $$\operatorname{Cl}_{3,0}^+(\mathbb R)$$ and $$\mathbb H$$ are isomorphic. One natural identification is

$$1 \mapsto 1\,, \quad \mathbf{i} \mapsto - \sigma_2 \sigma_3\,, \quad \mathbf{j} \mapsto - \sigma_3 \sigma_1\,, \quad \mathbf{k} \mapsto - \sigma_1 \sigma_2\,,$$

and it is straightforward to confirm that this preserves the Hamilton relations

$$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i \,j \,k} = -1~.$$

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions $M(4,R)$ and $a + bi + cj + dk$, there is only one bivector basis element $c = d = 0$, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements $b = d = 0$, $SU(2)$, $\{±1, ±i, ±j, ±k\}$, so three imaginaries.

This reasoning extends further. In the Clifford algebra $$\operatorname{Cl}_{4,0}(\mathbb R),$$ there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, called rotors, can be very useful for applications involving homogeneous coordinates. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a pseudovector.

There are several advantages for placing quaternions in this wider setting:
 * Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
 * In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
 * Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
 * In the conformal model of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
 * Rotor-encoded transformations make interpolation particularly straightforward.
 * Rotors carry over naturally to pseudo-Euclidean spaces, for example, the Minkowski space of special relativity. In such spaces rotors can be used to efficiently represent Lorentz boosts, and to interpret formulas involving the gamma matrices.

For further detail about the geometrical uses of Clifford algebras, see Geometric algebra.

Brauer group
The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers is Brauer equivalent to either the real numbers or the quaternions. Explicitly, the Brauer group of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.

CSAs – finite dimensional rings over a field, which are simple algebras (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of extension fields, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.

Quotations
"I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to $\{±I, $\pm i \sigma_3$,$\pm i \sigma_2$, $\pm i \sigma_1$\}$ etc."

"Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. ... And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be."

"Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell."

"There was a time, indeed, when I, although recognizing the appropriateness of vector analysis in electromagnetic theory (and in mathematical physics generally), did think it was harder to understand and to work than the Cartesian analysis. But that was before I had thrown off the quaternionic old-man-of-the-sea who fastened himself about my shoulders when reading the only accessible treatise on the subject – Prof. Tait's Quaternions. But I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work. There is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). The vector analysis I use may be described either as a convenient and systematic abbreviation of Cartesian analysis; or else, as Quaternions without the quaternions, .... 'Quaternion' was, I think, defined by an American schoolgirl to be 'an ancient religious ceremony'. This was, however, a complete mistake. The ancients – unlike Prof. Tait – knew not, and did not worship Quaternions."

"Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in everyday life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols."

"... quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist."

Books and publications

 * Hamilton, William Rowan (1853), "Lectures on Quaternions". Royal Irish Academy.
 * Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
 * Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co.
 * Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.] : The University Press.
 * Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". Clarendon Press, Oxford.
 * Tait, Peter Guthrie (1886), "". M.A. Sec. R.S.E. Encyclopædia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript file)
 * (See section on quaternions.)
 * Michael J. Crowe (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * Michael J. Crowe (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * (review).
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by
 * For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by

Links and monographs
two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
 * Notices and materials related to Quaternion conference presentations
 * Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
 * 3D Raytraced Quaternion Julia Fractals
 * Great page explaining basic math with links to straight forward rotation conversion formulae.
 * David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * 3D Raytraced Quaternion Julia Fractals
 * Great page explaining basic math with links to straight forward rotation conversion formulae.
 * David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
 * Quaternions the Android app shows the quaternion corresponding to the orientation of the device.
 * Rotating Objects Using Quaternions article speaking to the use of Quaternions for rotation in video games/computer graphics.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Quaternion


text:
Zero divisor

In abstract algebra, an element $a$ of a ring $R$ is called a left zero divisor if there exists a nonzero $x$ in $R$ such that $ax = 0$, or equivalently if the map from $R$ to $R$ that sends $x$ to $ax$ is not injective. Similarly, an element $ax = ay$ of a ring is called a right zero divisor if there exists a nonzero $x$ in $y$ such that $a(x − y) = 0$. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element $a$ that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero $y$ such that $R$ may be different from the nonzero $ya = 0$ such that $a$). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.

Examples
$$\begin{pmatrix}1&1\\2&2\end{pmatrix}\begin{pmatrix}1&1\\-1&-1\end{pmatrix}=\begin{pmatrix}-2&1\\-2&1\end{pmatrix}\begin{pmatrix}1&1\\2&2\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix} ,$$ $$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} =\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix} =\begin{pmatrix}0&0\\0&0\end{pmatrix}.$$
 * In the ring $\mathbb{Z}/4\mathbb{Z}$, the residue class $$\overline{2}$$ is a zero divisor since $$\overline{2} \times \overline{2}=\overline{4}=\overline{0}$$.
 * The only zero divisor of the ring $$\mathbb{Z}$$ of integers is $$0$$.
 * A nilpotent element of a nonzero ring is always a two-sided zero divisor.
 * An idempotent element $$e\ne 1$$ of a ring is always a two-sided zero divisor, since $$e(1-e)=0=(1-e)e$$.
 * The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:
 * A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $$R_1 \times R_2$$ with each $$R_i$$ nonzero, $$(1,0)(0,1) = (0,0)$$, so $$(1,0)$$ is a zero divisor.
 * Let $$K$$ be a field and $$G$$ be a group. Suppose that $$G$$ has an element $$g$$ of finite order $$n > 1$$. Then in the group ring $$K[G]$$ one has $$(1-g)(1+g+ \cdots +g^{n-1})=1-g^{n}=0$$, with neither factor being zero, so $$1-g$$ is a nonzero zero divisor in $$K[G]$$.

One-sided zero-divisor

 * Consider the ring of (formal) matrices $$\begin{pmatrix}x&y\\0&z\end{pmatrix}$$ with $$x,z\in\mathbb{Z}$$ and $$y\in\mathbb{Z}/2\mathbb{Z}$$. Then $$\begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}a&b\\0&c\end{pmatrix}=\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}$$ and $$\begin{pmatrix}a&b\\0&c\end{pmatrix}\begin{pmatrix}x&y\\0&z\end{pmatrix}=\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}$$. If $$x\ne0\ne z$$, then $$\begin{pmatrix}x&y\\0&z\end{pmatrix}$$ is a left zero divisor if and only if $$x$$ is even, since $$\begin{pmatrix}x&y\\0&z\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}=\begin{pmatrix}0&x\\0&0\end{pmatrix}$$, and it is a right zero divisor if and only if $$z$$ is even for similar reasons. If either of $$x,z$$ is $$0$$, then it is a two-sided zero-divisor.
 * Here is another example of a ring with an element that is a zero divisor on one side only. Let $$S$$ be the set of all sequences of integers $$(a_1,a_2,a_3,...)$$.  Take for the ring all additive maps from $$S$$ to $$S$$, with pointwise addition and composition as the ring operations. (That is, our ring is $$\mathrm{End}(S)$$, the endomorphism ring of the additive group $$S$$.) Three examples of elements of this ring are the right shift $$R(a_1,a_2,a_3,...)=(0,a_1,a_2,...)$$, the left shift $$L(a_1,a_2,a_3,...)=(a_2,a_3,a_4,...)$$, and the projection map onto the first factor $$P(a_1,a_2,a_3,...)=(a_1,0,0,...)$$.  All three of these additive maps are not zero, and the composites $$LP$$ and $$PR$$ are both zero, so $$L$$ is a left zero divisor and $$R$$ is a right zero divisor in the ring of additive maps from $$S$$ to $$S$$.  However, $$L$$ is not a right zero divisor and $$R$$ is not a left zero divisor: the composite $$LR$$ is the identity. $$RL$$ is a two-sided zero-divisor since $$RLP=0=PRL$$, while $$LR=1$$ is not in any direction.

Non-examples

 * The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field.
 * More generally, a division ring has no nonzero zero divisors.
 * A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.

Properties

 * In the ring of $n$ × $n$ matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of $n$ × $n$ matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.
 * Left or right zero divisors can never be units, because if $x$ is invertible and $ax = 0$ for some nonzero $y$, then $ya = 0$, a contradiction.
 * An element is cancellable on the side on which it is regular. That is, if $a$ is a left regular, $ax = 0$ implies that $x$, and similarly for right regular.

Zero as a zero divisor
There is no need for a separate convention for the case $0 = a^{−1}0 = a^{−1}ax = x$, because the definition applies also in this case:
 * If $a$ is a ring other than the zero ring, then $ax = ay$ is a (two-sided) zero divisor, because any nonzero element $x$ satisfies $x = y$.
 * If $a = 0$ is the zero ring, in which $R$, then $0$ is not a zero divisor, because there is no nonzero element that when multiplied by $0x = 0 = x 0$ yields $R$.

Some references include or exclude $0 = 1$ as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:
 * In a commutative ring $0$, the set of non-zero-divisors is a multiplicative set in $R$. (This, in turn, is important for the definition of the total quotient ring.)  The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
 * In a commutative noetherian ring $0$, the set of zero divisors is the union of the associated prime ideals of $0$.

Zero divisor on a module
Let $R$ be a commutative ring, let $M$ be an $R$-module, and let $a$ be an element of $R$. One says that $a$ is $M$-regular if the "multiplication by $a$" map $$M \,\stackrel{a}\to\, M$$ is injective, and that $a$ is a zero divisor on $M$ otherwise. The set of $M$-regular elements is a multiplicative set in $R$.

Specializing the definitions of "$M$-regular" and "zero divisor on $M$" to the case $0$ recovers the definitions of "regular" and "zero divisor" given earlier in this article.

title: Zero divisor


text:
Zorn's lemma

Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.

The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.

Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two. An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.

Motivation
To prove the existence of a mathematical object that can be viewed as a maximal element in some partially ordered set in some way, one can try proving the existence of such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the conditions a situation needs to satisfy in order for such an argument to work and enables mathematicians to not have to repeat the transfinite induction argument by hand each time, but just check the conditions of Zorn's lemma.

"If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you."

- William Timothy Gowers

Statement of the lemma
Preliminary notions:
 * A set P equipped with a binary relation ≤ that is reflexive (x ≤ x for every x), antisymmetric (if both x ≤ y and y ≤ x hold, then x = y), and transitive (if x ≤ y and y ≤ z then x ≤ z) is said to be (partially) ordered by ≤. Given two elements x and y of P with x ≤ y, y is said to be greater than or equal to x. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation, that is, in a partially ordered set P with order relation ≤ there may be elements x and y with neither x ≤ y nor y ≤ x. An ordered set in which every pair of elements is comparable is called totally ordered.
 * Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
 * An element m of a partially ordered set P with order relation ≤ is maximal (with respect to ≤) if there is no other element of P greater than m, that is, if there is no s in P with s ≠ m and m ≤ s. Depending on the order relation, a partially ordered set may have any number of maximal elements. However, a totally ordered set can have at most one maximal element.
 * Given a subset S of a partially ordered set P, an element u of P is an upper bound of S if it is greater than or equal to every element of S. Here, S is not required to be a chain, and u is required to be comparable to every element of S but need not itself be an element of S.

Zorn's lemma can then be stated as:

Variants of this formulation are sometimes used, such as requiring that the set P and the chains be non-empty.

Although this formulation appears to be formally weaker (since it places on P the additional condition of being non-empty, but obtains the same conclusion about P), in fact the two formulations are equivalent. To verify this, suppose first that P satisfies the condition that every chain in P has an upper bound in P. Then the empty subset of P is a chain, as it satisfies the definition vacuously; so the hypothesis implies that this subset must have an upper bound in P, and this upper bound shows that P is in fact non-empty. Conversely, if P is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in P, then P also satisfies the condition that every chain has an upper bound, as an arbitrary element of P serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).

The difference may seem subtle, but in many proofs that invoke Zorn's lemma one takes unions of some sort to produce an upper bound, and so the case of the empty chain may be overlooked; that is, the verification that all chains have upper bounds may have to deal with empty and non-empty chains separately. So many authors prefer to verify the non-emptiness of the set P rather than deal with the empty chain in the general argument.

Every vector space has a basis
Zorn's lemma can be used to show that every vector space V has a basis.

If V = {0}, then the empty set is a basis for V. Now, suppose that V ≠ {0}. Let P be the set consisting of all linearly independent subsets of V. Since V is not the zero vector space, there exists a nonzero element v of V, so P contains the linearly independent subset {v}. Furthermore, P is partially ordered by set inclusion (see inclusion order). Finding a maximal linearly independent subset of V is the same as finding a maximal element in P.

To apply Zorn's lemma, take a chain T in P (that is, T is a subset of P that is totally ordered). If T is the empty set, then {v} is an upper bound for T in P. Suppose then that T is non-empty. We need to show that T has an upper bound, that is, there exists a linearly independent subset B of V containing all the members of T.

Take B to be the union of all the sets in T. We wish to show that B is an upper bound for T in P. To do this, it suffices to show that B is a linearly independent subset of V.

Suppose otherwise, that B is not linearly independent. Then there exists vectors v1, v2, ..., vk ∈ B and scalars a1, a2, ..., ak, not all zero, such that
 * $$a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \cdots + a_k\mathbf{v}_k = \mathbf{0}.$$

Since B is the union of all the sets in T, there are some sets S1, S2, ..., Sk ∈ T such that vi ∈ Si for every i = 1, 2, ..., k. As T is totally ordered, one of the sets S1, S2, ..., Sk must contain the others, so there is some set Si that contains all of v1, v2, ..., vk. This tells us there is a linearly dependent set of vectors in Si, contradicting that Si is linearly independent (because it is a member of P).

The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal linearly independent subset B of V.

Finally, we show that B is indeed a basis of V. It suffices to show that B is a spanning set of V. Suppose for the sake of contradiction that B is not spanning. Then there exists some v ∈ V not covered by the span of B. This says that B ∪ {v} is a linearly independent subset of V that is larger than B, contradicting the maximality of B. Therefore, B is a spanning set of V, and thus, a basis of V.

Every nontrivial ring with unity contains a maximal ideal
Zorn's lemma can be used to show that every nontrivial ring R with unity contains a maximal ideal.

Let P be the set consisting of all proper ideals in R (that is, all ideals in R except R itself). Since R is non-trivial, the set P contains the trivial ideal {0}. Furthermore, P is partially ordered by set inclusion. Finding a maximal ideal in R is the same as finding a maximal element in P.

To apply Zorn's lemma, take a chain T in P. If T is empty, then the trivial ideal {0} is an upper bound for T in P. Assume then that T is non-empty. It is necessary to show that T has an upper bound, that is, there exists an ideal I ⊆ R containing all the members of T but still smaller than R (otherwise it would not be a proper ideal, so it is not in P).

Take I to be the union of all the ideals in T. We wish to show that I is an upper bound for T in P. We will first show that I is an ideal of R. For I to be an ideal, it must satisfy three conditions:

 * 1) I is a nonempty subset of R,
 * 2) For every x, y ∈ I, the sum x + y is in I,
 * 3) For every r ∈ R and every x ∈ I, the product rx is in I.

#1 - I is a nonempty subset of R.

Because T contains at least one element, and that element contains at least 0, the union I contains at least 0 and is not empty. Every element of T is a subset of R, so the union I only consists of elements in R.

#2 - For every x, y ∈ I, the sum x + y is in I.

Suppose x and y are elements of I. Then there exist two ideals J, K ∈ T such that x is an element of J and y is an element of K. Since T is totally ordered, we know that J ⊆ K or K ⊆ J. Without loss of generality, assume the first case. Both x and y are members of the ideal K, therefore their sum x + y is a member of K, which shows that x + y is a member of I.

#3 - For every r ∈ R and every x ∈ I, the product rx is in I.

Suppose x is an element of I. Then there exists an ideal J ∈ T such that x is in J. If r ∈ R, then rx is an element of J and hence an element of I. Thus, I is an ideal in R.

Now, we show that I is a proper ideal. An ideal is equal to R if and only if it contains 1. (It is clear that if it is R then it contains 1; on the other hand, if it contains 1 and r is an arbitrary element of R, then r1 = r is an element of the ideal, and so the ideal is equal to R.) So, if I were equal to R, then it would contain 1, and that means one of the members of T would contain 1 and would thus be equal to R – but R is explicitly excluded from P.

The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in P, in other words a maximal ideal in R.

Proof sketch
A sketch of the proof of Zorn's lemma follows, assuming the axiom of choice. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and that for every element in P there is another element bigger than it. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually define the function b, we need to employ the axiom of choice (explicitly: let $$ B(T) = \{b \in P : \forall t \in T, b \ge t\} $$, that is, the set of upper bounds for T. The axiom of choice furnishes $$ b: b(T) \in B(T) $$).

Using the function b, we are going to define elements a0 < a1 < a2 < a3 < ... < aω < aω+1 <…, in P. This uncountable sequence is really long: the indices are not just the natural numbers, but all ordinals. In fact, the sequence is too long for the set P; there are too many ordinals (a proper class), more than there are elements in any set (in other words, given any set of ordinals, there exists a larger ordinal), and the set P will be exhausted before long and then we will run into the desired contradiction.

The ai are defined by transfinite recursion: we pick a0 in P arbitrary (this is possible, since P contains an upper bound for the empty set and is thus not empty) and for any other ordinal w we set aw = b({av : v < w}). Because the av are totally ordered, this is a well-founded definition.

The above proof can be formulated without explicitly referring to ordinals by considering the initial segments {av : v < w} as subsets of P. Such sets can be easily characterized as well-ordered chains S ⊆ P where each x ∈ S satisfies x = b({y ∈ S : y < x}). Contradiction is reached by noting that we can always find a "next" initial segment either by taking the union of all such S (corresponding to the limit ordinal case) or by appending b(S) to the "last" S (corresponding to the successor ordinal case).

This proof shows that actually a slightly stronger version of Zorn's lemma is true: $$

History
The Hausdorff maximal principle is an early statement similar to Zorn's lemma.

Kazimierz Kuratowski proved in 1922 a version of the lemma close to its modern formulation (it applies to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by Max Zorn in 1935, who proposed it as a new axiom of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.

The name "Zorn's lemma" appears to be due to John Tukey, who used it in his book Convergence and Uniformity in Topology in 1940. Bourbaki's Théorie des Ensembles of 1939 refers to a similar maximal principle as "le théorème de Zorn". The name "Kuratowski–Zorn lemma" prevails in Poland and Russia.

Equivalent forms of Zorn's lemma
Zorn's lemma is equivalent (in ZF) to three main results:
 * 1) Hausdorff maximal principle
 * 2) Axiom of choice
 * 3) Well-ordering theorem.

A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

Zorn's lemma is also equivalent to the strong completeness theorem of first-order logic.

Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
 * 1) Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
 * 2) Every vector space has a basis, a result from linear algebra (to which it is equivalent ). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis.
 * 3) Every commutative unital ring has a maximal ideal, a result from ring theory known as Krull's theorem, to which Zorn's lemma is equivalent
 * 4) Tychonoff's theorem in topology (to which it is also equivalent )
 * 5) Every proper filter is contained in an ultrafilter, a result that yields the completeness theorem of first-order logic

In this sense, we see how Zorn's lemma can be seen as a powerful tool, applicable to many areas of mathematics.

Analogs under weakenings of the axiom of choice
A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the axiom of dependent choice). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element.

More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities. In the limit where we allow arbitrarily large ordinals, we recover the proof of the full Zorn's lemma using the axiom of choice in the preceding section.

In popular culture
The 1970 film Zorns Lemma is named after the lemma.

The lemma was referenced on The Simpsons in the episode "Bart's New Friend".

title: Zorn's lemma