Quaternions are a mathematical entity invented by William Rowan Hamilton. This article is written exclusively in Hamilton's original notation using his original definitions of terms. It uses only primary and secondary sources written on or before 1901. Works on quaternions now span three centuries, and this artificial distinction is an agreed upon editorial device, used to organize material on the subject. Mathematically speaking, the quaternions discussed in this article are the same quaternions that are used in almost all modern applications.

Some other writers use different notations, define terms differently and also define a quaternion differently than Hamilton. The question as to whether the mathematical entity defined by Hamilton is the same as the mathematical entities that other writers have called quaternions, over the centuries, has been at times a hotly debated subject; however, comparison with other notations systems is a topic for other articles.

Short history

In classical quaternion notation, a unit of distance squared is equal to a negative scalar quantity. The Pythagorean theorem, where B = 3i and C = 4j are the sides of a right triangle and A is the hypotenuse would look like:

A^{2}=B^{2}+C^{2}\,

-25=-9-16\,

To put it in classical quaternion terminology: the square of every vector is a negative scalar.^[1] In 1835, years before he discovered quaternions, Hamilton wrote an article entitled "Algebra the Science of Pure Time", which expressed the view that time worked like a real number or scalar.^[2]

Hamilton's discovery of quaternions has sometimes been linked a 19th century version of the modern concept of spacetime.^[3] In the words of John Baez, Hamilton gave quaternions a "cosmic significance"^[4]. As time progressed, Hamilton devoted more and more of his efforts to pure mathematics, continuing his research into quaternions from the time of his discovery of them until his death in 1865.

Classical elements of a quaternion

Four Operations

Four operations are of fundamental importance in quaternion notation.^[5]

$+-\div \times$

In particular it is important to understand that there is a single operation of multiplication, a single operation of division, and a single operations of addition and subtraction. This single multiplication operator can operate on any of the types of mathematical entities. Likewise every kind of entity can be can be divided, added or subtracted from any other type of entity. Understanding the meaning of the subtraction symbol is critical in quaternion theory, because it leads to an understanding of the concept of a vector.

Ordinal operators

The two ordinal operations in classical quaternion notation were addition and subtraction or + and -.

These signs are called addition and subtraction.

Subtraction is a type of analysis called ordinal analysis^[6]

Cardinal operations

The two Cardinal operations^[7] in quaternion notation are geometric multiplication and geometric division and can be written:

$\div \,\times$

It is not required to learn the following more advanced terms in order to use division and multiplication.

Division is a kind of analysis called cardinal analysis.^[8]Multiplication is a kind of synthesis called cardinal synthesis^[9]

Vector

One approach to defining a vector is to view it as the subtraction of two different coordinates.^[10]The subtraction symbol, when applied between two different coordinates results in a new type of mathematical entity called a vector which represents the act of moving from one point to another. For example if one point is the earth, and the other point is the sun, subtracting the position of the sun to the earth results in a vector, which represents going from the earth to the sun.^[11] It can be represented as an arrow drawn from the earth to the sun. It has both magnitude and direction.

Hamilton introduced the world to the concept of a vector in the 1840s. In his first lecture article 15, Hamilton introduces the word vector,^[12] from the Latin vection, or to move.

Tensor

The word tensor^[13] is defined as a positive, or more properly signless, number. ^[14]^[15]^[16] It can represent a "stretching factor"^[17]

A tensor is not a negative quantity, which can be thought of as an abridged notation for zero minus a quantity; for example^[18]

$0-5=-5\,$

Scalar

Hamilton invented the term scalars for the real numbers, because they span the "scale of progression from positive to negative infinity".^[19] The scalar of a quaternion q is its first component, or real part, denoted Sq.

Versor

The versor of a quaternion which can be written as

$\mathbf {U} q$

is another special type of quaternion with useful properties.^[20]

The tensor of a versor is always equal to one.

$\mathbf {TU} q=1$

In general a Versor can be associated with a plane, an axis and an angle.^[21]

A versor can also in general be represented by a unique great circle arc.^[22] This arc is greater than zero and less than 180 degrees. This is because the shortest distance between any two points of a sphere has a maximum limit of an arc corresponding to 180 degrees.

When a versor and a vector which lies in the plane of the versor are multiplied the result is a new vector of the same length but turned by the angle of the versor.

Right versor

When the arc of a versor has the magnitude of a right angle, then it is called a right versor, a right radial or quadrantal versor.

Like all quaternions a versor can be decomposed into the product of its tensor and its versor.

The versor of a versor is the same as the versor:

$\mathbf {UU} q=\mathbf {U} q$

Like other quaternions, a versor consists of the sum of a scalar and a vector.

Radial Quotient

The ratio of two vectors of equal length is called a radial quotient or a radial.^[23] A versor may also be viewed as the quotient of two vectors which are equal in length. In this case the arc can be visualized as the arc connecting the two vectors when they are placed tail to tail. In this representation the plan of the versor is the plane of the two vectors and the axis of the versor is a unit vector perpendicular to the plane. If the two vectors in the quotient are at right angles to each other then the quaternion is called a right radial quotient.

Degenerate forms

The scalar number One was sometimes called the nonversor^[24]^[25] and the scalar minus one sometimes called the inversor.^[26] These two scalars are special limiting cases, corresponding to versors with angle approaching either zero or π.

Zero and π are then two special scalar points of singularity.

The nonversor and the inversor, when multiplied with vectors, either have no effect or reverse the direction of the vector.^[27]

Unlike other versors these two cannot be represented by a unique arc. The arc of one is a single point, and minus one can be represented by an infinite number of arcs, because there are an infinite number of shortest lines between antipodal points of a sphere.

Quadrantal versor

A quadrantal versor has the effect of rotating a vector perpendicular to it by 90 degrees. Hence i × j = k. Here i represents an operator on j rotating it by 90 degrees.^[28] Using i as an operator again, i × k = −j. Classical notation viewed this as i operating on k to produce another rotation of 90 degrees. Note the logical consistency here; if it was true that i × (i × j) = −k then it should also be true that (i × i) × j = −k and so i × i must equal minus one.^[29]

In multiplication Minus one was called an inversor, having the effect on any vector of reversing it by 180 degrees to point in the opposite direction. Classical reasoning was that two successive rotations of 90 degrees in the same plane should produce the same effect as one rotation of 180 degrees. Quadrantal versors were therefore called semi-inversors. Quadrantal versors have a zero scalar component since the scalar component of a versor is the cosine of the angle of the versor.

Quaternion

Hamilton defined a quaternion as the quotient of two directed lines in tridimensional space^[30]; or, more simply, as the quotient of two vectors.^[31] A quaternion which could be represented as the sum of a vector and a scalar.

A quaternion could be decomposed into a scalar and a vector, or into a tensor and a versor.

Every quaternion can be decomposed into a scalar and a vector.

$q=\mathbf {S} (q)+\mathbf {V} (q)$

These two operations S and V are called "take the Scalar of" and "take the vector of" a quaternion. The vector part of a quaternion is also called the right part.^[32].

In abridged notation, parentheses are not required and were not normally used. In the above expression Vq and Sq could be written without ambiguity. The operation of taking the vector of a quaternion took priority over the operation of raising to a power, unless a dot was placed between the operation and the rest of the expression, as in the relations below.

$\mathbf {V} q^{2}=(\mathbf {V} q)^{2}$

$\mathbf {V} .q^{2}=\mathbf {V} (q^{2})\,$ ^[33]

The operations "take the tensor of" and "take the versor of" could then decompose the vector of a quaternion V(q) further into a tensor and a unit vector. Like all vectors, this unit vector has the property that its square equals the scalar minus one.

The first of these operations would be written s=T(v). The second operation, taking the versor of the vector, returns a unit vector u=U(v). A unit vector is also a special type of versor with an angle of 90 degrees; hence a unit vector can rightfully be called a special type of versor called a right versor.

$\mathbf {V} q=(\mathbf {T} .\mathbf {V} q)(\mathbf {U} .\mathbf {V} q)\,$

Right Quaternion

A right quaternion is quaternion whose scalar component is zero, $S(q)=0$ . The angle of a right quaternion is 90 degrees.

Right quaternions may be put in what was called the standard trinomial form. For example, if Q is a right quaternion, it may be written as:

Q=xi+yj+zk\,

^[34]

The study of this important subclass of quaternions, called right quaternions, is essentially modern vector analysis. It can be proven that every vector function is a function of a right quaternion.

Product of two Right Quaternion

The product of two Right Quaternions is generally a quaternion. Two very useful operations in Hamilton's calculus were taking the scalar and the vector of the product of two Right Quaternions.

Let α and β be the right quaternions that result from taking the vectors of two quaternions:

\alpha =\mathbf {V} p

\beta =\mathbf {V} q

Their product in general is then a new quaternion represented here by r. This product is not ambiguous because classical notation has only one product.

r=\,\alpha \beta ;

Like all quaternions r may now naturally be decomposed into its vector and scalar parts.

r=\mathbf {S} r+\mathbf {V} r

The terms on the right are called scalar of the product, and the vector of the product^[35] of two right quaternions.

In hermaphroditical^[36] three dimensional notation systems featuring more than one product, these two characteristics are often given their own symbols. It should be noted that, in some notation systems, operations related to the scalar of the product differ in sign from that of the classical operation.

Operators in detail

Multiplication

Classical quaternion notation system had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation.

Multiplication of a scalar and the vector of a quaternion was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.

Division

Classical quaternion notation had an operation called division. In fact most classical books on quaternions first introduce the quaternion as the ratio of two vectors. This was sometimes called a Geometric Fraction.

If OA and OB represent two vectors drawn from the origin O to two other points A and B, then the geometric fraction was written as

$OA:OB\,$

Alternately if the two vectors are represented by α and β the quotient was written as

$\alpha \div \beta$ or ${\frac {\alpha }{\beta }}$

Hamilton spends a great deal of time on the development of the concept of a vector and is already 110 pages into Elements of Quaternions before he even introduces the word quaternion. At the end of article 112 Hamilton reaches the important conclusion he has been working up to: "The quotient of two vectors is generally a quaternion".^[37]

Lectures on Quaternions also first introduces the concept of a quaternion as the quotient of two vectors, if

$q={\alpha }\times {\beta }.$

Logically and by way of definition^[38]

${q}\times {\beta }=\alpha .$ .

when

${\frac {\alpha }{\beta }}=q$

Notice that the order of the variables is of great importance. If the order of q and β were to be reversed the result would not in general be α. This is because the product in Hamilton's calculus is not commutative. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, what they used to call an act of version and then changing the length of it, which is what used to be call an act of tension. Also by definition the quotient of two vectors is equal to the numerator times the reciprocal of the denominator. Since multiplication of vectors is not commutative, the order can not be changed in the following expression.

${\frac {\alpha }{\beta }}=\,{\alpha }\times {\frac {1}{\beta }}$

Again the order of the two quantities on the right hand side of the equation is an important part of the classical definition of division.

Hardy^[39] presents the definition of division in terms of pneumonic cancellation rules. "Canceling being performed by an upward right hand stroke".

Other important operations

Taking the scalar and vector of a quaternion

Two important operations in two the classical quaternion notation system were S(q) and V(q) which meant take the scalar part of, and take the imaginary part, what Hamilton called the vector part of the quaternion. Here S and V are operators acting on q. Parenthesis can be omitted in these kinds of expressions without ambiguity.

In the classical era this is what the notation looked like:

$q=\,\mathbf {S} q+\mathbf {V} q$

Here, q is a quaternion. Sq is the scalar of the quaternion while Vq is the vector of the quaternion.

Taking the tensor and versor of a quaternion

Another important pair of classical quaternion operations were deconstructing a quaternion into a tensor and versor:^[40]^[41]

q=\mathbf {T} q\cdot \mathbf {U} q\,

The formula for the tensor of a quaternion is:^[42]

\mathbf {T} q={\sqrt {w^{2}+x^{2}+y^{2}+z^{2}}}\,

Another way to obtain the tensor of a quaternion is from the common norm, defined as the product of a quaternion and its conjugate. The square root of the common norm of a quaternion is equal to its tensor.

\mathbf {T} q={\sqrt {qKq}}\,

Taking the conjugate

The K(q) operator means, take the conjugate. The conjugate of a quaternion is another quaternion obtained by multiplying the vector part of the first quaternion by minus one.

If

q=\,\mathbf {S} q+\mathbf {V} q

then

\mathbf {K} q=\mathbf {S} \,q-\mathbf {V} q

.

The expression

r=\,\mathbf {K} q

,

means, assign the quaternion r the value of the conjugate of the quaternion q.

Axis and angle of a quaternion

Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π.^[43]^[44]

When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector pointing perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule.^[45] The angle is the angle between the two vectors.

In symbols

$u=Ax.q\,$

$\theta =\angle q$

Reciprocal of a quaternion

if

q={\frac {\alpha }{\beta }}

Then its reciprocal is defined as

${\frac {1}{q}}=q^{-1}={\frac {\beta }{\alpha }}$

The expression:

${q}\times {\alpha }\times {\frac {1}{q}}$

Has many important applications^[46]^[47] for example rotations, particularly when q is the special type of quaternion called a versor. A versor has an easy formula for its reciprocal.^[48]

${\frac {1}{(\mathbf {U} q)}}=\mathbf {S.U} q-\mathbf {V.U} q=\mathbf {K.U} q$

In words this says that the reciprocal of a versor is equal to its conjugate. The dots between operators show the order to take the operations in, and also help to indicate that S and U for example are two different operations rather than a single operation named SU.

Common norm

The product of a quaternion with its conjugate was called the common norm.^[49]

The operation of taking the common norm of a quaternion is represented with the letter N. By definition the common norm is the product of a quaternion with its conjugate. It can be proven^[50] that common norm is equal to the square of the tensor of a quaternion. However this proof does not constitute a definition. Hamilton gives an exact definition both the common norm and the tensor, which do not depend on each other. This norm was adopted as suggested from the theory of numbers however to quote Hamilton "they will not often be wanted". The tensor is generally of greater utility. The word norm does not appear at all in Lectures on Quaternions, and only appears twice in the table of contents of Elements of Quaternions.

In symbols:

$\mathbf {N} q=\,q\mathbf {K} q=\,(\mathbf {T} q)^{2}$

The common norm of a versor is always equal to positive unity.^[51]

$\mathbf {NU} q=\mathbf {U} q.\mathbf {KU} q=1\,$

Cardinal Operations in Detail

Division in Detail

Division of the Unit Vectors i,j,k

The results of the using the division operator on i,j and k was as follows. ^[52]

{\frac {k}{j}}=i

{\frac {i}{k}}=j

{\frac {j}{i}}=k

{\frac {-k}{i}}=j

{\frac {-i}{j}}=k

{\frac {-j}{k}}=i

{\frac {-k}{-j}}=i

{\frac {j}{-k}}=i

{\frac {-j}{-i}}=k

{\frac {i}{-j}}=k

{\frac {-i}{-k}}=j

{\frac {k}{-i}}=j

Division of two parallel Vectors

While in general the quotient of two vectors is a quaternion, If α and β are two parallel vectors then the quotient of these two vectors is a scalar. For example if

$\alpha =ai\,$ ,

and $\beta =bi\,$ then

\alpha \div \beta ={\frac {\alpha }{\beta }}={\frac {ai}{bi}}={\frac {a}{b}}

Where a/b is a scalar.^[53]

Division of two non-parallel Vectors

The quotient of two vectors is in general the quaternion:

$q={\frac {\alpha }{\beta }}$ $={\frac {T\alpha }{T\beta }}(\cos \phi +\epsilon \sin \phi )$

Where α and β are two non-parallel vectors, φ is that angle between them, and e is a unit vector perpendicular to the plane of the vectors α and β, with its direction given by the standard right hand rule.^[54]

Multiplication in Detail

Distributive

In the classical notation system, the operation of multiplication was distributive. Understanding this makes it simple to see why the product of two vectors in classical notation produced a quaternion.

$q=(ai+bj+ck)\times (ei+fj+gk)\,$

$q=ae({i}\times {i})+af({i}\times {j})+ag({i}\times {k})+be({j}\times {i})+bf({j}\times {j})+bg({j}\times {k})+ce({k}\times {i})+cf({k}\times {j})+cg({k}\times {k})$

Using the quaternion multiplication table we have:

$q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)\,$

Then collecting terms:

$q=-ae-bf-cg+(bg-cf)i+(ce-ag)j+(af-be)k\,$

The first three terms are a scalar.

Letting

$w=-ae-bf-cg\,$

$x=(bg-cf)\,$

$y=(ce-ag)\,$

$z=(af-be)\,$

So that the product of two vectors is a quaternion, and can be written in the form:

$q=w+xi+yj+zk\,$

Geometrically Real and Geometrically Imaginary Numbers

In classical quaternion literature the equation

 $q^{2}=-1\,$

was thought to have infinitely many solutions that were called geometrically real. These solutions are the unit vectors that form the surface of a unit sphere.

The term Geometrically Real roots of the above equation refers to quantities that can be written as a linear combination of the i,j and k, with the condition the sum of the squares of the coefficients of the expression add up to one. Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real ones. Given the existence of the imaginary scalar a number of expressions can be written and given proper names. All of these were part of Hamilton's original quaternion calculus.

$q+q'{\sqrt {-1}}$

where q and q' were real quaternions, and the square root of minus one was understood to be the imaginary of ordinary algebra, and called an imaginary or symbolical roots^[55] and not a geometrically real vector quantity.

Imaginary Scalar

Geometrically Imaginary quantities are additional roots of the above equation of a purely symbolic nature. In article 214 or elements Hamilton proves that if there is an i j and k there also has to be another quantity h which is an imaginary scalar, which he observes should have already occured to anyone who hard read the preceding articles with attention.^[56]Article 149 of elements of quaternions is an important article about Geometrically Imaginary numbers and includes a footnote introducing the term biquaterion.^[57] The term imaginary of ordinary algebra and scalar imaginary are sometimes used to refer to these geometrically imaginary quantities.

Geometrically Imaginary' roots to an equation were interpreted in classical thinking as geometrically impossible situations. Article 214 of elements of quaternions explores the example of the equation of a line and a circle that do not intersect, as being indicated by the equation having only a geometrically imaginary root.^[58]

In Hamilton's later writings he proposed using the letter h do denote the imaginary scalar^[59]^[60]^[61]

Bi Scalar

Bi Vector

Article 214 also defines a Bivector as the product of a vector and and the imaginary of ordnary algebra,

Biquaternion

A Biquaternion is by definition the quotient of a bivector and a vector. It can also be written in this same form.

Other double quaternions

Hamilton invented the term associative to distinguish between the both commutative and associative imaginary scalar, and four other possible roots of negative unity. These he suggested should be given the designations L M N and O, and they are discussed very briefly in appendix B of Lectures on Quaternions, and in private letters however non-associative roots of minus one do not appear in Elements of Quaternions. Hamilton's life ended before he ever had a chance to work on these strange entities, they are a bow for another Ulysses^[62]

Footnotes

^ Hamilton (1853), Lecture 3 Article 85 pg 81
^ Hamilton (1853), Page 2 of preface, paragraph 3
^ See relevant section on history of space time
^ AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 2, Pages 145–205 S 0273-0979(01)00934-X Article electronically published on December 21, 2001
^ Hamilton 1853 pg 4
^ Hamilton pg 33
^ Hamilton (1853), pg 35, Hamilton first introduces cardinal operations
^ Hamilton 1953 pg.36 Division defined as cardinal analysis
^ Hamilton 1853 pg 37
^ Hamilton 1953 pg 15 vector defined as a past analysis and a future synthesis
^ Hamilton 1853 pg 15 example of traveling from earth to moon given
^ Hamilton 1853 pg.15 page scan where introduces the word "vector"
^ Cayley (1890), pg 146
^ Hamilton 1853 pg 57
^ Hardy 1881 pg 5
^ Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number
^ Tait (1890), pg 32
^ Hamilton 1853 pg 51
^ Hamilton, in the Philosophical magazine, as cited in the OED.
^ Hamilton (1899), art 156 pg 135, introduction of term versor
^ Hamilton (1899), Section 8 article 151 pg 133
^ Hardy (1881), art. 49 pg 71-72
^ Hamilton (1899), Article 146 pg 129
^ Hamilton (1853), page 240
^ Hamilton 1853 Pg 54-55 explains non-version
^ Hamilton (1899), art. 153 pg 54
^ See Hamilton (1853), Art. 60 starting on page 53
^ Hardy (1881), page 41
^ Hamilton (1853), pg 70
^ Hamilton 1853 pg. 60
^ Hardy 1881 pg. 32
^ See Elements of Quaternions Section 13 starting on page 190
^ See Elements of Quaternions bottom of page 195
^ Hamilton (1899), Section 14 article 221 on page 233
^ Hardy (1887) scalar of the product vector of the product defined, pg 57
^ An infamous remark by Tait
^ Hamilton (1899), Article 112 page 110
^ Hardy (1881), pg 32
^ Hardy (1881), pg 46
^ Hardy (1881) pg 34
^ Hamilton (1853), art 90 pg 87-89
^ Hardy (1881), pg 65
^ Hamilton (1853), pg 164, art 148
^ Hamilton (1899), pg 118
^ Hamilton (1899), pg 117
^ See Goldstein (1980) Chapter 7 for the same function written in matrix notation
^ Lorentz Transforms Hamilton (1853), pg 268 1853
^ Hardy (1881), pg 71
^ Hamilton (1899), pg 128 -129
^ See foot note at bottom of page, were word proven is highlighted.
^ Hamilton 1899 pg 138
^ Hardy (1881), page 40-41
^ Elements of Quaternions, book one.
^ Hardy (1881), pg 39 article 25
^ See Elements of Quaternions Articles 256 and 257
^ Hamilton Elements article 214 infamous remark...as would already have occured to anyone who had read the preceding articles with attention
^ Elements of Quaternions Article 49
^ See elements of quaternions article 214
^ Hamilton Elements of Quatenions pg 276 Example of h notation for imaginary scalar
^ Hamilton Elements Article 274 pg 300 Example of use of h notation
^ Hamilton Elements article 274 pg. 300 Example of h denoting imaginary of ordinary algebra
^ Bow for another Ulysses is a famous remark made by Hamilton's son, in the preface of Elements 1866

+=References=+

W.R. Hamilton (1853), Lectures on Quaternions, Dublin: Hodges and Smith
W.R. Hamilton (1899), Elements of Quaternions, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.
A.S. Hardy (1887), Elements of Quaternions
P.G. Tait (1890), An Elementary Treatise on Quaternions, Cambridge: C.J. Clay and Sons
Herbert Goldstein(1980), Classical Mechanics, 2nd edition, Library of congress catalog number QA805.G6 1980