5682:. By definition the common norm is the product of a quaternion with its conjugate. It can be proven that common norm is equal to the square of the tensor of a quaternion. However this proof does not constitute a definition. Hamilton gives exact, independent definitions of both the common norm and the tensor. 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
3628:
5912:
add up to one. Hamilton demonstrated that there had to be additional roots of this equation in addition to the geometrically real roots. 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
657:
In particular it is important to understand that there is a single operation of multiplication, a single operation of division, and a single operation each of addition and subtraction. This single multiplication operator can operate on any of the types of mathematical entities. Likewise every kind of
695:
of that progression. ...I am led to regard the word "Minus," or the mark −, in geometry, as the sign or characteristic of analysis of one geometric position (in space), as compared with another (such) position. The comparison of one mathematical point with another with a view to the determination of
256:
it seems convenient to enlarge by definition the signification of the new word tensor, so as to render it capable of including also those other cases in which we operate on a line by diminishing instead of increasing its length; and generally by altering that length in any definite ratio. We shall
5331:
When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector perpendicular to the plane of the two vectors in this original quotient, in a direction specified by the right hand rule. The angle is the angle between the two vectors.
278:
Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is the ratio of the lengths of these vectors.
4448:(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. Classical notation:
5137:
It can also be proven from this definition that another formula 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.
964:, i.e., 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. The quaternion q can be thought of as an operator that changes β into α, by first rotating it, formerly an act of
362:
Unlike other versors, these two cannot be represented by a unique arc. The arc of 1 is a single point, and –1 can be represented by an infinite number of arcs, because there are an infinite number of shortest lines between antipodal points of a sphere.
3361:
3236:
Multiplication of a scalar and a vector 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.
5666:
In words the reciprocal of a versor is equal to its conjugate. The dots between operators show the order of the operations, and also help to indicate that S and U for example, are two different operations rather than a single operation named SU.
5983:
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 occurred to anyone who had read the preceding articles with attention. Article 149 of
148:
Hamilton conceived a vector as the "difference of its two extreme points." For
Hamilton, a vector was always a three-dimensional entity, having three co-ordinates relative to any given co-ordinate system, including but not limited to both
5661:
3102:
2482:
Because a unit vector and its reciprocal are parallel to each other but point in opposite directions, the product of a unit vector and its reciprocal have a special case commutative property, for example if a is any unit vector then:
121:, because they span the "scale of progression from positive to negative infinity" or because they represent the "comparison of positions upon one common scale". Hamilton regarded ordinary scalar algebra as the science of pure time.
1384:
732:
A vector plus a scalar is always a quaternion even if the scalar is zero. If the scalar added to the vector is zero then the new quaternion produced is called a right quaternion. It has an angle characteristic of 90 degrees.
1456:
658:
entity 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.
3232:
Classical quaternion notation 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.
1194:
5041:
3843:
1041:
3218:
2782:
5266:
198:
or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.
5763:
5512:
5132:
4901:
3223:
Where α and β are two non-parallel vectors, φ is that angle between them, and ε is a unit vector perpendicular to the plane of the vectors α and β, with its direction given by the standard right hand rule.
2577:
171:, then extended this to division by an integer, and multiplication (and division) of a vector by a rational number. Finally, by taking limits, he defined the result of multiplying a vector α by any scalar
5828:
700:
The first example of subtraction is to take the point A to represent the earth, and the point B to represent the sun, then an arrow drawn from A to B represents the act of moving or vection from A to B.
4624:
5560:
1278:
3980:
2951:
426:
672:"...characteristics of synthesis and analysis of a state of progression, according as this state is considered as being derived from, or compared with, some other state of that progression."
4570:
4492:
7065:
244:
as a positive numerical quantity, or, more properly, signless number. A tensor can be thought of as a positive scalar. The "tensor" can be thought of as representing a "stretching factor."
4405:
3353:
1316:
5311:
4983:
536:
5447:
5180:
3149:
1086:
955:
914:
587:
The angle of a right quaternion is 90 degrees. So a right quaternion has only a vector part and no scalar part. Right quaternions may be put in standard trinomial form. For example, if
865:
2850:
2475:
7122:
4941:
5955:
4284:
4318:
3623:{\displaystyle 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})}
1128:
4774:
4357:
2333:
2165:
1997:
832:
4731:
4663:
2413:
2245:
2077:
1909:
1835:
1761:
5403:
4827:
2650:
1687:
1621:
1555:
2617:
491:
460:
4239:
4039:
6063:) which is both commutative and associative, and four other possible roots of negative unity which he designated L, M, N and O, mentioning them briefly in appendix B of
2992:
639:
4946:
A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these two vectors. In symbols:
4180:
4133:
4086:
3024:
5880:
2289:
2121:
1953:
2963:
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
2374:
2206:
2038:
1216:
582:
303:
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.
5371:
1870:
1796:
1722:
1236:
800:
4906:
The tensor of a unit vector is one. Since the versor of a vector is a unit vector, the tensor of the versor of any vector is always equal to unity. Symbolically:
1653:
1587:
1521:
431:
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.
6010:
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.
5575:
3032:
5185:
A useful identity is that the square of the tensor of a quaternion is equal to the tensor of the square of a quaternion, so that the parentheses may be omitted.
48:, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more
7441:
273:; because, in the operation here considered, we abstract from the directions (as well as from the situations) of the lines which are compared or operated on.
3254:
When two quantities are multiplied the first quantity is called the factor, the second quantity is called the faciend and the result is called the factum.
2584:
However, in the more general case involving more than one vector (whether or not it is a unit vector) the commutative property does not hold. For example:
1327:
6090:
1399:
1135:
4523:
is the conjugate operator. The conjugate of a quaternion is a quaternion obtained by multiplying the vector part of the first quaternion by minus one.
4991:
3639:
997:
4323:
Their product in general is a new quaternion represented here by r. This product is not ambiguous because classical notation has only one product.
3153:
2665:
5191:
1049:
Hardy presents the definition of division in terms of mnemonic cancellation rules. "Canceling being performed by an upward right hand stroke".
355:
There are two special degenerate versor cases, called the unit-scalars. These two scalars (negative and positive unity) can be thought of as
7239:
Hamilton
Elements article 214 infamous remark...as would already have occurred to anyone who had read the preceding articles with attention
5703:
6071:. Hamilton died before he worked on these strange entities. His son claimed them to be "bows reserved for the hands of another Ulysses".
5461:
5056:
4838:
7354:
2491:
713:
this represents the first example in
Hamilton's lectures of a vector. In this case the act of traveling from the earth to the sun.
5774:
767:
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
5566:
4581:
546:
A real multiple of a right versor is a right quaternion, thus a right quaternion is a quaternion whose scalar component is zero,
293:
A versor is a quaternion with a tensor of 1. Alternatively, a versor can be defined as the quotient of two equal-length vectors.
6080:
323:, connecting these two points, drawn from the divisor or lower part of quotient, to the dividend or upper part of the quotient.
7446:
7436:
5522:
729:
Vectors and scalars can be added. When a vector is added to a scalar, a completely different entity, a quaternion is created.
1245:
258:
3854:
6386:
Hamilton 1898 section 8 pg 133 art 151 On the versor of a quaternion or a vector and some general formula of transformation
741:
The two
Cardinal operations in quaternion notation are geometric multiplication and geometric division and can be written:
56:
properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.
2866:
434:
Every quaternion is equal to a versor multiplied by the tensor of the quaternion. Denoting the versor of a quaternion by
3266:. Understanding this makes it simple to see why the product of two vectors in classical notation produced a quaternion.
377:
251:
in his first book, Lectures on
Quaternions, based on lectures he gave shortly after his invention of the quaternions:
4532:
4454:
135:
Hamilton defined a vector as "a right line ... having not only length but also direction". Hamilton derived the word
6178:
Hamilton 1853, pg 2 paragraph 3 of introduction. Refers to his early article "Algebra as the
Science of pure time".
6006:
roots to an equation were interpreted in classical thinking as geometrically impossible situations. Article 214 of
4368:
3272:
1283:
5277:
4952:
502:
5419:
5144:
3121:
1058:
971:
921:
886:
991:. Since multiplication of vectors is not commutative, the order cannot be changed in the following expression.
843:
229:
69:
2798:
2428:
4912:
150:
4258:
764:
Classically, the quaternion was viewed as the ratio of two vectors, sometimes called a geometric fraction.
749:
It is not required to learn the following more advanced terms in order to use division and multiplication.
7340:
5453:
4292:
1094:
984:
965:
359:. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π.
41:
31:
6437:
Hamilton 1898 section 9 art 162 pg 142 Vector Arcs considered as representative of versors of quaternions
4739:
4329:
3263:
2295:
2127:
1959:
811:
4699:
4636:
2380:
2212:
2044:
1876:
1802:
1728:
5919:
5379:
4785:
2622:
1659:
1593:
1527:
666:
The two ordinal operations in classical quaternion notation were addition and subtraction or + and −.
5979:
quantities are additional roots of the above equation of a purely symbolic nature. In article 214 of
2592:
315:, and since a versor can be thought of as the quotient of two vectors, a versor has a representative
108:
5328:
Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π.
471:
440:
233:
195:
4191:
6040:
5317:
3994:
17:
7344:
2968:
597:
4141:
4094:
4047:
3000:
691:...let space be now regarded as the field of progression which is to be studied, and POINTS as
7350:
5849:
2253:
2085:
1917:
1389:
An important way to think of q is as an operator that changes β into α, by first rotating it (
297:
4252:
Let α and β be the right quaternions that result from taking the vectors of two quaternions:
2341:
2173:
2005:
1201:
552:
6044:
5656:{\displaystyle {\frac {1}{(\mathbf {U} q)}}=\mathbf {S.U} q-\mathbf {V.U} q=\mathbf {K.U} q}
5341:
3097:{\displaystyle \alpha \div \beta ={\frac {\alpha }{\beta }}={\frac {ai}{bi}}={\frac {a}{b}}}
685:
161:
7188:
See
Hamilton 1898 pg. 169 art. 190 for proof of relationship between tensor and common norm
6013:
In
Hamilton's later writings he proposed using the letter h to denote the imaginary scalar
1843:
1769:
1695:
1221:
773:
1629:
1563:
1497:
756:
called cardinal analysis. Multiplication is a kind of synthesis called cardinal synthesis
7324:
Hamilton
Elements article 274 pg. 300 Example of h denoting imaginary of ordinary algebra
6607:
Hamilton 1853 pg 15 introduction of the term vector as the difference between two points.
167:
By adding a vector to itself multiple times, he defined multiplication of a vector by an
6060:
6032:
5961:
5678:
The operation of taking the common norm of a quaternion is represented with the letter
5047:
4690:
4689:
The tensor of a positive scalar is that scalar. The tensor of a negative scalar is the
4423:
164:
of the second vector at the end of the first. He went on to define vector subtraction.
5988:
is about Geometrically Imaginary numbers and includes a footnote introducing the term
805:
Alternately if the two vectors are represented by α and β the quotient was written as
7430:
6335:
Tait 1890 pg.31 explains Hamilton's older definition of a tensor as a positive number
6067:
and in private letters. However, non-associative roots of minus one do not appear in
5569:, particularly when q is a versor. A versor has an easy formula for its reciprocal.
1379:{\displaystyle \gamma =(\gamma \div \beta )\times (\beta \div \alpha )\times \alpha }
7049:
Hamilton 1898 pg 167 art. 187 equation 12 Tensors of conjugate quaternions are equal
4185:
So that the product of two vectors is a quaternion, and can be written in the form:
257:
thus (as was hinted at the end of the article in question) have fractional and even
7400:
7383:
7328:
7311:
7294:
7277:
7260:
7243:
7226:
7209:
7192:
7175:
7158:
7141:
7101:
7084:
7053:
7036:
7019:
7002:
6985:
6968:
6951:
6934:
6917:
6900:
6883:
6866:
6849:
6832:
6827:
6815:
6798:
6781:
6764:
6747:
6730:
6713:
6696:
6679:
6662:
6645:
6628:
6611:
6594:
6577:
6560:
6543:
6526:
6509:
6492:
6475:
6458:
6441:
6424:
6407:
6390:
6373:
6356:
6339:
6322:
6305:
6288:
6271:
6254:
6236:
6219:
6202:
6182:
6165:
6131:
6114:
6022:
4779:
The tensor of a vector is by definition the length of the vector. For example, if:
1451:{\displaystyle \gamma \div \alpha =(\gamma \div \beta )\times (\beta \div \alpha )}
316:
130:
118:
7290:
Hamilton Elements of Quaternions pg 276 Example of h notation for imaginary scalar
6035:
coefficients. The scalar part of a biquaternion is then a complex number called a
1189:{\displaystyle {\frac {\alpha }{\beta }}.\beta =\alpha \beta ^{-1}.\beta =\alpha }
877:
also first introduces the concept of a quaternion as the quotient of two vectors:
7394:
7391:
7377:
7374:
7323:
7306:
7289:
7272:
7255:
7238:
7221:
7204:
7187:
7170:
7153:
7136:
7096:
7079:
7048:
7031:
7014:
6997:
6980:
6946:
6929:
6912:
6895:
6878:
6861:
6844:
6810:
6793:
6776:
6759:
6742:
6725:
6708:
6691:
6674:
6657:
6640:
6623:
6606:
6589:
6572:
6555:
6538:
6521:
6504:
6487:
6436:
6419:
6402:
6385:
6351:
6334:
6317:
6300:
6283:
6266:
6249:
6231:
6214:
6197:
6177:
6160:
6126:
6109:
696:
what may be called their ordinal relation, or their relative position in space...
5909:
5036:{\displaystyle \mathbf {T} q={\frac {\mathbf {T} \alpha }{\mathbf {T} \beta }}.}
3838:{\displaystyle q=ae(-1)+af(+k)+ag(-j)+be(-k)+bf(-1)+bg(+i)+ce(+j)+cf(-i)+cg(-1)}
988:
961:
332:
312:
296:
In general a versor defines all of the following: a directional axis; the plane
208:
154:
7112:
See Goldstein (1980) Chapter 7 for the same function written in matrix notation
5889:. These solutions are the unit vectors that form the surface of a unit sphere.
4669:
means, assign the quaternion r the value of the conjugate of the quaternion q.
4362:
Like all quaternions r may now be decomposed into its vector and scalar parts.
1036:{\displaystyle {\frac {\alpha }{\beta }}=\,{\alpha }\times {\frac {1}{\beta }}}
6215:
Hamilton (1853) Lecture I Article 15, introduction of term vector, from vehere
4440:
Two important operations in two the classical quaternion notation system were
1046:
Again the order of the two quantities on the right hand side is significant.
45:
35:
7032:
Hamilton 1898 pg 169 art 190 Tensor of the square is the square of the tensor
6470:
5960:
where q and q′ are real quaternions, and the square root of minus one is the
3213:{\displaystyle ={\frac {T\alpha }{T\beta }}(\cos \phi +\epsilon \sin \phi )}
980:
7403:, 2nd edition, edited by Charles Jasper Joly, Longmans Green & Company.
6963:
6453:
2777:{\displaystyle {\frac {k}{i}}=k{\frac {1}{i}}=ki^{-1}=k(-i)=-(ki)=-(j)=-j}
6368:
6085:
5261:{\displaystyle (\mathbf {T} q)^{2}=\mathbf {T} (q^{2})=\mathbf {T} q^{2}}
753:
681:
211:
is a vector of length one. Examples of unit vectors include i, j and k.
191:
65:
49:
6624:
Hamilton 1853 pg.19 Hamilton associates plus sign with ordinal synthesis
160:
Hamilton defined addition of vectors in geometric terms, by placing the
140:
6043:
consisting of three complex components. The biquaternions are then the
168:
53:
649:
Four operations are of fundamental importance in quaternion notation.
6641:
Hamilton (1853), pg 35, Hamilton first introduces cardinal operations
288:
261:
tensors, which will simply be numerical multipliers, and will all be
7349:. London, New York, and Bombay: Longmans, Green, and Co. p. v.
5758:{\displaystyle \mathbf {N} q=\,q\mathbf {K} q=\,(\mathbf {T} q)^{2}}
5675:
The product of a quaternion with its conjugate is its common norm.
224:
by Hamilton does not coincide with modern terminology. Hamilton's
7307:
Hamilton Elements Article 274 pg 300 Example of use of h notation
7171:
See foot note at bottom of page, were word proven is highlighted.
6964:(1887) scalar of the product vector of the product defined, pg 57
6000:
are sometimes used for these geometrically imaginary quantities.
5896:
quaternion is one that can be written as a linear combination of
4249:
The product of two right quaternions is generally a quaternion.
7423:, 2nd edition, Library of congress catalog number QA805.G6 1980
6059:
to distinguish between the imaginary scalar (known by now as a
5885:
was thought to have infinitely many solutions that were called
5768:
The common norm of a versor is always equal to positive unity.
5507:{\displaystyle {\frac {1}{q}}=q^{-1}={\frac {\beta }{\alpha }}}
5127:{\displaystyle \mathbf {T} q={\sqrt {w^{2}+x^{2}+y^{2}+z^{2}}}}
4896:{\displaystyle \mathbf {T} \alpha ={\sqrt {x^{2}+y^{2}+z^{2}}}}
4426:
of two vectors up to the change of sign (multiplication to −1).
979:
Also by definition the quotient of two vectors is equal to the
371:
Every quaternion can be decomposed into a scalar and a vector.
6981:
Hamilton 1898 pg164 Tensor of the versor of a vector is unity.
6352:
Hamilton 1989 pg 165, refers to a tensor as a positive scalar.
6232:
Hamilton (1853) Lecture I Article 17 vector is natural triplet
4693:
of the scalar (i.e., without the negative sign). For example:
2572:{\displaystyle {\frac {1}{a}}a=(-a)a=1=a(-a)=a{\frac {1}{a}}.}
1052:
If alpha and beta are vectors and q is a quaternion such that
970:
and then changing the length of it, formerly called an act of
5823:{\displaystyle \mathbf {NU} q=\mathbf {U} q.\mathbf {KU} q=1}
4680:
is the tensor operator. It returns a kind of number called a
6403:
Hamilton (1899), art 156 pg 135, introduction of term versor
6488:
See Elements of Quaternions Section 13 starting on page 190
4619:{\displaystyle \mathbf {K} q=\mathbf {S} \,q-\mathbf {V} q}
271:
unclothed with the algebraic signs of positive and negative
72:
space; or, more generally, as the quotient of two vectors.
5565:
Reciprocals have many important applications, for example
3115:
The quotient of two vectors is in general the quaternion:
311:
Since every unit vector can be thought of as a point on a
6658:
Hamilton 1953 pg.36 Division defined as cardinal analysis
721:
Addition is a type of analysis called ordinal synthesis.
157:
systems. He therefore referred to vectors as "triplets".
6031:
Hamilton defines a biquaternion to be a quaternion with
5555:{\displaystyle {q}\times {\alpha }\times {\frac {1}{q}}}
2420:
The reciprocal of a unit vector is the vector reversed.
872:"The quotient of two vectors is generally a quaternion".
52:
than the modern approach, which emphasizes quaternions'
6930:
Hamilton 1853 pg. 27 explains Factor Faciend and Factum
4422:
Note: "Scalar of the product" corresponds to Euclidean
1273:{\displaystyle \beta \div \alpha \times \alpha =\beta }
5839:
Geometrically real and geometrically imaginary numbers
5271:
Also, the tensors of conjugate quaternions are equal.
3975:{\displaystyle q=-ae-bf-cg+(bg-cf)i+(ce-ag)j+(af-be)k}
5922:
5852:
5777:
5706:
5578:
5525:
5464:
5422:
5382:
5344:
5280:
5194:
5147:
5059:
4994:
4955:
4915:
4841:
4788:
4742:
4702:
4639:
4584:
4535:
4457:
4371:
4332:
4295:
4261:
4194:
4144:
4097:
4050:
3997:
3857:
3642:
3364:
3275:
3156:
3124:
3035:
3003:
2971:
2869:
2801:
2668:
2625:
2595:
2494:
2431:
2383:
2344:
2298:
2256:
2215:
2176:
2130:
2088:
2047:
2008:
1962:
1920:
1879:
1846:
1805:
1772:
1731:
1698:
1662:
1632:
1596:
1566:
1530:
1500:
1402:
1330:
1286:
1248:
1224:
1204:
1138:
1097:
1061:
1000:
924:
889:
846:
814:
776:
600:
555:
505:
474:
443:
380:
6505:
Hamilton (1899), Section 14 article 221 on page 233
5046:From this definition it can be shown that a useful
3633:Using the quaternion multiplication table we have:
2946:{\displaystyle {\frac {k}{i}}i=(-j)i=-(ji)=-(-k)=k}
87:. It can also be represented as the product of its
34:. For a more general treatment of quaternions, see
5949:
5874:
5822:
5757:
5655:
5554:
5506:
5441:
5397:
5365:
5305:
5260:
5174:
5126:
5035:
4977:
4935:
4895:
4821:
4768:
4725:
4657:
4618:
4564:
4486:
4399:
4351:
4312:
4278:
4233:
4174:
4127:
4080:
4033:
3974:
3837:
3622:
3347:
3212:
3143:
3096:
3018:
2986:
2945:
2844:
2776:
2644:
2611:
2571:
2469:
2407:
2368:
2327:
2283:
2239:
2200:
2159:
2115:
2071:
2032:
1991:
1947:
1903:
1864:
1829:
1790:
1755:
1716:
1681:
1647:
1615:
1581:
1549:
1515:
1450:
1378:
1310:
1272:
1230:
1210:
1188:
1122:
1080:
1035:
949:
908:
859:
826:
794:
633:
576:
530:
485:
454:
420:
7123:"Lorentz Transforms Hamilton (1853), pg 268 1853"
421:{\displaystyle q=\mathbf {S} (q)+\mathbf {V} (q)}
7222:See Elements of Quaternions Articles 256 and 257
5843:In classical quaternion literature the equation
331:When the arc of a versor has the magnitude of a
75:A quaternion can be represented as the sum of a
4565:{\displaystyle q=\,\mathbf {S} q+\mathbf {V} q}
4487:{\displaystyle q=\,\mathbf {S} q+\mathbf {V} q}
6471:Elements of Quaternions Article 147 pg 130 130
5968:and not a geometrically real vector quantity.
1477:The results of using the division operator on
175:as a vector β with the same direction as α if
6420:Hamilton (1899), Section 8 article 151 pg 133
5690:, and only twice in the table of contents of
5316:The tensor of a quaternion is now called its
4400:{\displaystyle r=\mathbf {S} r+\mathbf {V} r}
3348:{\displaystyle q=(ai+bj+ck)\times (ei+fj+gk)}
2657:This is because k/i is carefully defined as:
1311:{\displaystyle q\times \alpha \div \alpha =q}
591:is a right quaternion, it may be written as:
8:
6284:Hamilton (1866) Book I Chapter II Article 19
6161:Hamilton (1866) Book I Chapter II Article 17
5306:{\displaystyle \mathbf {TK} q=\mathbf {T} q}
4978:{\displaystyle q={\frac {\alpha }{\beta }}.}
531:{\displaystyle q=\mathbf {T} q\mathbf {U} q}
232:on the quaternion algebra, which makes it a
179:is positive; the opposite direction to α if
27:Hamilton's original treatment of quaternions
6267:Hamilton (1866) Book I Chapter I Article 15
5442:{\displaystyle q={\frac {\alpha }{\beta }}}
5175:{\displaystyle \mathbf {T} q={\sqrt {qKq}}}
3144:{\displaystyle q={\frac {\alpha }{\beta }}}
1081:{\displaystyle {\frac {\alpha }{\beta }}=q}
950:{\displaystyle {q}\times {\beta }=\alpha .}
909:{\displaystyle {\frac {\alpha }{\beta }}=q}
6250:Hamilton (1866) Book I Chapter I Article 6
6198:Hamilton (1866) Book I Chapter I Article 1
6192:
6190:
3262:In classical notation, multiplication was
1393:) and then changing its length (tension).
960:In Hamilton's calculus the product is not
6039:. The vector part of a biquaternion is a
5937:
5921:
5857:
5851:
5803:
5792:
5778:
5776:
5749:
5737:
5733:
5722:
5718:
5707:
5705:
5639:
5622:
5605:
5588:
5579:
5577:
5542:
5534:
5526:
5524:
5494:
5482:
5465:
5463:
5429:
5421:
5381:
5343:
5295:
5281:
5279:
5252:
5243:
5231:
5219:
5210:
5198:
5193:
5159:
5148:
5146:
5116:
5103:
5090:
5077:
5071:
5060:
5058:
5019:
5009:
5006:
4995:
4993:
4962:
4954:
4916:
4914:
4885:
4872:
4859:
4853:
4842:
4840:
4787:
4743:
4741:
4703:
4701:
4647:
4646:
4638:
4608:
4601:
4596:
4585:
4583:
4554:
4543:
4542:
4534:
4476:
4465:
4464:
4456:
4389:
4378:
4370:
4339:
4331:
4302:
4294:
4268:
4260:
4193:
4143:
4096:
4049:
3996:
3856:
3641:
3612:
3604:
3584:
3576:
3556:
3548:
3528:
3520:
3500:
3492:
3472:
3464:
3444:
3436:
3416:
3408:
3388:
3380:
3363:
3274:
3160:
3155:
3131:
3123:
3084:
3061:
3048:
3034:
3002:
2970:
2870:
2868:
2805:
2800:
2705:
2685:
2669:
2667:
2626:
2624:
2599:
2594:
2556:
2495:
2493:
2449:
2432:
2430:
2384:
2382:
2343:
2299:
2297:
2255:
2216:
2214:
2175:
2131:
2129:
2087:
2048:
2046:
2007:
1963:
1961:
1919:
1880:
1878:
1845:
1806:
1804:
1771:
1732:
1730:
1697:
1663:
1661:
1631:
1597:
1595:
1565:
1531:
1529:
1499:
1401:
1329:
1285:
1247:
1223:
1203:
1165:
1139:
1137:
1105:
1096:
1062:
1060:
1023:
1015:
1014:
1001:
999:
933:
925:
923:
890:
888:
860:{\displaystyle {\frac {\alpha }{\beta }}}
847:
845:
813:
775:
599:
554:
520:
512:
504:
475:
473:
444:
442:
404:
387:
379:
6794:Hamilton Lectures on quaternions page 41
6726:Hamilton Lectures on Quaternions page 37
2845:{\displaystyle i{\frac {k}{i}}=i(-j)=-k}
2470:{\displaystyle {\frac {1}{i}}=i^{-1}=-i}
300:to that axis; and an angle of rotation.
7273:See elements of quaternions article 214
6101:
6811:Hamilton Lectures on quaternions pg 42
6777:Hamilton Lectures On Quaternions pg 38
5048:formula for the tensor of a quaternion
4936:{\displaystyle \mathbf {TU} \alpha =1}
4508:is the scalar of the quaternion while
7414:An Elementary Treatise on Quaternions
6692:Hamilton (1899), Article 112 page 110
4279:{\displaystyle \alpha =\mathbf {V} p}
64:Hamilton defined a quaternion as the
7:
6047:of the original (real) quaternions.
4313:{\displaystyle \beta =\mathbf {V} q}
3985:The first three terms are a scalar.
3111:Division of two non-parallel vectors
1123:{\displaystyle \alpha \beta ^{-1}=q}
30:For the history of quaternions, see
7442:Historical treatment of quaternions
7256:Elements of Quaternions Article 149
4512:q is the vector of the quaternion.
183:is negative; and a length that is |
7066:"Hamilton (1853), pg 164, art 148"
6896:Elements of Quaternions, book one.
5389:
4769:{\displaystyle \mathbf {T} (-5)=5}
4410:The terms on the right are called
4352:{\displaystyle r=\,\alpha \beta ;}
2328:{\displaystyle {\frac {-i}{-k}}=j}
2160:{\displaystyle {\frac {-j}{-i}}=k}
1992:{\displaystyle {\frac {-k}{-j}}=i}
1238:are inverse operations, such that:
827:{\displaystyle \alpha \div \beta }
465:and the tensor of a quaternion by
60:Classical elements of a quaternion
25:
4726:{\displaystyle \mathbf {T} (5)=5}
4658:{\displaystyle r=\,\mathbf {K} q}
2408:{\displaystyle {\frac {k}{-i}}=j}
2240:{\displaystyle {\frac {i}{-j}}=k}
2072:{\displaystyle {\frac {j}{-k}}=i}
1904:{\displaystyle {\frac {-j}{k}}=i}
1830:{\displaystyle {\frac {-i}{j}}=k}
1756:{\displaystyle {\frac {-k}{i}}=j}
18:Classical hamiltonian quaternions
5950:{\displaystyle q+q'{\sqrt {-1}}}
5807:
5804:
5793:
5782:
5779:
5738:
5723:
5708:
5646:
5643:
5640:
5629:
5626:
5623:
5612:
5609:
5606:
5589:
5398:{\displaystyle \theta =\angle q}
5296:
5285:
5282:
5244:
5220:
5199:
5149:
5061:
5020:
5010:
4996:
4920:
4917:
4843:
4822:{\displaystyle \alpha =xi+yj+zk}
4744:
4704:
4648:
4609:
4597:
4586:
4555:
4544:
4477:
4466:
4390:
4379:
4303:
4269:
4245:Product of two right quaternions
2959:Division of two parallel vectors
2645:{\displaystyle {\frac {k}{i}}i.}
1682:{\displaystyle {\frac {j}{i}}=k}
1616:{\displaystyle {\frac {i}{k}}=j}
1550:{\displaystyle {\frac {k}{j}}=i}
521:
513:
476:
445:
405:
388:
7416:, Cambridge: C.J. Clay and Sons
6998:Elements of Quaternions, Ch. 11
5908:, such that the squares of the
2612:{\displaystyle i{\frac {k}{i}}}
725:Addition of vectors and scalars
6913:Hardy (1881), pg 39 article 25
5746:
5734:
5596:
5585:
5237:
5224:
5207:
5195:
4757:
4748:
4714:
4708:
4169:
4151:
4122:
4104:
4075:
4057:
3966:
3948:
3939:
3921:
3912:
3894:
3832:
3823:
3811:
3802:
3790:
3781:
3769:
3760:
3748:
3739:
3727:
3718:
3706:
3697:
3685:
3676:
3664:
3655:
3617:
3601:
3589:
3573:
3561:
3545:
3533:
3517:
3505:
3489:
3477:
3461:
3449:
3433:
3421:
3405:
3393:
3377:
3342:
3315:
3309:
3282:
3207:
3180:
2934:
2925:
2916:
2907:
2895:
2886:
2830:
2821:
2762:
2756:
2747:
2738:
2729:
2720:
2547:
2538:
2520:
2511:
2357:
2348:
2269:
2260:
2189:
2180:
2101:
2092:
2021:
2012:
1933:
1924:
1445:
1433:
1427:
1415:
1367:
1355:
1349:
1337:
565:
559:
415:
409:
398:
392:
1:
5994:imaginary of ordinary algebra
5966:imaginary or symbolical roots
5962:imaginary of ordinary algebra
1462:Division of the unit vectors
880:Logically and by definition,
486:{\displaystyle \mathbf {T} q}
455:{\displaystyle \mathbf {U} q}
247:Hamilton introduced the term
7154:Hamilton (1899), pg 128 -129
6760:Tait Treaties on Quaternions
4234:{\displaystyle q=w+xi+yj+zk}
265:or (to speak more properly)
68:of two directed lines in tri
6862:Hardy 1887 pg 45 formula 30
6845:Hardy 1887 pg 45 formula 29
6539:Hamilton 1853 art 5 pg 4 -5
6454:(1881), art. 49 pg 71-72 71
6081:Cayley–Dickson construction
6055:Hamilton invented the term
4681:
4034:{\displaystyle w=-ae-bf-cg}
113:Hamilton invented the term
94:
88:
82:
76:
7463:
6020:
4418:of two right quaternions.
3241:Factor, Faciend and Factum
2987:{\displaystyle \alpha =ai}
634:{\displaystyle Q=xi+yj+zk}
286:
220:Note: The use of the word
128:
106:
29:
7419:Herbert Goldstein(1980),
6947:Hamilton 1898 section 103
6590:see Hamilton 1853 pg 8-15
4431:Other operators in detail
4175:{\displaystyle z=(af-be)}
4128:{\displaystyle y=(ce-ag)}
4081:{\displaystyle x=(bg-cf)}
3248:Factor × Faciend = Factum
3019:{\displaystyle \beta =bi}
680:Subtraction is a type of
187:| times the length of α.
7386:Dublin: Hodges and Smith
6828:Hardy (1881), page 40-41
6051:Other double quaternions
5875:{\displaystyle q^{2}=-1}
2284:{\displaystyle j(-k)=-i}
2116:{\displaystyle k(-i)=-j}
1948:{\displaystyle i(-j)=-k}
7408:Elements of Quaternions
7395:Elements of Quaternions
7378:Lectures on Quaternions
7346:Elements of Quaternions
7341:Hamilton, William Rowan
7097:Hamilton (1899), pg 118
7080:Hamilton (1899), pg 118
6743:Elements of quaternions
6069:Elements of Quaternions
6065:Lectures on Quaternions
6029:Elements of Quaternions
6008:Elements of Quaternions
6004:Geometrically Imaginary
5977:Geometrically Imaginary
5692:Elements of Quaternions
5688:Lectures on Quaternions
3848:Then collecting terms:
3107:Where a/b is a scalar.
2369:{\displaystyle j(-i)=k}
2201:{\displaystyle k(-j)=i}
2033:{\displaystyle i(-k)=j}
1211:{\displaystyle \times }
875:Lectures on Quaternions
577:{\displaystyle S(q)=0.}
7447:William Rowan Hamilton
7437:History of mathematics
7389:W.R. Hamilton (1866),
7372:W.R. Hamilton (1853),
6144:Philosophical magazine
5951:
5913:calculus. In symbols:
5876:
5824:
5759:
5657:
5556:
5508:
5443:
5399:
5367:
5366:{\displaystyle u=Ax.q}
5307:
5262:
5176:
5128:
5037:
4979:
4937:
4897:
4823:
4770:
4727:
4659:
4620:
4566:
4488:
4401:
4353:
4314:
4280:
4235:
4176:
4129:
4082:
4035:
3976:
3839:
3624:
3349:
3214:
3145:
3098:
3020:
2988:
2947:
2846:
2778:
2646:
2613:
2573:
2471:
2409:
2370:
2329:
2285:
2241:
2202:
2161:
2117:
2073:
2034:
1993:
1949:
1905:
1866:
1831:
1792:
1757:
1718:
1683:
1649:
1617:
1583:
1551:
1517:
1452:
1380:
1312:
1274:
1232:
1212:
1190:
1124:
1082:
1037:
951:
910:
861:
828:
796:
752:Division is a kind of
698:
635:
578:
532:
487:
456:
422:
335:, then it is called a
42:William Rowan Hamilton
32:history of quaternions
5952:
5877:
5825:
5760:
5658:
5557:
5509:
5444:
5400:
5368:
5308:
5263:
5177:
5129:
5038:
4980:
4938:
4898:
4824:
4771:
4728:
4660:
4621:
4567:
4489:
4416:vector of the product
4412:scalar of the product
4402:
4354:
4315:
4281:
4236:
4177:
4130:
4083:
4036:
3977:
3840:
3625:
3350:
3215:
3146:
3099:
3021:
2989:
2948:
2847:
2779:
2647:
2614:
2574:
2472:
2410:
2371:
2330:
2286:
2242:
2203:
2162:
2118:
2074:
2035:
1994:
1950:
1906:
1867:
1865:{\displaystyle ik=-j}
1832:
1793:
1791:{\displaystyle kj=-i}
1758:
1719:
1717:{\displaystyle ji=-k}
1684:
1650:
1618:
1584:
1552:
1518:
1453:
1381:
1313:
1275:
1233:
1231:{\displaystyle \div }
1213:
1191:
1125:
1083:
1038:
952:
911:
862:
829:
797:
795:{\displaystyle OA:OB}
689:
636:
579:
533:
488:
457:
423:
7205:Hamilton 1899 pg 138
6573:Hamilton 1853 pg 5-6
6110:Hamilton 1853 pg. 60
5964:, and are called an
5920:
5850:
5775:
5704:
5576:
5523:
5462:
5420:
5380:
5342:
5278:
5192:
5145:
5057:
4992:
4953:
4913:
4839:
4786:
4740:
4700:
4637:
4582:
4533:
4455:
4369:
4330:
4293:
4259:
4192:
4142:
4095:
4048:
3995:
3855:
3640:
3362:
3273:
3154:
3122:
3033:
3001:
2969:
2867:
2799:
2666:
2623:
2593:
2492:
2429:
2381:
2342:
2296:
2254:
2213:
2174:
2128:
2086:
2045:
2006:
1960:
1918:
1877:
1844:
1803:
1770:
1729:
1696:
1660:
1648:{\displaystyle ki=j}
1630:
1594:
1582:{\displaystyle jk=i}
1564:
1528:
1516:{\displaystyle ij=k}
1498:
1400:
1328:
1284:
1246:
1222:
1202:
1136:
1095:
1059:
998:
922:
887:
844:
812:
774:
598:
553:
503:
472:
441:
378:
109:Scalar (mathematics)
7421:Classical Mechanics
7406:A.S. Hardy (1887),
7137:Hardy (1881), pg 71
7015:Hardy (1881), pg 65
6709:Hardy (1881), pg 32
6675:Hamilton 1853 pg 37
6301:Hamilton 1853 pg 57
5686:does not appear in
737:Cardinal operations
234:normed vector space
7412:P.G. Tait (1890),
6522:Hamilton 1853 pg 4
6146:, as cited in the
5947:
5894:geometrically real
5887:geometrically real
5872:
5820:
5755:
5653:
5552:
5504:
5439:
5395:
5363:
5303:
5258:
5172:
5124:
5033:
4975:
4933:
4893:
4819:
4766:
4723:
4655:
4616:
4562:
4484:
4397:
4349:
4310:
4276:
4231:
4172:
4125:
4078:
4031:
3972:
3835:
3620:
3345:
3210:
3141:
3094:
3016:
2984:
2943:
2842:
2774:
2642:
2609:
2569:
2467:
2405:
2366:
2325:
2281:
2237:
2198:
2157:
2113:
2069:
2030:
1989:
1945:
1901:
1862:
1827:
1788:
1753:
1714:
1679:
1645:
1613:
1579:
1547:
1513:
1448:
1376:
1308:
1270:
1228:
1208:
1186:
1120:
1078:
1033:
947:
906:
870:Hamilton asserts:
857:
824:
792:
631:
574:
528:
483:
452:
418:
357:scalar quaternions
6142:Hamilton, in the
6127:Hardy 1881 pg. 32
6091:Frobenius theorem
5945:
5600:
5550:
5502:
5473:
5437:
5170:
5122:
5028:
4970:
4891:
4501:is a quaternion.
4436:Scalar and vector
3178:
3139:
3092:
3079:
3056:
2878:
2813:
2693:
2677:
2634:
2607:
2564:
2503:
2440:
2418:
2417:
2397:
2317:
2229:
2149:
2061:
1981:
1893:
1819:
1745:
1671:
1605:
1539:
1147:
1070:
1031:
1009:
898:
855:
669:These marks are:
662:Ordinal operators
345:quadrantal versor
240:Hamilton defined
16:(Redirected from
7454:
7361:
7360:
7337:
7331:
7320:
7314:
7303:
7297:
7286:
7280:
7269:
7263:
7252:
7246:
7235:
7229:
7218:
7212:
7201:
7195:
7184:
7178:
7167:
7161:
7150:
7144:
7133:
7127:
7126:
7119:
7113:
7110:
7104:
7093:
7087:
7076:
7070:
7069:
7062:
7056:
7045:
7039:
7028:
7022:
7011:
7005:
6994:
6988:
6977:
6971:
6960:
6954:
6943:
6937:
6926:
6920:
6909:
6903:
6892:
6886:
6879:Hardy 1887 pg 46
6875:
6869:
6858:
6852:
6841:
6835:
6824:
6818:
6807:
6801:
6790:
6784:
6773:
6767:
6756:
6750:
6739:
6733:
6722:
6716:
6705:
6699:
6688:
6682:
6671:
6665:
6654:
6648:
6637:
6631:
6620:
6614:
6603:
6597:
6586:
6580:
6569:
6563:
6552:
6546:
6535:
6529:
6518:
6512:
6501:
6495:
6484:
6478:
6467:
6461:
6450:
6444:
6433:
6427:
6416:
6410:
6399:
6393:
6382:
6376:
6369:(1890), pg 32 31
6365:
6359:
6348:
6342:
6331:
6325:
6314:
6308:
6297:
6291:
6280:
6274:
6263:
6257:
6245:
6239:
6228:
6222:
6211:
6205:
6194:
6185:
6174:
6168:
6157:
6151:
6140:
6134:
6123:
6117:
6106:
6045:complexification
5998:scalar imaginary
5972:Imaginary scalar
5956:
5954:
5953:
5948:
5946:
5938:
5936:
5881:
5879:
5878:
5873:
5862:
5861:
5829:
5827:
5826:
5821:
5810:
5796:
5785:
5764:
5762:
5761:
5756:
5754:
5753:
5741:
5726:
5711:
5662:
5660:
5659:
5654:
5649:
5632:
5615:
5601:
5599:
5592:
5580:
5561:
5559:
5558:
5553:
5551:
5543:
5538:
5530:
5516:The expression:
5513:
5511:
5510:
5505:
5503:
5495:
5490:
5489:
5474:
5466:
5448:
5446:
5445:
5440:
5438:
5430:
5404:
5402:
5401:
5396:
5372:
5370:
5369:
5364:
5312:
5310:
5309:
5304:
5299:
5288:
5267:
5265:
5264:
5259:
5257:
5256:
5247:
5236:
5235:
5223:
5215:
5214:
5202:
5181:
5179:
5178:
5173:
5171:
5160:
5152:
5133:
5131:
5130:
5125:
5123:
5121:
5120:
5108:
5107:
5095:
5094:
5082:
5081:
5072:
5064:
5042:
5040:
5039:
5034:
5029:
5027:
5023:
5017:
5013:
5007:
4999:
4984:
4982:
4981:
4976:
4971:
4963:
4942:
4940:
4939:
4934:
4923:
4902:
4900:
4899:
4894:
4892:
4890:
4889:
4877:
4876:
4864:
4863:
4854:
4846:
4828:
4826:
4825:
4820:
4775:
4773:
4772:
4767:
4747:
4732:
4730:
4729:
4724:
4707:
4664:
4662:
4661:
4656:
4651:
4625:
4623:
4622:
4617:
4612:
4600:
4589:
4571:
4569:
4568:
4563:
4558:
4547:
4493:
4491:
4490:
4485:
4480:
4469:
4406:
4404:
4403:
4398:
4393:
4382:
4358:
4356:
4355:
4350:
4319:
4317:
4316:
4311:
4306:
4285:
4283:
4282:
4277:
4272:
4240:
4238:
4237:
4232:
4181:
4179:
4178:
4173:
4134:
4132:
4131:
4126:
4087:
4085:
4084:
4079:
4040:
4038:
4037:
4032:
3981:
3979:
3978:
3973:
3844:
3842:
3841:
3836:
3629:
3627:
3626:
3621:
3616:
3608:
3588:
3580:
3560:
3552:
3532:
3524:
3504:
3496:
3476:
3468:
3448:
3440:
3420:
3412:
3392:
3384:
3354:
3352:
3351:
3346:
3219:
3217:
3216:
3211:
3179:
3177:
3169:
3161:
3150:
3148:
3147:
3142:
3140:
3132:
3103:
3101:
3100:
3095:
3093:
3085:
3080:
3078:
3070:
3062:
3057:
3049:
3025:
3023:
3022:
3017:
2993:
2991:
2990:
2985:
2952:
2950:
2949:
2944:
2879:
2871:
2851:
2849:
2848:
2843:
2814:
2806:
2783:
2781:
2780:
2775:
2713:
2712:
2694:
2686:
2678:
2670:
2651:
2649:
2648:
2643:
2635:
2627:
2618:
2616:
2615:
2610:
2608:
2600:
2578:
2576:
2575:
2570:
2565:
2557:
2504:
2496:
2476:
2474:
2473:
2468:
2457:
2456:
2441:
2433:
2414:
2412:
2411:
2406:
2398:
2396:
2385:
2375:
2373:
2372:
2367:
2334:
2332:
2331:
2326:
2318:
2316:
2308:
2300:
2290:
2288:
2287:
2282:
2246:
2244:
2243:
2238:
2230:
2228:
2217:
2207:
2205:
2204:
2199:
2166:
2164:
2163:
2158:
2150:
2148:
2140:
2132:
2122:
2120:
2119:
2114:
2078:
2076:
2075:
2070:
2062:
2060:
2049:
2039:
2037:
2036:
2031:
1998:
1996:
1995:
1990:
1982:
1980:
1972:
1964:
1954:
1952:
1951:
1946:
1910:
1908:
1907:
1902:
1894:
1889:
1881:
1871:
1869:
1868:
1863:
1836:
1834:
1833:
1828:
1820:
1815:
1807:
1797:
1795:
1794:
1789:
1762:
1760:
1759:
1754:
1746:
1741:
1733:
1723:
1721:
1720:
1715:
1688:
1686:
1685:
1680:
1672:
1664:
1654:
1652:
1651:
1646:
1622:
1620:
1619:
1614:
1606:
1598:
1588:
1586:
1585:
1580:
1556:
1554:
1553:
1548:
1540:
1532:
1522:
1520:
1519:
1514:
1492:
1491:
1489:was as follows.
1457:
1455:
1454:
1449:
1385:
1383:
1382:
1377:
1317:
1315:
1314:
1309:
1279:
1277:
1276:
1271:
1237:
1235:
1234:
1229:
1217:
1215:
1214:
1209:
1195:
1193:
1192:
1187:
1173:
1172:
1148:
1140:
1129:
1127:
1126:
1121:
1113:
1112:
1087:
1085:
1084:
1079:
1071:
1063:
1042:
1040:
1039:
1034:
1032:
1024:
1019:
1010:
1002:
956:
954:
953:
948:
937:
929:
915:
913:
912:
907:
899:
891:
866:
864:
863:
858:
856:
848:
833:
831:
830:
825:
801:
799:
798:
793:
686:ordinal analysis
640:
638:
637:
632:
583:
581:
580:
575:
542:Right quaternion
537:
535:
534:
529:
524:
516:
492:
490:
489:
484:
479:
461:
459:
458:
453:
448:
427:
425:
424:
419:
408:
391:
351:Degenerate forms
267:SignLess Numbers
228:is actually the
21:
7462:
7461:
7457:
7456:
7455:
7453:
7452:
7451:
7427:
7426:
7369:
7364:
7357:
7339:
7338:
7334:
7321:
7317:
7304:
7300:
7287:
7283:
7270:
7266:
7253:
7249:
7236:
7232:
7219:
7215:
7202:
7198:
7185:
7181:
7168:
7164:
7151:
7147:
7134:
7130:
7121:
7120:
7116:
7111:
7107:
7094:
7090:
7077:
7073:
7064:
7063:
7059:
7046:
7042:
7029:
7025:
7012:
7008:
6995:
6991:
6978:
6974:
6961:
6957:
6944:
6940:
6927:
6923:
6910:
6906:
6893:
6889:
6876:
6872:
6859:
6855:
6842:
6838:
6825:
6821:
6808:
6804:
6791:
6787:
6774:
6770:
6757:
6753:
6740:
6736:
6723:
6719:
6706:
6702:
6689:
6685:
6672:
6668:
6655:
6651:
6638:
6634:
6621:
6617:
6604:
6600:
6587:
6583:
6570:
6566:
6553:
6549:
6536:
6532:
6519:
6515:
6502:
6498:
6485:
6481:
6468:
6464:
6451:
6447:
6434:
6430:
6417:
6413:
6400:
6396:
6383:
6379:
6366:
6362:
6349:
6345:
6332:
6328:
6318:Hardy 1881 pg 5
6315:
6311:
6298:
6294:
6281:
6277:
6264:
6260:
6246:
6242:
6229:
6225:
6212:
6208:
6195:
6188:
6175:
6171:
6158:
6154:
6141:
6137:
6124:
6120:
6107:
6103:
6099:
6077:
6053:
6027:On page 665 of
6025:
6019:
5974:
5929:
5918:
5917:
5853:
5848:
5847:
5841:
5836:
5773:
5772:
5745:
5702:
5701:
5673:
5584:
5574:
5573:
5521:
5520:
5478:
5460:
5459:
5418:
5417:
5411:
5378:
5377:
5340:
5339:
5326:
5276:
5275:
5248:
5227:
5206:
5190:
5189:
5143:
5142:
5112:
5099:
5086:
5073:
5055:
5054:
5018:
5008:
4990:
4989:
4951:
4950:
4911:
4910:
4881:
4868:
4855:
4837:
4836:
4784:
4783:
4738:
4737:
4698:
4697:
4675:
4635:
4634:
4630:The expression
4580:
4579:
4531:
4530:
4518:
4453:
4452:
4438:
4433:
4367:
4366:
4328:
4327:
4291:
4290:
4257:
4256:
4247:
4190:
4189:
4140:
4139:
4093:
4092:
4046:
4045:
3993:
3992:
3853:
3852:
3638:
3637:
3360:
3359:
3271:
3270:
3260:
3243:
3230:
3170:
3162:
3152:
3151:
3120:
3119:
3113:
3071:
3063:
3031:
3030:
2999:
2998:
2967:
2966:
2961:
2865:
2864:
2797:
2796:
2701:
2664:
2663:
2621:
2620:
2591:
2590:
2490:
2489:
2445:
2427:
2426:
2389:
2379:
2378:
2340:
2339:
2309:
2301:
2294:
2293:
2252:
2251:
2221:
2211:
2210:
2172:
2171:
2141:
2133:
2126:
2125:
2084:
2083:
2053:
2043:
2042:
2004:
2003:
1973:
1965:
1958:
1957:
1916:
1915:
1882:
1875:
1874:
1842:
1841:
1808:
1801:
1800:
1768:
1767:
1734:
1727:
1726:
1694:
1693:
1658:
1657:
1628:
1627:
1592:
1591:
1562:
1561:
1526:
1525:
1496:
1495:
1475:
1398:
1397:
1326:
1325:
1282:
1281:
1244:
1243:
1220:
1219:
1200:
1199:
1161:
1134:
1133:
1101:
1093:
1092:
1057:
1056:
996:
995:
920:
919:
885:
884:
842:
841:
810:
809:
772:
771:
762:
739:
727:
719:
678:
664:
647:
645:Four operations
596:
595:
551:
550:
544:
501:
500:
470:
469:
439:
438:
376:
375:
369:
353:
329:
309:
291:
285:
259:incommensurable
217:
205:
139:from the Latin
133:
127:
111:
105:
62:
39:
28:
23:
22:
15:
12:
11:
5:
7460:
7458:
7450:
7449:
7444:
7439:
7429:
7428:
7425:
7424:
7417:
7410:
7404:
7387:
7368:
7365:
7363:
7362:
7355:
7332:
7315:
7298:
7281:
7264:
7247:
7230:
7213:
7196:
7179:
7162:
7145:
7128:
7114:
7105:
7088:
7071:
7057:
7040:
7023:
7006:
6989:
6972:
6955:
6938:
6921:
6904:
6887:
6870:
6853:
6836:
6819:
6802:
6785:
6768:
6751:
6734:
6717:
6700:
6683:
6666:
6649:
6632:
6615:
6598:
6581:
6564:
6556:Hamilton pg 33
6547:
6530:
6513:
6496:
6479:
6462:
6445:
6428:
6411:
6394:
6377:
6360:
6343:
6326:
6309:
6292:
6275:
6258:
6240:
6223:
6206:
6186:
6169:
6152:
6135:
6118:
6100:
6098:
6095:
6094:
6093:
6088:
6083:
6076:
6073:
6061:complex number
6052:
6049:
6033:complex number
6021:Main article:
6018:
6015:
5973:
5970:
5958:
5957:
5944:
5941:
5935:
5932:
5928:
5925:
5883:
5882:
5871:
5868:
5865:
5860:
5856:
5840:
5837:
5835:
5832:
5831:
5830:
5819:
5816:
5813:
5809:
5806:
5802:
5799:
5795:
5791:
5788:
5784:
5781:
5766:
5765:
5752:
5748:
5744:
5740:
5736:
5732:
5729:
5725:
5721:
5717:
5714:
5710:
5672:
5669:
5664:
5663:
5652:
5648:
5645:
5642:
5638:
5635:
5631:
5628:
5625:
5621:
5618:
5614:
5611:
5608:
5604:
5598:
5595:
5591:
5587:
5583:
5563:
5562:
5549:
5546:
5541:
5537:
5533:
5529:
5501:
5498:
5493:
5488:
5485:
5481:
5477:
5472:
5469:
5456:is defined as
5450:
5449:
5436:
5433:
5428:
5425:
5410:
5407:
5406:
5405:
5394:
5391:
5388:
5385:
5374:
5373:
5362:
5359:
5356:
5353:
5350:
5347:
5325:
5324:Axis and angle
5322:
5314:
5313:
5302:
5298:
5294:
5291:
5287:
5284:
5269:
5268:
5255:
5251:
5246:
5242:
5239:
5234:
5230:
5226:
5222:
5218:
5213:
5209:
5205:
5201:
5197:
5183:
5182:
5169:
5166:
5163:
5158:
5155:
5151:
5135:
5134:
5119:
5115:
5111:
5106:
5102:
5098:
5093:
5089:
5085:
5080:
5076:
5070:
5067:
5063:
5044:
5043:
5032:
5026:
5022:
5016:
5012:
5005:
5002:
4998:
4986:
4985:
4974:
4969:
4966:
4961:
4958:
4944:
4943:
4932:
4929:
4926:
4922:
4919:
4904:
4903:
4888:
4884:
4880:
4875:
4871:
4867:
4862:
4858:
4852:
4849:
4845:
4830:
4829:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4777:
4776:
4765:
4762:
4759:
4756:
4753:
4750:
4746:
4734:
4733:
4722:
4719:
4716:
4713:
4710:
4706:
4691:absolute value
4674:
4671:
4667:
4666:
4654:
4650:
4645:
4642:
4628:
4627:
4615:
4611:
4607:
4604:
4599:
4595:
4592:
4588:
4573:
4572:
4561:
4557:
4553:
4550:
4546:
4541:
4538:
4517:
4514:
4495:
4494:
4483:
4479:
4475:
4472:
4468:
4463:
4460:
4437:
4434:
4432:
4429:
4428:
4427:
4424:scalar product
4408:
4407:
4396:
4392:
4388:
4385:
4381:
4377:
4374:
4360:
4359:
4348:
4345:
4342:
4338:
4335:
4321:
4320:
4309:
4305:
4301:
4298:
4287:
4286:
4275:
4271:
4267:
4264:
4246:
4243:
4242:
4241:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4183:
4182:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4136:
4135:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4089:
4088:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4042:
4041:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3983:
3982:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3846:
3845:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3631:
3630:
3619:
3615:
3611:
3607:
3603:
3600:
3597:
3594:
3591:
3587:
3583:
3579:
3575:
3572:
3569:
3566:
3563:
3559:
3555:
3551:
3547:
3544:
3541:
3538:
3535:
3531:
3527:
3523:
3519:
3516:
3513:
3510:
3507:
3503:
3499:
3495:
3491:
3488:
3485:
3482:
3479:
3475:
3471:
3467:
3463:
3460:
3457:
3454:
3451:
3447:
3443:
3439:
3435:
3432:
3429:
3426:
3423:
3419:
3415:
3411:
3407:
3404:
3401:
3398:
3395:
3391:
3387:
3383:
3379:
3376:
3373:
3370:
3367:
3356:
3355:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3259:
3256:
3252:
3251:
3250:
3249:
3242:
3239:
3229:
3228:Multiplication
3226:
3221:
3220:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3176:
3173:
3168:
3165:
3159:
3138:
3135:
3130:
3127:
3112:
3109:
3105:
3104:
3091:
3088:
3083:
3077:
3074:
3069:
3066:
3060:
3055:
3052:
3047:
3044:
3041:
3038:
3015:
3012:
3009:
3006:
2983:
2980:
2977:
2974:
2960:
2957:
2956:
2955:
2954:
2953:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2877:
2874:
2856:
2855:
2854:
2853:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2812:
2809:
2804:
2788:
2787:
2786:
2785:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2711:
2708:
2704:
2700:
2697:
2692:
2689:
2684:
2681:
2676:
2673:
2655:
2654:
2653:
2652:
2641:
2638:
2633:
2630:
2606:
2603:
2598:
2582:
2581:
2580:
2579:
2568:
2563:
2560:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2502:
2499:
2480:
2479:
2478:
2477:
2466:
2463:
2460:
2455:
2452:
2448:
2444:
2439:
2436:
2416:
2415:
2404:
2401:
2395:
2392:
2388:
2376:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2336:
2335:
2324:
2321:
2315:
2312:
2307:
2304:
2291:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2248:
2247:
2236:
2233:
2227:
2224:
2220:
2208:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2168:
2167:
2156:
2153:
2147:
2144:
2139:
2136:
2123:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2080:
2079:
2068:
2065:
2059:
2056:
2052:
2040:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2000:
1999:
1988:
1985:
1979:
1976:
1971:
1968:
1955:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1912:
1911:
1900:
1897:
1892:
1888:
1885:
1872:
1861:
1858:
1855:
1852:
1849:
1838:
1837:
1826:
1823:
1818:
1814:
1811:
1798:
1787:
1784:
1781:
1778:
1775:
1764:
1763:
1752:
1749:
1744:
1740:
1737:
1724:
1713:
1710:
1707:
1704:
1701:
1690:
1689:
1678:
1675:
1670:
1667:
1655:
1644:
1641:
1638:
1635:
1624:
1623:
1612:
1609:
1604:
1601:
1589:
1578:
1575:
1572:
1569:
1558:
1557:
1546:
1543:
1538:
1535:
1523:
1512:
1509:
1506:
1503:
1474:
1460:
1459:
1458:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1387:
1386:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1319:
1318:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1240:
1239:
1227:
1207:
1185:
1182:
1179:
1176:
1171:
1168:
1164:
1160:
1157:
1154:
1151:
1146:
1143:
1119:
1116:
1111:
1108:
1104:
1100:
1089:
1088:
1077:
1074:
1069:
1066:
1044:
1043:
1030:
1027:
1022:
1018:
1013:
1008:
1005:
946:
943:
940:
936:
932:
928:
905:
902:
897:
894:
868:
867:
854:
851:
835:
834:
823:
820:
817:
803:
802:
791:
788:
785:
782:
779:
761:
758:
747:
746:
738:
735:
726:
723:
718:
715:
711:
710:
709:
708:
677:
674:
663:
660:
655:
654:
646:
643:
642:
641:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
585:
584:
573:
570:
567:
564:
561:
558:
543:
540:
539:
538:
527:
523:
519:
515:
511:
508:
494:
493:
482:
478:
463:
462:
451:
447:
429:
428:
417:
414:
411:
407:
403:
400:
397:
394:
390:
386:
383:
368:
365:
352:
349:
328:
325:
319:arc, called a
308:
305:
287:Main article:
284:
281:
276:
275:
238:
237:
230:absolute value
216:
213:
204:
201:
126:
123:
107:Main article:
104:
101:
61:
58:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7459:
7448:
7445:
7443:
7440:
7438:
7435:
7434:
7432:
7422:
7418:
7415:
7411:
7409:
7405:
7402:
7398:
7397:
7393:
7392:
7388:
7385:
7381:
7380:
7376:
7375:
7371:
7370:
7366:
7358:
7356:9780828402194
7352:
7348:
7347:
7342:
7336:
7333:
7330:
7326:
7325:
7319:
7316:
7313:
7309:
7308:
7302:
7299:
7296:
7292:
7291:
7285:
7282:
7279:
7275:
7274:
7268:
7265:
7262:
7258:
7257:
7251:
7248:
7245:
7241:
7240:
7234:
7231:
7228:
7224:
7223:
7217:
7214:
7211:
7207:
7206:
7200:
7197:
7194:
7190:
7189:
7183:
7180:
7177:
7173:
7172:
7166:
7163:
7160:
7156:
7155:
7149:
7146:
7143:
7139:
7138:
7132:
7129:
7124:
7118:
7115:
7109:
7106:
7103:
7099:
7098:
7092:
7089:
7086:
7082:
7081:
7075:
7072:
7067:
7061:
7058:
7055:
7051:
7050:
7044:
7041:
7038:
7034:
7033:
7027:
7024:
7021:
7017:
7016:
7010:
7007:
7004:
7000:
6999:
6993:
6990:
6987:
6983:
6982:
6976:
6973:
6970:
6966:
6965:
6959:
6956:
6953:
6949:
6948:
6942:
6939:
6936:
6932:
6931:
6925:
6922:
6919:
6915:
6914:
6908:
6905:
6902:
6898:
6897:
6891:
6888:
6885:
6881:
6880:
6874:
6871:
6868:
6864:
6863:
6857:
6854:
6851:
6847:
6846:
6840:
6837:
6834:
6830:
6829:
6823:
6820:
6817:
6813:
6812:
6806:
6803:
6800:
6796:
6795:
6789:
6786:
6783:
6779:
6778:
6772:
6769:
6766:
6762:
6761:
6755:
6752:
6749:
6745:
6744:
6738:
6735:
6732:
6728:
6727:
6721:
6718:
6715:
6711:
6710:
6704:
6701:
6698:
6694:
6693:
6687:
6684:
6681:
6677:
6676:
6670:
6667:
6664:
6660:
6659:
6653:
6650:
6647:
6643:
6642:
6636:
6633:
6630:
6626:
6625:
6619:
6616:
6613:
6609:
6608:
6602:
6599:
6596:
6592:
6591:
6585:
6582:
6579:
6575:
6574:
6568:
6565:
6562:
6558:
6557:
6551:
6548:
6545:
6541:
6540:
6534:
6531:
6528:
6524:
6523:
6517:
6514:
6511:
6507:
6506:
6500:
6497:
6494:
6490:
6489:
6483:
6480:
6477:
6473:
6472:
6466:
6463:
6460:
6456:
6455:
6449:
6446:
6443:
6439:
6438:
6432:
6429:
6426:
6422:
6421:
6415:
6412:
6409:
6405:
6404:
6398:
6395:
6392:
6388:
6387:
6381:
6378:
6375:
6371:
6370:
6364:
6361:
6358:
6354:
6353:
6347:
6344:
6341:
6337:
6336:
6330:
6327:
6324:
6320:
6319:
6313:
6310:
6307:
6303:
6302:
6296:
6293:
6290:
6286:
6285:
6279:
6276:
6273:
6269:
6268:
6262:
6259:
6256:
6252:
6251:
6244:
6241:
6238:
6234:
6233:
6227:
6224:
6221:
6217:
6216:
6210:
6207:
6204:
6200:
6199:
6193:
6191:
6187:
6184:
6180:
6179:
6173:
6170:
6167:
6163:
6162:
6156:
6153:
6149:
6145:
6139:
6136:
6133:
6129:
6128:
6122:
6119:
6116:
6112:
6111:
6105:
6102:
6096:
6092:
6089:
6087:
6084:
6082:
6079:
6078:
6074:
6072:
6070:
6066:
6062:
6058:
6050:
6048:
6046:
6042:
6038:
6034:
6030:
6024:
6016:
6014:
6011:
6009:
6005:
6001:
5999:
5995:
5991:
5987:
5982:
5978:
5971:
5969:
5967:
5963:
5942:
5939:
5933:
5930:
5926:
5923:
5916:
5915:
5914:
5911:
5907:
5903:
5899:
5895:
5890:
5888:
5869:
5866:
5863:
5858:
5854:
5846:
5845:
5844:
5838:
5834:Biquaternions
5833:
5817:
5814:
5811:
5800:
5797:
5789:
5786:
5771:
5770:
5769:
5750:
5742:
5730:
5727:
5719:
5715:
5712:
5700:
5699:
5698:
5695:
5693:
5689:
5685:
5681:
5676:
5670:
5668:
5650:
5636:
5633:
5619:
5616:
5602:
5593:
5581:
5572:
5571:
5570:
5568:
5547:
5544:
5539:
5535:
5531:
5527:
5519:
5518:
5517:
5514:
5499:
5496:
5491:
5486:
5483:
5479:
5475:
5470:
5467:
5457:
5455:
5434:
5431:
5426:
5423:
5416:
5415:
5414:
5408:
5392:
5386:
5383:
5376:
5375:
5360:
5357:
5354:
5351:
5348:
5345:
5338:
5337:
5336:
5333:
5329:
5323:
5321:
5319:
5300:
5292:
5289:
5274:
5273:
5272:
5253:
5249:
5240:
5232:
5228:
5216:
5211:
5203:
5188:
5187:
5186:
5167:
5164:
5161:
5156:
5153:
5141:
5140:
5139:
5117:
5113:
5109:
5104:
5100:
5096:
5091:
5087:
5083:
5078:
5074:
5068:
5065:
5053:
5052:
5051:
5049:
5030:
5024:
5014:
5003:
5000:
4988:
4987:
4972:
4967:
4964:
4959:
4956:
4949:
4948:
4947:
4930:
4927:
4924:
4909:
4908:
4907:
4886:
4882:
4878:
4873:
4869:
4865:
4860:
4856:
4850:
4847:
4835:
4834:
4833:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4782:
4781:
4780:
4763:
4760:
4754:
4751:
4736:
4735:
4720:
4717:
4711:
4696:
4695:
4694:
4692:
4687:
4685:
4684:
4679:
4672:
4670:
4652:
4643:
4640:
4633:
4632:
4631:
4613:
4605:
4602:
4593:
4590:
4578:
4577:
4576:
4559:
4551:
4548:
4539:
4536:
4529:
4528:
4527:
4524:
4522:
4515:
4513:
4511:
4507:
4505:
4500:
4481:
4473:
4470:
4461:
4458:
4451:
4450:
4449:
4447:
4443:
4435:
4430:
4425:
4421:
4420:
4419:
4417:
4413:
4394:
4386:
4383:
4375:
4372:
4365:
4364:
4363:
4346:
4343:
4340:
4336:
4333:
4326:
4325:
4324:
4307:
4299:
4296:
4289:
4288:
4273:
4265:
4262:
4255:
4254:
4253:
4250:
4244:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4188:
4187:
4186:
4166:
4163:
4160:
4157:
4154:
4148:
4145:
4138:
4137:
4119:
4116:
4113:
4110:
4107:
4101:
4098:
4091:
4090:
4072:
4069:
4066:
4063:
4060:
4054:
4051:
4044:
4043:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3991:
3990:
3989:
3986:
3969:
3963:
3960:
3957:
3954:
3951:
3945:
3942:
3936:
3933:
3930:
3927:
3924:
3918:
3915:
3909:
3906:
3903:
3900:
3897:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3851:
3850:
3849:
3829:
3826:
3820:
3817:
3814:
3808:
3805:
3799:
3796:
3793:
3787:
3784:
3778:
3775:
3772:
3766:
3763:
3757:
3754:
3751:
3745:
3742:
3736:
3733:
3730:
3724:
3721:
3715:
3712:
3709:
3703:
3700:
3694:
3691:
3688:
3682:
3679:
3673:
3670:
3667:
3661:
3658:
3652:
3649:
3646:
3643:
3636:
3635:
3634:
3613:
3609:
3605:
3598:
3595:
3592:
3585:
3581:
3577:
3570:
3567:
3564:
3557:
3553:
3549:
3542:
3539:
3536:
3529:
3525:
3521:
3514:
3511:
3508:
3501:
3497:
3493:
3486:
3483:
3480:
3473:
3469:
3465:
3458:
3455:
3452:
3445:
3441:
3437:
3430:
3427:
3424:
3417:
3413:
3409:
3402:
3399:
3396:
3389:
3385:
3381:
3374:
3371:
3368:
3365:
3358:
3357:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3312:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3279:
3276:
3269:
3268:
3267:
3265:
3257:
3255:
3247:
3246:
3245:
3244:
3240:
3238:
3234:
3227:
3225:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3174:
3171:
3166:
3163:
3157:
3136:
3133:
3128:
3125:
3118:
3117:
3116:
3110:
3108:
3089:
3086:
3081:
3075:
3072:
3067:
3064:
3058:
3053:
3050:
3045:
3042:
3039:
3036:
3029:
3028:
3027:
3013:
3010:
3007:
3004:
2995:
2981:
2978:
2975:
2972:
2964:
2958:
2940:
2937:
2931:
2928:
2922:
2919:
2913:
2910:
2904:
2901:
2898:
2892:
2889:
2883:
2880:
2875:
2872:
2863:
2862:
2861:
2860:
2859:
2839:
2836:
2833:
2827:
2824:
2818:
2815:
2810:
2807:
2802:
2795:
2794:
2793:
2792:
2791:
2771:
2768:
2765:
2759:
2753:
2750:
2744:
2741:
2735:
2732:
2726:
2723:
2717:
2714:
2709:
2706:
2702:
2698:
2695:
2690:
2687:
2682:
2679:
2674:
2671:
2662:
2661:
2660:
2659:
2658:
2639:
2636:
2631:
2628:
2604:
2601:
2596:
2589:
2588:
2587:
2586:
2585:
2566:
2561:
2558:
2553:
2550:
2544:
2541:
2535:
2532:
2529:
2526:
2523:
2517:
2514:
2508:
2505:
2500:
2497:
2488:
2487:
2486:
2485:
2484:
2464:
2461:
2458:
2453:
2450:
2446:
2442:
2437:
2434:
2425:
2424:
2423:
2422:
2421:
2402:
2399:
2393:
2390:
2386:
2377:
2363:
2360:
2354:
2351:
2345:
2338:
2337:
2322:
2319:
2313:
2310:
2305:
2302:
2292:
2278:
2275:
2272:
2266:
2263:
2257:
2250:
2249:
2234:
2231:
2225:
2222:
2218:
2209:
2195:
2192:
2186:
2183:
2177:
2170:
2169:
2154:
2151:
2145:
2142:
2137:
2134:
2124:
2110:
2107:
2104:
2098:
2095:
2089:
2082:
2081:
2066:
2063:
2057:
2054:
2050:
2041:
2027:
2024:
2018:
2015:
2009:
2002:
2001:
1986:
1983:
1977:
1974:
1969:
1966:
1956:
1942:
1939:
1936:
1930:
1927:
1921:
1914:
1913:
1898:
1895:
1890:
1886:
1883:
1873:
1859:
1856:
1853:
1850:
1847:
1840:
1839:
1824:
1821:
1816:
1812:
1809:
1799:
1785:
1782:
1779:
1776:
1773:
1766:
1765:
1750:
1747:
1742:
1738:
1735:
1725:
1711:
1708:
1705:
1702:
1699:
1692:
1691:
1676:
1673:
1668:
1665:
1656:
1642:
1639:
1636:
1633:
1626:
1625:
1610:
1607:
1602:
1599:
1590:
1576:
1573:
1570:
1567:
1560:
1559:
1544:
1541:
1536:
1533:
1524:
1510:
1507:
1504:
1501:
1494:
1493:
1490:
1488:
1484:
1480:
1473:
1469:
1465:
1461:
1442:
1439:
1436:
1430:
1424:
1421:
1418:
1412:
1409:
1406:
1403:
1396:
1395:
1394:
1392:
1373:
1370:
1364:
1361:
1358:
1352:
1346:
1343:
1340:
1334:
1331:
1324:
1323:
1322:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1242:
1241:
1225:
1205:
1198:
1197:
1196:
1183:
1180:
1177:
1174:
1169:
1166:
1162:
1158:
1155:
1152:
1149:
1144:
1141:
1130:
1117:
1114:
1109:
1106:
1102:
1098:
1075:
1072:
1067:
1064:
1055:
1054:
1053:
1050:
1047:
1028:
1025:
1020:
1016:
1011:
1006:
1003:
994:
993:
992:
990:
986:
982:
977:
975:
974:
969:
968:
963:
958:
944:
941:
938:
934:
930:
926:
916:
903:
900:
895:
892:
881:
878:
876:
873:
852:
849:
840:
839:
838:
821:
818:
815:
808:
807:
806:
789:
786:
783:
780:
777:
770:
769:
768:
765:
759:
757:
755:
750:
744:
743:
742:
736:
734:
730:
724:
722:
716:
714:
706:
705:
704:
703:
702:
697:
694:
688:
687:
683:
675:
673:
670:
667:
661:
659:
652:
651:
650:
644:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
594:
593:
592:
590:
571:
568:
562:
556:
549:
548:
547:
541:
525:
517:
509:
506:
499:
498:
497:
480:
468:
467:
466:
449:
437:
436:
435:
432:
412:
401:
395:
384:
381:
374:
373:
372:
366:
364:
360:
358:
350:
348:
346:
342:
338:
334:
326:
324:
322:
318:
314:
306:
304:
301:
299:
294:
290:
282:
280:
274:
272:
268:
264:
260:
254:
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245:
243:
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231:
227:
223:
219:
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214:
212:
210:
202:
200:
197:
193:
188:
186:
182:
178:
174:
170:
165:
163:
158:
156:
152:
146:
144:
143:
138:
132:
124:
122:
120:
116:
110:
102:
100:
98:
97:
92:
91:
86:
85:
80:
79:
73:
71:
67:
59:
57:
55:
51:
47:
43:
37:
33:
19:
7420:
7413:
7407:
7401:Google Books
7396:
7390:
7384:Google Books
7379:
7373:
7345:
7335:
7329:Google Books
7322:
7318:
7312:Google Books
7305:
7301:
7295:Google Books
7288:
7284:
7278:Google Books
7271:
7267:
7261:Google Books
7254:
7250:
7244:Google Books
7237:
7233:
7227:Google Books
7220:
7216:
7210:Google Books
7203:
7199:
7193:Google Books
7186:
7182:
7176:Google Books
7169:
7165:
7159:Google Books
7152:
7148:
7142:Google Books
7135:
7131:
7117:
7108:
7102:Google Books
7095:
7091:
7085:Google Books
7078:
7074:
7060:
7054:Google Books
7047:
7043:
7037:Google Books
7030:
7026:
7020:Google Books
7013:
7009:
7003:Google Books
6996:
6992:
6986:Google Books
6979:
6975:
6969:Google Books
6962:
6958:
6952:Google Books
6945:
6941:
6935:Google Books
6928:
6924:
6918:Google Books
6911:
6907:
6901:Google Books
6894:
6890:
6884:Google Books
6877:
6873:
6867:Google Books
6860:
6856:
6850:Google Books
6843:
6839:
6833:Google Books
6826:
6822:
6816:Google Books
6809:
6805:
6799:Google Books
6792:
6788:
6782:Google Books
6775:
6771:
6765:Google Books
6758:
6754:
6748:Google Books
6741:
6737:
6731:Google Books
6724:
6720:
6714:Google Books
6707:
6703:
6697:Google Books
6690:
6686:
6680:Google Books
6673:
6669:
6663:Google Books
6656:
6652:
6646:Google Books
6639:
6635:
6629:Google Books
6622:
6618:
6612:Google Books
6605:
6601:
6595:Google Books
6588:
6584:
6578:Google Books
6571:
6567:
6561:Google Books
6554:
6550:
6544:Google Books
6537:
6533:
6527:Google Books
6520:
6516:
6510:Google Books
6503:
6499:
6493:Google Books
6486:
6482:
6476:Google Books
6469:
6465:
6459:Google Books
6452:
6448:
6442:Google Books
6435:
6431:
6425:Google Books
6418:
6414:
6408:Google Books
6401:
6397:
6391:Google Books
6384:
6380:
6374:Google Books
6367:
6363:
6357:Google Books
6350:
6346:
6340:Google Books
6333:
6329:
6323:Google Books
6316:
6312:
6306:Google Books
6299:
6295:
6289:Google Books
6282:
6278:
6272:Google Books
6265:
6261:
6255:Google Books
6248:
6243:
6237:Google Books
6230:
6226:
6220:Google Books
6213:
6209:
6203:Google Books
6196:
6183:Google Books
6176:
6172:
6166:Google Books
6159:
6155:
6147:
6143:
6138:
6132:Google Books
6125:
6121:
6115:Google Books
6108:
6104:
6068:
6064:
6056:
6054:
6036:
6028:
6026:
6023:Biquaternion
6017:Biquaternion
6012:
6007:
6003:
6002:
5997:
5993:
5992:. The terms
5990:biquaternion
5989:
5985:
5980:
5976:
5975:
5965:
5959:
5910:coefficients
5905:
5901:
5897:
5893:
5891:
5886:
5884:
5842:
5767:
5697:In symbols:
5696:
5691:
5687:
5683:
5679:
5677:
5674:
5665:
5564:
5515:
5458:
5451:
5412:
5335:In symbols,
5334:
5330:
5327:
5315:
5270:
5184:
5136:
5045:
4945:
4905:
4831:
4778:
4688:
4682:
4677:
4676:
4668:
4629:
4574:
4525:
4520:
4519:
4509:
4503:
4502:
4498:
4496:
4445:
4441:
4439:
4415:
4411:
4409:
4361:
4322:
4251:
4248:
4184:
3987:
3984:
3847:
3632:
3264:distributive
3261:
3258:Distributive
3253:
3235:
3231:
3222:
3114:
3106:
2996:
2965:
2962:
2857:
2789:
2656:
2583:
2481:
2419:
1486:
1482:
1478:
1476:
1471:
1467:
1463:
1390:
1388:
1320:
1131:
1090:
1051:
1048:
1045:
978:
972:
966:
959:
917:
882:
879:
874:
871:
869:
836:
804:
766:
763:
751:
748:
740:
731:
728:
720:
712:
699:
692:
690:
679:
671:
668:
665:
656:
648:
588:
586:
545:
495:
464:
433:
430:
370:
361:
356:
354:
344:
341:right radial
340:
337:right versor
336:
330:
327:Right versor
320:
317:great circle
310:
302:
295:
292:
277:
270:
266:
262:
255:
248:
246:
241:
239:
225:
221:
206:
189:
184:
180:
176:
172:
166:
159:
147:
145:, to carry.
141:
136:
134:
131:Vector space
119:real numbers
114:
112:
95:
89:
83:
77:
74:
63:
40:
6057:associative
5671:Common norm
989:denominator
962:commutative
676:Subtraction
333:right angle
313:unit sphere
269:, that is,
209:unit vector
203:Unit vector
155:rectangular
70:dimensional
46:quaternions
7431:Categories
7367:References
5454:reciprocal
5409:Reciprocal
4414:, and the
985:reciprocal
983:times the
367:Quaternion
321:vector arc
307:Vector arc
129:See also:
36:quaternion
6097:Footnotes
6086:Octonions
5940:−
5867:−
5620:−
5567:rotations
5540:×
5536:α
5532:×
5500:α
5497:β
5484:−
5452:then its
5435:β
5432:α
5390:∠
5384:θ
5025:β
5015:α
4968:β
4965:α
4925:α
4848:α
4790:α
4752:−
4606:−
4516:Conjugate
4344:β
4341:α
4297:β
4263:α
4161:−
4114:−
4067:−
4023:−
4014:−
4005:−
3958:−
3931:−
3904:−
3883:−
3874:−
3865:−
3827:−
3806:−
3743:−
3722:−
3701:−
3659:−
3610:×
3582:×
3554:×
3526:×
3498:×
3470:×
3442:×
3414:×
3386:×
3313:×
3205:ϕ
3202:
3196:ϵ
3190:ϕ
3187:
3175:β
3167:α
3137:β
3134:α
3054:β
3051:α
3043:β
3040:÷
3037:α
3005:β
2973:α
2929:−
2923:−
2905:−
2890:−
2837:−
2825:−
2790:So that:
2769:−
2754:−
2736:−
2724:−
2707:−
2542:−
2515:−
2462:−
2451:−
2391:−
2352:−
2311:−
2303:−
2276:−
2264:−
2223:−
2184:−
2143:−
2135:−
2108:−
2096:−
2055:−
2016:−
1975:−
1967:−
1940:−
1928:−
1884:−
1857:−
1810:−
1783:−
1736:−
1709:−
1443:α
1440:÷
1437:β
1431:×
1425:β
1422:÷
1419:γ
1410:α
1407:÷
1404:γ
1374:α
1371:×
1365:α
1362:÷
1359:β
1353:×
1347:β
1344:÷
1341:γ
1332:γ
1300:α
1297:÷
1294:α
1291:×
1268:β
1262:α
1259:×
1256:α
1253:÷
1250:β
1226:÷
1206:×
1184:α
1178:β
1167:−
1163:β
1159:α
1153:β
1145:β
1142:α
1107:−
1103:β
1099:α
1068:β
1065:α
1029:β
1021:×
1017:α
1007:β
1004:α
981:numerator
942:α
935:β
931:×
896:β
893:α
853:β
850:α
822:β
819:÷
816:α
54:algebraic
50:geometric
44:invented
7343:(1899).
6075:See also
6041:bivector
6037:biscalar
5986:Elements
5981:Elements
5934:′
4444:(q) and
3988:Letting
2858:however
760:Division
754:analysis
717:Addition
682:analysis
496:we have
263:positive
196:parallel
192:quotient
117:for the
93:and its
66:quotient
1391:version
987:of the
973:tension
967:version
684:called
653:+ − ÷ ×
194:of two
169:integer
115:scalars
7353:
4683:tensor
4673:Tensor
4497:Here,
1485:, and
693:states
298:normal
289:versor
283:Versor
249:tensor
242:tensor
226:tensor
222:tensor
215:Tensor
162:origin
142:vehere
137:vector
125:Vector
103:Scalar
96:versor
90:tensor
84:vector
81:and a
78:scalar
4832:Then
4575:then
3026:then
1091:then
918:then
707:B − A
151:polar
7351:ISBN
5996:and
5904:and
5684:norm
5318:norm
5050:is:
2997:and
1321:and
1280:and
1218:and
1132:and
745:÷, ×
339:, a
190:The
153:and
7399:at
7382:at
7327:at
7310:at
7293:at
7276:at
7259:at
7242:at
7225:at
7208:at
7191:at
7174:at
7157:at
7140:at
7100:at
7083:at
7052:at
7035:at
7018:at
7001:at
6984:at
6967:at
6950:at
6933:at
6916:at
6899:at
6882:at
6865:at
6848:at
6831:at
6814:at
6797:at
6780:at
6763:at
6746:at
6729:at
6712:at
6695:at
6678:at
6661:at
6644:at
6627:at
6610:at
6593:at
6576:at
6559:at
6542:at
6525:at
6508:at
6491:at
6474:at
6457:at
6440:at
6423:at
6406:at
6389:at
6372:at
6355:at
6338:at
6321:at
6304:at
6287:at
6270:at
6253:at
6235:at
6218:at
6201:at
6181:at
6164:at
6148:OED
6130:at
6113:at
5413:If
4526:If
3199:sin
3184:cos
883:if
837:or
343:or
7433::
6189:^
5900:,
5892:A
5694:.
5320:.
4686:.
2994:,
2619:≠
1481:,
1470:,
1466:,
976:.
957:.
572:0.
347:.
207:A
99:.
7359:.
7125:.
7068:.
6247:a
6150:.
5943:1
5931:q
5927:+
5924:q
5906:k
5902:j
5898:i
5870:1
5864:=
5859:2
5855:q
5818:1
5815:=
5812:q
5808:U
5805:K
5801:.
5798:q
5794:U
5790:=
5787:q
5783:U
5780:N
5751:2
5747:)
5743:q
5739:T
5735:(
5731:=
5728:q
5724:K
5720:q
5716:=
5713:q
5709:N
5680:N
5651:q
5647:U
5644:.
5641:K
5637:=
5634:q
5630:U
5627:.
5624:V
5617:q
5613:U
5610:.
5607:S
5603:=
5597:)
5594:q
5590:U
5586:(
5582:1
5548:q
5545:1
5528:q
5492:=
5487:1
5480:q
5476:=
5471:q
5468:1
5427:=
5424:q
5393:q
5387:=
5361:q
5358:.
5355:x
5352:A
5349:=
5346:u
5301:q
5297:T
5293:=
5290:q
5286:K
5283:T
5254:2
5250:q
5245:T
5241:=
5238:)
5233:2
5229:q
5225:(
5221:T
5217:=
5212:2
5208:)
5204:q
5200:T
5196:(
5168:q
5165:K
5162:q
5157:=
5154:q
5150:T
5118:2
5114:z
5110:+
5105:2
5101:y
5097:+
5092:2
5088:x
5084:+
5079:2
5075:w
5069:=
5066:q
5062:T
5031:.
5021:T
5011:T
5004:=
5001:q
4997:T
4973:.
4960:=
4957:q
4931:1
4928:=
4921:U
4918:T
4887:2
4883:z
4879:+
4874:2
4870:y
4866:+
4861:2
4857:x
4851:=
4844:T
4817:k
4814:z
4811:+
4808:j
4805:y
4802:+
4799:i
4796:x
4793:=
4764:5
4761:=
4758:)
4755:5
4749:(
4745:T
4721:5
4718:=
4715:)
4712:5
4709:(
4705:T
4678:T
4665:,
4653:q
4649:K
4644:=
4641:r
4626:.
4614:q
4610:V
4603:q
4598:S
4594:=
4591:q
4587:K
4560:q
4556:V
4552:+
4549:q
4545:S
4540:=
4537:q
4521:K
4510:V
4506:q
4504:S
4499:q
4482:q
4478:V
4474:+
4471:q
4467:S
4462:=
4459:q
4446:V
4442:S
4395:r
4391:V
4387:+
4384:r
4380:S
4376:=
4373:r
4347:;
4337:=
4334:r
4308:q
4304:V
4300:=
4274:p
4270:V
4266:=
4229:k
4226:z
4223:+
4220:j
4217:y
4214:+
4211:i
4208:x
4205:+
4202:w
4199:=
4196:q
4170:)
4167:e
4164:b
4158:f
4155:a
4152:(
4149:=
4146:z
4123:)
4120:g
4117:a
4111:e
4108:c
4105:(
4102:=
4099:y
4076:)
4073:f
4070:c
4064:g
4061:b
4058:(
4055:=
4052:x
4029:g
4026:c
4020:f
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4011:e
4008:a
4002:=
3999:w
3970:k
3967:)
3964:e
3961:b
3955:f
3952:a
3949:(
3946:+
3943:j
3940:)
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3922:(
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3901:g
3898:b
3895:(
3892:+
3889:g
3886:c
3880:f
3877:b
3871:e
3868:a
3862:=
3859:q
3833:)
3830:1
3824:(
3821:g
3818:c
3815:+
3812:)
3809:i
3803:(
3800:f
3797:c
3794:+
3791:)
3788:j
3785:+
3782:(
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3752:+
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3734:b
3731:+
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3725:k
3719:(
3716:e
3713:b
3710:+
3707:)
3704:j
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3695:g
3692:a
3689:+
3686:)
3683:k
3680:+
3677:(
3674:f
3671:a
3668:+
3665:)
3662:1
3656:(
3653:e
3650:a
3647:=
3644:q
3618:)
3614:k
3606:k
3602:(
3599:g
3596:c
3593:+
3590:)
3586:j
3578:k
3574:(
3571:f
3568:c
3565:+
3562:)
3558:i
3550:k
3546:(
3543:e
3540:c
3537:+
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3518:(
3515:g
3512:b
3509:+
3506:)
3502:j
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3490:(
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3481:+
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3431:g
3428:a
3425:+
3422:)
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3406:(
3403:f
3400:a
3397:+
3394:)
3390:i
3382:i
3378:(
3375:e
3372:a
3369:=
3366:q
3343:)
3340:k
3337:g
3334:+
3331:j
3328:f
3325:+
3322:i
3319:e
3316:(
3310:)
3307:k
3304:c
3301:+
3298:j
3295:b
3292:+
3289:i
3286:a
3283:(
3280:=
3277:q
3208:)
3193:+
3181:(
3172:T
3164:T
3158:=
3129:=
3126:q
3090:b
3087:a
3082:=
3076:i
3073:b
3068:i
3065:a
3059:=
3046:=
3014:i
3011:b
3008:=
2982:i
2979:a
2976:=
2941:k
2938:=
2935:)
2932:k
2926:(
2920:=
2917:)
2914:i
2911:j
2908:(
2902:=
2899:i
2896:)
2893:j
2887:(
2884:=
2881:i
2876:i
2873:k
2852:,
2840:k
2834:=
2831:)
2828:j
2822:(
2819:i
2816:=
2811:i
2808:k
2803:i
2784:.
2772:j
2766:=
2763:)
2760:j
2757:(
2751:=
2748:)
2745:i
2742:k
2739:(
2733:=
2730:)
2727:i
2721:(
2718:k
2715:=
2710:1
2703:i
2699:k
2696:=
2691:i
2688:1
2683:k
2680:=
2675:i
2672:k
2640:.
2637:i
2632:i
2629:k
2605:i
2602:k
2597:i
2567:.
2562:a
2559:1
2554:a
2551:=
2548:)
2545:a
2539:(
2536:a
2533:=
2530:1
2527:=
2524:a
2521:)
2518:a
2512:(
2509:=
2506:a
2501:a
2498:1
2465:i
2459:=
2454:1
2447:i
2443:=
2438:i
2435:1
2403:j
2400:=
2394:i
2387:k
2364:k
2361:=
2358:)
2355:i
2349:(
2346:j
2323:j
2320:=
2314:k
2306:i
2279:i
2273:=
2270:)
2267:k
2261:(
2258:j
2235:k
2232:=
2226:j
2219:i
2196:i
2193:=
2190:)
2187:j
2181:(
2178:k
2155:k
2152:=
2146:i
2138:j
2111:j
2105:=
2102:)
2099:i
2093:(
2090:k
2067:i
2064:=
2058:k
2051:j
2028:j
2025:=
2022:)
2019:k
2013:(
2010:i
1987:i
1984:=
1978:j
1970:k
1943:k
1937:=
1934:)
1931:j
1925:(
1922:i
1899:i
1896:=
1891:k
1887:j
1860:j
1854:=
1851:k
1848:i
1825:k
1822:=
1817:j
1813:i
1786:i
1780:=
1777:j
1774:k
1751:j
1748:=
1743:i
1739:k
1712:k
1706:=
1703:i
1700:j
1677:k
1674:=
1669:i
1666:j
1643:j
1640:=
1637:i
1634:k
1611:j
1608:=
1603:k
1600:i
1577:i
1574:=
1571:k
1568:j
1545:i
1542:=
1537:j
1534:k
1511:k
1508:=
1505:j
1502:i
1487:k
1483:j
1479:i
1472:k
1468:j
1464:i
1446:)
1434:(
1428:)
1416:(
1413:=
1368:)
1356:(
1350:)
1338:(
1335:=
1306:q
1303:=
1288:q
1265:=
1181:=
1175:.
1170:1
1156:=
1150:.
1118:q
1115:=
1110:1
1076:q
1073:=
1026:1
1012:=
945:.
939:=
927:q
904:q
901:=
790:B
787:O
784::
781:A
778:O
629:k
626:z
623:+
620:j
617:y
614:+
611:i
608:x
605:=
602:Q
589:Q
569:=
566:)
563:q
560:(
557:S
526:q
522:U
518:q
514:T
510:=
507:q
481:q
477:T
450:q
446:U
416:)
413:q
410:(
406:V
402:+
399:)
396:q
393:(
389:S
385:=
382:q
236:.
185:x
181:x
177:x
173:x
38:.
20:)
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