text:
BREAKING NEWS
Quaternion

## Summary

Quaternion multiplication table
1 i j k
1 1 i j k
i i −1 k j
j j k −1 i
k k j i −1
Cayley Q8 graph showing the 6 cycles of multiplication by i, j and k. (In the SVG file, hover over or click a cycle to highlight it.)

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843[1][2] and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space,[3] or, equivalently, as the quotient of two vectors.[4] Multiplication of quaternions is noncommutative.

Quaternions are generally represented in the form

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

where a, b, c, and d are real numbers; and i, j, and k are the basic quaternions.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis.[5] They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by ${\displaystyle \mathbb {H} .}$ It can also be given by the Clifford algebra classifications ${\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).}$ In fact, it was the first noncommutative division algebra to be discovered.

According to the Frobenius theorem, the algebra ${\displaystyle \mathbb {H} }$ is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisors and so cannot be a normed division algebra.)[6]

The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).

Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of {1, i, j, k}. The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually i  j = −(j  i).
• In blue:
• 1  i = i (1/i plane)
• i  j = k (i/k plane)
• In red:
• 1  j = j (1/j plane)
• j  i = k (j/k plane)

## History

Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i2 = j2 = k2 = i j k = −1
& cut it on a stone of this bridge

Quaternions were introduced by Hamilton in 1843.[7] Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.[8][9] Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.[10][11]

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

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

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

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

On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science;[12] Hamilton states:

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.[12]

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions,[13] was 800 pages long; it was edited by his son and published shortly after his death.

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

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

However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles, they are not susceptible to "gimbal lock". For this reason, quaternions are used in computer graphics,[14][15] computer vision, robotics,[16] control theory, signal processing, attitude control, physics, bioinformatics,[17][18] molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.[19]

### Quaternions in physics

P.R. Girard's 1984 essay The quaternion group and modern physics[20] discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely SO(3), the Lorentz group, the general theory of relativity group, the Clifford algebra SU(2) and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion. Next he used complex quaternions (biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions (split-biquaternions) as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.[21]

The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "Plate trick").[22][23] As of 2018, their use has not overtaken rotation groups.[a]

## Definition

A quaternion is an expression of the form

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

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

Hamilton describes a quaternion ${\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$, as consisting of a scalar part and a vector part. The quaternion ${\displaystyle b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$ is called the vector part (sometimes imaginary part) of q, and a is the scalar part (sometimes real part) of q. A quaternion that equals its real part (that is, its vector part is zero) is called a scalar or real quaternion, and is identified with the corresponding real number. That is, the real numbers are embedded in the quaternions. (More properly, the field of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.)[24] A quaternion that equals its vector part is called a vector quaternion.

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

${\displaystyle (a_{1}+b_{1}\,\mathbf {i} +c_{1}\,\mathbf {j} +d_{1}\,\mathbf {k} )+(a_{2}+b_{2}\,\mathbf {i} +c_{2}\,\mathbf {j} +d_{2}\,\mathbf {k} )=(a_{1}+a_{2})+(b_{1}+b_{2})\,\mathbf {i} +(c_{1}+c_{2})\,\mathbf {j} +(d_{1}+d_{2})\,\mathbf {k} \,,}$

and the componentwise scalar multiplication

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

A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:

• The real quaternion 1 is the identity element.
• The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
• The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the distributive property and the center property of the real quaternions. The Hamilton product is not commutative, but is associative, thus the quaternions form an associative algebra over the real numbers.
• Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product:
${\displaystyle (a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} )^{-1}={\frac {1}{a^{2}+b^{2}+c^{2}+d^{2}}}\,(a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} ).}$

Thus the quaternions form a division algebra.

### Multiplication of basis elements

Multiplication table
× 1 i j k
1 1 i j k
i i −1 k j
j j k −1 i
k k j i −1
Non commutativity is emphasized by colored squares

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

${\displaystyle \mathbf {i} \,1=1\,\mathbf {i} =\mathbf {i} ,\qquad \mathbf {j} \,1=1\,\mathbf {j} =\mathbf {j} ,\qquad \mathbf {k} \,1=1\,\mathbf {k} =\mathbf {k} \,.}$

The other products of basis elements are defined from the product rules for ${\displaystyle \mathbf {i} }$ and ${\displaystyle \mathbf {j} \,\colon }$

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

and

{\displaystyle {\begin{aligned}\mathbf {i\,j} &=\mathbf {k} \,,\quad &\mathbf {j\,i} &=-\mathbf {k} \,.\end{aligned}}}

Then, the other product rules are obtained by replacing ${\displaystyle \mathbf {k} }$ by ${\displaystyle \mathbf {i\,j} ,}$ and applying associativity and the anticommutativity of ${\displaystyle \mathbf {i} }$ and ${\displaystyle \mathbf {j} }$ (that is, ${\displaystyle \mathbf {i\,j} =-\mathbf {j\,i} }$), which gives

{\displaystyle {\begin{aligned}\mathbf {k} ^{2}&=-1\,\\\mathbf {j\,k} &=\mathbf {i} \,,\quad &\mathbf {k\,j} &=-\mathbf {i} \,,\\\mathbf {k\,i} &=\mathbf {j} \,,\quad &\mathbf {i\,k} &=-\mathbf {j} \,,\\\mathbf {i\,j\,k} &=-1\,.\end{aligned}}}

### Center

The center of a noncommutative ring is the subring of elements c such that cx = xc for every x. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if q = a + b i + c j + d k belongs to the center, then

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

and c = d = 0. A similar computation with j instead of i shows that one has also b = 0. Thus q = a is a real quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a polynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation z2 + 1 = 0, has infinitely many quaternion solutions, which are the quaternions z = b i + c j + d k such that b2 + c2 + d2 = 1. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.

### Hamilton product

For two elements a1 + b1i + c1j + d1k and a2 + b2i + c2j + d2k, their product, called the Hamilton product (a1 + b1i + c1j + d1k) (a2 + b2i + c2j + d2k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

{\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&+a_{1}b_{2}\mathbf {i} &&+a_{1}c_{2}\mathbf {j} &&+a_{1}d_{2}\mathbf {k} \\{}+{}&b_{1}a_{2}\mathbf {i} &&+b_{1}b_{2}\mathbf {i} ^{2}&&+b_{1}c_{2}\mathbf {ij} &&+b_{1}d_{2}\mathbf {ik} \\{}+{}&c_{1}a_{2}\mathbf {j} &&+c_{1}b_{2}\mathbf {ji} &&+c_{1}c_{2}\mathbf {j} ^{2}&&+c_{1}d_{2}\mathbf {jk} \\{}+{}&d_{1}a_{2}\mathbf {k} &&+d_{1}b_{2}\mathbf {ki} &&+d_{1}c_{2}\mathbf {kj} &&+d_{1}d_{2}\mathbf {k} ^{2}\end{alignedat}}}

Now the basis elements can be multiplied using the rules given above to get:[7]

{\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&-b_{1}b_{2}&&-c_{1}c_{2}&&-d_{1}d_{2}\\{}+{}(&a_{1}b_{2}&&+b_{1}a_{2}&&+c_{1}d_{2}&&-d_{1}c_{2})\mathbf {i} \\{}+{}(&a_{1}c_{2}&&-b_{1}d_{2}&&+c_{1}a_{2}&&+d_{1}b_{2})\mathbf {j} \\{}+{}(&a_{1}d_{2}&&+b_{1}c_{2}&&-c_{1}b_{2}&&+d_{1}a_{2})\mathbf {k} \end{alignedat}}}

The product of two rotation quaternions[25] will be equivalent to the rotation a2 + b2i + c2j + d2k followed by the rotation a1 + b1i + c1j + d1k.

### Scalar and vector parts

A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c or d is nonzero, is called a vector quaternion. If a + b i + c j + d k is any quaternion, then a is called its scalar part and b i + c j + d k is called its vector part. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space ${\displaystyle \mathbb {R} ^{3}.}$[b]

Hamilton also called vector quaternions right quaternions[27][28] and real numbers (considered as quaternions with zero vector part) scalar quaternions.

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

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

then the formulas for addition and multiplication are

${\displaystyle (r_{1},\ {\vec {v}}_{1})+(r_{2},\ {\vec {v}}_{2})=(r_{1}+r_{2},\ {\vec {v}}_{1}+{\vec {v}}_{2}),}$
${\displaystyle (r_{1},\ {\vec {v}}_{1})(r_{2},\ {\vec {v}}_{2})=(r_{1}r_{2}-{\vec {v}}_{1}\cdot {\vec {v}}_{2},\ r_{1}{\vec {v}}_{2}+r_{2}{\vec {v}}_{1}+{\vec {v}}_{1}\times {\vec {v}}_{2}),}$

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

## Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let ${\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }$ be a quaternion. The conjugate of q is the quaternion ${\displaystyle q^{*}=a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} }$. It is denoted by q, qt, ${\displaystyle {\tilde {q}}}$, or q.[7] Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq) = qp, not pq.

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

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

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is 1/2(p + p) , and the vector part of p is 1/2(pp) .

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

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

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

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

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

${\displaystyle \lVert pq\rVert =\lVert p\rVert \,\lVert q\rVert }$

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

${\displaystyle \det {\Bigl (}{\begin{array}{cc}a+ib&id+c\\id-c&a-ib\end{array}}{\Bigr )}=a^{2}+b^{2}+c^{2}+d^{2},}$

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

This norm makes it possible to define the distance d(p, q) between p and q as the norm of their difference:

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

This makes ${\displaystyle \mathbb {H} }$ a metric space. Addition and multiplication are continuous in the metric topology. Indeed, for any scalar, positive a it holds

${\displaystyle \lVert (p+ap_{1}+q+aq_{1})-(p+q)\rVert =a\lVert p_{1}+q_{1}\rVert \,.}$

Continuity follows from taking a to zero in the limit. Continuity for multiplication holds similarly.

### Unit quaternion

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:

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

Every quaternion has a polar decomposition ${\displaystyle q=\lVert q\rVert \cdot \mathbf {U} q}$.

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

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

This makes it possible to divide two quaternions p and q in two different ways (when q is non-zero). That is, their quotient can be either p q−1 or q−1p ; in general, those products are different, depending on the order of multiplication, except for the special case that p and q are scalar multiples of each other (which includes the case where p = 0). Hence, the notation p/q is ambiguous because it does not specify whether q divides on the left or the right (whether  q−1 multiplies p on its left or its right).

## Algebraic properties

Cayley graph of Q8. The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j.

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

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

Three dimensional graph of Q8. Red, green and blue arrows represent multiplication by i, j, and k, respectively. Multiplication by negative numbers are omitted for clarity.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±i, ±j, ±k} forms a group under multiplication. This non-abelian group is called the quaternion group and is denoted Q8.[29] The real group ring of Q8 is a ring ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]}$ which is also an eight-dimensional vector space over ${\displaystyle \mathbb {R} .}$ It has one basis vector for each element of ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}].}$ The quaternions are isomorphic to the quotient ring of ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]}$ by the ideal generated by the elements 1 + (−1), i + (−i), j + (−j), and k + (−k). Here the first term in each of the differences is one of the basis elements 1, i, j, and k, and the second term is one of basis elements −1, −i, −j, and k, not the additive inverses of 1, i, j, and k.

## Quaternions and the space geometry

The vector part of a quaternion can be interpreted as a coordinate vector in ${\displaystyle \mathbb {R} ^{3};}$ therefore, the algebraic operations of the quaternions reflect the geometry of ${\displaystyle \mathbb {R} ^{3}.}$ Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics.[14]

For the remainder of this section, i, j, and k will denote both the three imaginary[30] basis vectors of ${\displaystyle \mathbb {H} }$ and a basis for ${\displaystyle \mathbb {R} ^{3}.}$ Replacing i by i, j by j, and k by k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

For two vector quaternions p = b1i + c1j + d1k and q = b2i + c2j + d2k their dot product, by analogy to vectors in ${\displaystyle \mathbb {R} ^{3},}$ is

${\displaystyle p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}~.}$

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

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

This is equal to the scalar parts of the products pq, qp, pq, and qp. Note that their vector parts are different.

The cross product of p and q relative to the orientation determined by the ordered basis i, j, and k is

${\displaystyle p\times q=(c_{1}d_{2}-d_{1}c_{2})\mathbf {i} +(d_{1}b_{2}-b_{1}d_{2})\mathbf {j} +(b_{1}c_{2}-c_{1}b_{2})\mathbf {k} \,.}$

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product pq (as quaternions), as well as the vector part of qp. It also has the formula

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

For the commutator, [p, q] = pqqp, of two vector quaternions one obtains

${\displaystyle [p,q]=2p\times q.}$

In general, let p and q be quaternions and write

${\displaystyle p=p_{\text{s}}+p_{\text{v}},}$
${\displaystyle q=q_{\text{s}}+q_{\text{v}},}$

where ps and qs are the scalar parts, and pv and qv are the vector parts of p and q. Then we have the formula

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

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton[31] showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.

Unit quaternions can be identified with rotations in ${\displaystyle \mathbb {R} ^{3}}$ and were called versors by Hamilton.[31] Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.

See Hanson (2005)[32] for visualization of quaternions.

## Matrix representations

Just as complex numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related representations. In the terminology of abstract algebra, these are injective homomorphisms from ${\displaystyle \mathbb {H} }$ to the matrix rings M(2,ℂ) and M(4,ℝ), respectively.

Using 2 × 2 complex matrices, the quaternion a + bi + cj + dk can be represented as

${\displaystyle {\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.}$

Note that the "i" of the complex numbers is distinct from the "i" of the quaternions.

This representation has the following properties:

• Constraining any two of b, c and d to zero produces a representation of complex numbers. For example, setting c = d = 0 produces a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation.
• The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.[33]
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
• By restriction this representation yields an isomorphism between the subgroup of unit quaternions and their image SU(2). Topologically, the unit quaternions are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group SU(2) is important for describing spin in quantum mechanics; see Pauli matrices.
• There is a strong relation between quaternion units and Pauli matrices. Obtain the eight quaternion unit matrices by taking a, b, c and d, set three of them at zero and the fourth at 1 or −1. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for −1. One obtains −1 via i2 = j2 = k2 = i j k = −1; e.g. the last equality is
{\displaystyle {\begin{aligned}ijk=\sigma _{1}\sigma _{2}\sigma _{3}\sigma _{1}\sigma _{2}\sigma _{3}=-1\end{aligned}}}

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

${\displaystyle {\begin{bmatrix}a&-b&-c&-d\\b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{bmatrix}}=a{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}+b{\begin{bmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}+c{\begin{bmatrix}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{bmatrix}}+d{\begin{bmatrix}0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}.}$

However, the representation of quaternions in M(4,ℝ) is not unique. For example, the same quaternion can also be represented as

${\displaystyle {\begin{bmatrix}a&d&-b&-c\\-d&a&c&-b\\b&-c&a&-d\\c&b&d&a\end{bmatrix}}=a{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}+b{\begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{bmatrix}}+c{\begin{bmatrix}0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}}+d{\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}}.}$

There exist 48 distinct matrix representations of this form in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48 sets of quadruples of matrices with these symmetry constraints such that a function sending 1, i, j, and k to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.[34] In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting c = d = 0.

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

× a d b c
a a d −b −c
−d −d a c −b
b b c a d
c c b d a

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

× 1 k i j
1 1 k i j
k k 1 j i
i i j 1 k
j j i k 1

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

## Lagrange’s four-square theorem

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

## Quaternions as pairs of complex numbers

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

Let ${\displaystyle \mathbb {C} ^{2}}$ be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in ${\displaystyle \mathbb {C} ^{2}}$ can be written in terms of the basis elements 1 and j as

${\displaystyle (a+bi)1+(c+di)\mathbf {j} \,.}$

If we define j2 = −1 and i j = −j i, then we can multiply two vectors using the distributive law. Using k as an abbreviated notation for the product i j leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion a + b i + c j + d k . If we write the elements of ${\displaystyle \mathbb {C} ^{2}}$ as ordered pairs and quaternions as quadruples, then the correspondence is

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

## Square roots

### Square roots of −1

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

${\displaystyle a^{2}-b^{2}-c^{2}-d^{2}=-1,}$
${\displaystyle 2ab=0,}$
${\displaystyle 2ac=0,}$
${\displaystyle 2ad=0.}$

To satisfy the last three equations, either a = 0 or b, c, and d are all 0. The latter is impossible because a is a real number and the first equation would imply that a2 = −1. Therefore, a = 0 and b2 + c2 + d2 = 1. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere.

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

#### As a union of complex planes

Each pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If q2 = −1, then the copy is determined by the function

${\displaystyle a+b{\sqrt {-1\,}}\mapsto a+bq\,.}$

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

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

${\displaystyle q=q_{s}+{\vec {q}}_{v}.}$

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

${\displaystyle q=q_{s}+\lVert {\vec {q}}_{v}\rVert \cdot \mathbf {U} {\vec {q}}_{v}=q_{s}+{\frac {q_{v}}{\|q_{v}\|}}.}$

(Note that this is not the same as ${\displaystyle q_{s}+\lVert q\rVert \cdot \mathbf {U} q}$.) The versor of the vector part of q, ${\displaystyle \mathbf {U} {\vec {q}}_{v}}$, is a right versor with –1 as its square. A straightforward verification shows that

${\displaystyle a+b{\sqrt {-1\,}}\mapsto a+b\mathbf {U} {\vec {q}}_{v}}$

defines an injective homomorphism of normed algebras from ${\displaystyle \mathbb {C} }$ into the quaternions. Under this homomorphism, q is the image of the complex number ${\displaystyle q_{s}+\lVert {\vec {q}}_{v}\rVert i}$.

As ${\displaystyle \mathbb {H} }$ is the union of the images of all these homomorphisms, this allows viewing the quaternions as a union of complex planes intersecting on the real line. Each of these complex planes contains exactly one pair of antipodal points of the sphere of square roots of minus one.

#### Commutative subrings

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

### Square roots of arbitrary quaternions

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

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

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

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

## Functions of a quaternion variable

The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3 dimensions. This Julia set is cross sectioned at the x y plane.

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4 dimensional space.[36]

### Exponential, logarithm, and power functions

Given a quaternion,

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

the exponential is computed as[37]

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

and the logarithm is[37]

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

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

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

where the angle ${\displaystyle \varphi }$[e]

${\displaystyle a=\|q\|\cos(\varphi )}$

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

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

Any unit quaternion may be expressed in polar form as:

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

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

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

### Geodesic norm

The geodesic distance dg(p, q) between unit quaternions p and q is defined as:

${\displaystyle d_{\text{g}}(p,q)=\lVert \ln(p^{-1}q)\rVert .}$[39]

and amounts to the absolute value of half the angle subtended by p and q along a great arc of the S3 sphere. This angle can also be computed from the quaternion dot product without the logarithm as:

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

## Three-dimensional and four-dimensional rotation groups

The word "conjugation", besides the meaning given above, can also mean taking an element a to r a r−1 where r is some non-zero quaternion. All elements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)

Thus the multiplicative group of non-zero quaternions acts by conjugation on the copy of ${\displaystyle \mathbb {R} ^{3}}$ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(φ) is a rotation by an angle 2φ, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

The set of all unit quaternions (versors) forms a 3-sphere S3 and a group (a Lie group) under multiplication, double covering the group SO(3,ℝ) of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See the plate trick.

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

The versors' group is isomorphic to SU(2), the group of complex unitary 2×2 matrices of determinant 1.

Let A be the set of quaternions of the form a + b i + c j + d k where a, b, c, and d are either all integers or all half-integers. The set A is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a regular 24 cell with Schläfli symbol {3,4,3}. They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600 cell with Schläfli symbol {3,3,5} can be taken as the unit icosians, corresponding to the double cover of the rotational symmetry group of the regular icosahedron. The double cover of the rotational symmetry group of the regular octahedron corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell.

## Quaternion algebras

The Quaternions can be generalized into further algebras called quaternion algebras. Take F to be any field with characteristic different from 2, and a and b to be elements of F; a four-dimensional unitary associative algebra can be defined over F with basis 1, i, j, and i j, where i2 = a, j2 = b and i j = −j i (so (i j)2 = −a b).

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

## Quaternions as the even part of Cl3,0(ℝ)

The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part ${\displaystyle \operatorname {Cl} _{3,0}^{+}(\mathbb {R} )}$ of the Clifford algebra ${\displaystyle \operatorname {Cl} _{3,0}(\mathbb {R} ).}$ This is an associative multivector algebra built up from fundamental basis elements σ1, σ2, σ3 using the product rules

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1,}$
${\displaystyle \sigma _{i}\sigma _{j}=-\sigma _{j}\sigma _{i}\qquad (j\neq i).}$

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

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

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

${\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}}$

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

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

In fact, the two are identical, if we make the identification

${\displaystyle \mathbf {k} =\sigma _{2}\sigma _{1}\,,\quad \mathbf {i} =\sigma _{3}\sigma _{2}\,,\quad \mathbf {j} =\sigma _{1}\sigma _{3}\,,}$

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

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

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to bivectors – quantities with magnitude and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions σ1 and σ2, there is only one bivector basis element σ1σ2, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ1σ2, σ2σ3, σ3σ1, so three imaginaries.

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

There are several advantages for placing quaternions in this wider setting:[40]

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

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

## Brauer group

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

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

## Quotations

I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc.

— William Rowan Hamilton[41]

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

— William Rowan Hamilton[42][full citation needed]

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

— W. Thompson, Lord Kelvin (1892)[citation needed]

I came later to see that, as far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work.

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

— Ludwik Silberstein (1924)[44][full citation needed]

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

— Simon L. Altmann (1986)[45]

## Notes

1. ^ A more personal view of quaternions was written by Joachim Lambek in 1995. He wrote in his essay If Hamilton had prevailed: quaternions in physics: "My own interest as a graduate student was raised by the inspiring book by Silberstein". He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics." Lambek, J. (1995). "If Hamilton had prevailed: Quaternions in physics". Math. Intelligencer. Vol. 17 no. 4. pp. 7–15. doi:10.1007/BF03024783.
2. ^ It is important to note that the vector part of a quaternion is, in truth, an "axial" vector or "pseudovector", not an ordinary or "polar" vector, as was formally proven by Altmann (1986).[26] A polar vector can be represented in calculations (for example, for rotation by a quaternion "similarity transform") by a pure imaginary quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.
3. ^ In comparison, the real numbers ${\displaystyle \mathbb {R} }$ have dimension 1, the complex numbers ${\displaystyle \mathbb {C} }$ have dimension 2, and the octonions ${\displaystyle \mathbb {O} }$ have dimension 8.
4. ^ The identification of the square roots of minus one in ${\displaystyle \mathbb {H} }$ was given by Hamilton[35] but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his three-page exposition included in Historical Topics in Algebra (page 39) published by the National Council of Teachers of Mathematics. More recently, the sphere of square roots of minus one is described in Ian R. Porteous's book Clifford Algebras and the Classical Groups (Cambridge, 1995) in proposition 8.13 on page 60.
5. ^ Books on applied mathematics, such as Corke (2017)[38] often use different notation with φ := 1/2θ — that is, another variable θ = 2φ.

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28. ^ Hardy (1881). "Elements of Quaternions". Science. library.cornell.edu. 2 (75): 65. doi:10.1126/science.os-2.75.564. PMID 17819877.
29. ^ "quaternion group". Wolframalpha.com.
30. ^ Gibbs, J. Willard; Wilson, Edwin Bidwell (1901). Vector Analysis. Yale University Press. p. 428. right tensor dyadic
31. ^ a b Hamilton, W.R. (1844–1850). "On quaternions or a new system of imaginaries in algebra". David R. Wilkins collection. Philosophical Magazine. Trinity College, Dublin.
32. ^ "Visualizing Quaternions". Morgan-Kaufmann/Elsevier. 2005.
33. ^ "[no title cited; determinant evaluation]". Wolframalpha.com.
34. ^ Farebrother, Richard William; Groß, Jürgen; Troschke, Sven-Oliver (2003). "Matrix representation of quaternions". Linear Algebra and Its Applications. 362: 251–255. doi:10.1016/s0024-3795(02)00535-9.
35. ^ Hamilton, W.R. (1899). Elements of Quaternions (2nd ed.). p. 244. ISBN 1-108-00171-8.
36. ^ "[no title cited]" (PDF). bridgesmathart.org. archive. Retrieved 19 August 2018.
37. ^ a b Särkkä, Simo (June 28, 2007). "Notes on Quaternions" (PDF). Lce.hut.fi. Archived from the original (PDF) on 5 July 2017.
38. ^ Corke, Peter (2017). Robotics, Vision, and Control – Fundamental Algorithms in MATLAB®. Springer. ISBN 978-3-319-54413-7.
39. ^ Park, F.C.; Ravani, Bahram (1997). "Smooth invariant interpolation of rotations". ACM Transactions on Graphics. 16 (3): 277–295. doi:10.1145/256157.256160. S2CID 6192031.
40. ^ "Quaternions and Geometric Algebra". geometricalgebra.net. Retrieved 2008-09-12. See also: Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007). Geometric Algebra for Computer Science. Morgan Kaufmann. ISBN 978-0-12-369465-2.
41. ^ Quoted from a letter from Tait to Cayley.[citation needed]
42. ^ Graves, R.P. Life of Sir William Rowan Hamilton.
43. ^ Heaviside, Oliver (1893). Electromagnetic Theory. I. London, UK: The Electrician Printing and Publishing Company. pp. 134–135.
44. ^ Ludwik Silberstein (1924). Notes on preparing the second edition of his Theory of Relativity.
45. ^ Altmann, Simon L. (1986). Rotations, quaternions, and double groups. Clarendon Press. ISBN 0-19-855372-2. LCCN 85013615.

### Books and publications

• Hamilton, William Rowan (1844). "On quaternions, or on a new system of imaginaries in algebra". Philosophical Magazine. 25 (3): 489–495. doi:10.1080/14786444408645047.*
• Hamilton, William Rowan (1853), "Lectures on Quaternions". Royal Irish Academy.
• Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
• Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co..
• Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.] : The University Press.
• Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". Clarendon Press, Oxford.
• Tait, Peter Guthrie (1886), ""Archived copy". Archived from the original on August 8, 2014. Retrieved June 26, 2005.CS1 maint: archived copy as title (link) CS1 maint: unfit URL (link)". M.A. Sec. R.S.E. Encyclopædia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript file)
• Joly, Charles Jasper (1905). A manual of quaternions. Macmillan. LCCN 05036137.
• Macfarlane, Alexander (1906). Vector analysis and quaternions (4th ed.). Wiley. LCCN 16000048.
• Chisholm, Hugh, ed. (1911). "Algebra" . Encyclopædia Britannica (11th ed.). Cambridge University Press. (See section on quaternions.)
• Finkelstein, David; Jauch, Josef M.; Schiminovich, Samuel; Speiser, David (1962). "Foundations of quaternion quantum mechanics". J. Math. Phys. 3 (2): 207–220. doi:10.1063/1.1703794.
• Du Val, Patrick (1964). Homographies, quaternions, and rotations. Oxford mathematical monographs. Clarendon Press. LCCN 64056979.
• Crowe, Michael J. (1967), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
• Altmann, Simon L. (1989). "Hamilton, Rodrigues, and the Quaternion Scandal". Mathematics Magazine. 62 (5): 291–308. doi:10.1080/0025570X.1989.11977459.
• Adler, Stephen L. (1995). Quaternionic quantum mechanics and quantum fields. International series of monographs on physics. 88. Oxford University Press. ISBN 0-19-506643-X. LCCN 94006306.
• Ward, J.P. (1997). Quaternions and Cayley Numbers: Algebra and Applications. Kluwer Academic. ISBN 0-7923-4513-4.
• Kantor, I.L.; Solodnikov, A.S. (1989). Hypercomplex numbers, an elementary introduction to algebras. Springer-Verlag. ISBN 0-387-96980-2.
• Gürlebeck, Klaus; Sprössig, Wolfgang (1997). Quaternionic and Clifford calculus for physicists and engineers. Mathematical methods in practice. 1. Wiley. ISBN 0-471-96200-7. LCCN 98169958.
• Kuipers, Jack (2002). Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality. Princeton University Press. ISBN 0-691-10298-8.
• Conway, John Horton; Smith, Derek A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A.K. Peters. ISBN 1-56881-134-9. (review).
• Jack, P.M. (2003). "Physical space as a quaternion structure, I: Maxwell equations. A brief Note". arXiv:math-ph/0307038.
• Kravchenko, Vladislav (2003). Applied Quaternionic Analysis. Heldermann Verlag. ISBN 3-88538-228-8.
• Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. 1. Springer. ISBN 1-4020-2690-0.
• Hanson, Andrew J. (2006). Visualizing Quaternions. Elsevier. ISBN 0-12-088400-3.
• Binz, Ernst; Pods, Sonja (2008). "1. The Skew Field of Quaternions". Geometry of Heisenberg Groups. American Mathematical Society. ISBN 978-0-8218-4495-3.
• Doran, Chris J.L.; Lasenby, Anthony N. (2003). Geometric Algebra for Physicists. Cambridge University Press. ISBN 978-0-521-48022-2.
• Vince, John A. (2008). Geometric Algebra for Computer Graphics. Springer. ISBN 978-1-84628-996-5.
• For molecules that can be regarded as classical rigid bodies molecular dynamics computer simulation employs quaternions. They were first introduced for this purpose by Evans, D.J. (1977). "On the Representation of Orientation Space". Mol. Phys. 34 (2): 317–325. doi:10.1080/00268977700101751.
• Zhang, Fuzhen (1997). "Quaternions and Matrices of Quaternions". Linear Algebra and Its Applications. 251: 21–57. doi:10.1016/0024-3795(95)00543-9.
• Ron Goldman (2010). Rethinking Quaternions: Theory and Computation. Morgan & Claypool. ISBN 978-1-60845-420-4.
• Eves, Howard (1976), An Introduction to the History of Mathematics (4th ed.), New York: Holt, Rinehart and Winston, ISBN 0-03-089539-1

• "Quaternion Notices". Notices and materials related to Quaternion conference presentations
• "Quaternion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Frequently Asked Questions". Matrix and Quaternion. 1.21.
• Sweetser, Doug. "Doing Physics with Quaternions".
• Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
• Gsponer, Andre; Hurni, Jean-Pierre (2002). "The Physical Heritage of Sir W. R. Hamilton". arXiv:math-ph/0201058.
• Wilkins, D.R. "Hamilton's Research on Quaternions".
• Grossman, David J. "Quaternion Julia Fractals".] 3D Raytraced Quaternion Julia Fractals
• "Quaternion Math and Conversions". Great page explaining basic math with links to straight forward rotation conversion formulae.
• Mathews, John H. "Bibliography for Quaternions". Archived from the original on 2006-09-02.
• "Quaternion powers". GameDev.net.
• Hanson, Andrew. "Visualizing Quaternions home page". Archived from the original on 2006-11-05.
• Karney, Charles F.F. (January 2007). "Quaternions in molecular modeling". J. Mol. Graph. Mod. 25 (5): 595–604. arXiv:physics/0506177. doi:10.1016/j.jmgm.2006.04.002. PMID 16777449. S2CID 6690718.
• Mebius, Johan E. (2005). "A matrix-based proof of the quaternion representation theorem for four-dimensional rotations". arXiv:math/0501249.
• Mebius, Johan E. (2007). "Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations". arXiv:math/0701759.
• "Hamilton Walk". Department of Mathematics, NUI Maynooth.
• "Using Quaternions to represent rotation". OpenGL:Tutorials. Archived from the original on 2007-12-15.
• David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
• Martinez, Alberto. "Negative Math, How Mathematical Rules Can Be Positively Bent". Department of History, University of Texas. Archived from the original on 2011-09-24.
• Stahlke, D. "Quaternions in Classical Mechanics" (PDF).
• Morier-Genoud, Sophie; Ovsienko, Valentin (2008). "Well, Papa, can you multiply triplets?". arXiv:0810.5562 [math.AC]. describes how the quaternions can be made into a skew-commutative algebra graded by Z/2 × Z/2 × Z/2.
• Joyce, Helen (November 2004). "Curious Quaternions". hosted by John Baez.
• Ibanez, Luis. "Tutorial on Quaternions. Part I" (PDF). Archived from the original (PDF) on 2012-02-04. Retrieved 2011-12-05. Part II (PDF; using Hamilton's terminology, which differs from the modern usage)
• Ghiloni, R.; Moretti, V.; Perotti, A. (2013). "Continuous slice functional calculus in quaternionic Hilbert spaces". Rev. Math. Phys. 25 (4): 1350006–126. arXiv:1207.0666. Bibcode:2013RvMaP..2550006G. doi:10.1142/S0129055X13500062. S2CID 119651315.
Ghiloni, R.; Moretti, V.; Perotti, A. (2017). "Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures". Rev. Math. Phys. 29: 1750034. arXiv:1602.02661. doi:10.1142/S0129055X17500349. two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
• Quaternions the Android app shows the quaternion corresponding to the orientation of the device.
• Rotating Objects Using Quaternions article speaking to the use of Quaternions for rotation in video games/computer graphics.

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CBSEtips.in

Thursday, 28 January 2021

CBSE Class 12 Maths - MCQ and Online Tests - Unit 13 - Probability

CBSE Class 12 Maths – MCQ and Online Tests – Unit 13 – Probability

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

CBSE Class 12 Maths – MCQ and Online Tests – Unit 13 – Probability

Question 1.
If P (a) = $$\frac{3}{8}$$, P(b) = $$\frac{1}{2}$$ and P(A∩B) = $$\frac{1}{4}$$ then P($$\frac{A’}{B’}$$) =
(a) $$\frac{1}{4}$$
(b) $$\frac{1}{3}$$
(c) $$\frac{3}{4}$$
(d) $$\frac{3}{8}$$

Answer: (b) $$\frac{1}{3}$$

Question 2.
If A and B are two events such that P(a) ≠ 0 and P($$\frac{B}{A}$$) = 1, then
(a) B ⊂ A
(b) B = φ
(c) A ⊂ B
(d) A ∩ B = φ

Question 3.
If A and B are any two events such that P(a) + P(b) – P(A∩B) = P(a) then
(a) P($$\frac{B}{A}$$) = 1
(b) P($$\frac{B}{A}$$) = 0
(c) P($$\frac{A}{B}$$) = 1
(d) P($$\frac{A}{B}$$) = 0

Answer: (c) P($$\frac{A}{B}$$) = 1

Question 4.
If A and B are events such that P (A∪B) = $$\frac{3}{4}$$. P(A∩B) = $$\frac{1}{4}$$, P(a) = $$\frac{2}{3}$$ then P(AB) is
(a) $$\frac{3}{8}$$
(b) $$\frac{5}{8}$$
(c) $$\frac{5}{12}$$
(d) $$\frac{1}{4}$$

Answer: (b) $$\frac{5}{8}$$

Question 5.
If one card is drawn out of 52 playing cards, the probability that it is an dice is
(a) $$\frac{1}{26}$$
(b) $$\frac{1}{13}$$
(c) $$\frac{1}{52}$$
(d) $$\frac{1}{4}$$

Answer: (b) $$\frac{1}{13}$$

Question 6.
The chance of getting a doublet with 2 dice is
(a) $$\frac{2}{3}$$
(b) $$\frac{1}{6}$$
(c) $$\frac{5}{6}$$
(d) $$\frac{5}{36}$$

Answer: (b) $$\frac{1}{6}$$

Question 7.
Two number are chosen, one by one without replacement from the set of number A = {1, 2, 3, 4, 5, 6} then the probability that minimum value of two number chosen is less than 4 is
(a) $$\frac{14}{15}$$
(b) $$\frac{1}{15}$$
(c) $$\frac{1}{5}$$
(d) $$\frac{8}{5}$$

Answer: (b) $$\frac{1}{15}$$

Question 8.
If P(x) = $$\frac{2}{15}$$; y = 1, 2, 3, 4, 5, 0 otherwise then P|x = 1 or 2| is
(a) $$\frac{1}{15}$$
(b) $$\frac{2}{15}$$
(c) $$\frac{1}{5}$$
(d) None of these

Answer: (c) $$\frac{1}{5}$$

Question 9.
The probability of an event is $$\frac{3}{7}$$. Then odd against the event is
(a) 4 : 3
(b) 7 : 3
(c) 3 : 7
(d) 3 : 4

Question 10.
Five horse are in a race. Mr. A select two of the horses at random and best on them. The probability that Mr. A select the winning horses is
(a) $$\frac{4}{5}$$
(b) $$\frac{3}{5}$$
(c) $$\frac{1}{5}$$
(d) $$\frac{2}{5}$$

Answer: (d) $$\frac{2}{5}$$

Question 11.
If A and B are two independent events, then
(a) P(A∩B) = P(a) × P(b)
(b) P(AB) = 1 – P(A’) P(B’)
(c) P(AB) = 1 + P (A’) P(B’) P(A’)
(d) P (AB) = $$\frac{P(A’)}{P(B’)}$$

Answer: (a) P(A∩B) = P(a) × P(b)

Question 12.
The probability of India w inning a test match against. West Indies is $$\frac{1}{2}$$. Assuming independence from match to match the probability that in a match series India second win occurs at the third test is
(a) $$\frac{1}{6}$$
(b) $$\frac{1}{4}$$
(c) $$\frac{1}{2}$$
(d) $$\frac{2}{3}$$

Answer: (b) $$\frac{1}{4}$$

Question 13.
P(A∩B) = $$\frac{3}{8}$$, P(b) = $$\frac{1}{2}$$ and P(a) = $$\frac{1}{4}$$ then P($$\frac{B’}{A’}$$) =
(a) $$\frac{3}{5}$$
(b) $$\frac{5}{8}$$
(c) $$\frac{3}{8}$$
(d) $$\frac{5}{6}$$

Answer: (d) $$\frac{5}{6}$$

Question 14.
Three distinct numbers.are selected from First 100 natural numbers. The probability divisible by 2 and 3 is
(a) $$\frac{9}{25}$$
(b) $$\frac{4}{35}$$
(c) $$\frac{4}{55}$$
(d) $$\frac{4}{1155}$$

Answer: (d) $$\frac{4}{1155}$$

Question 15.
A pair of dice are rolled. The probability of obtaining an even prime number on each die is
(a) $$\frac{1}{36}$$
(b) $$\frac{1}{12}$$
(c) $$\frac{1}{6}$$
(d) 0

Answer: (a) $$\frac{1}{36}$$

Question 16.
If P(a) = $$\frac{3}{8}$$, P(b) = $$\frac{1}{3}$$ and P(A∩B) = — then P (A’ ∩B’)
(a) $$\frac{13}{24}$$
(b) $$\frac{13}{8}$$
(c) $$\frac{13}{9}$$
(d) $$\frac{13}{4}$$

Answer: (a) $$\frac{13}{24}$$

Question 17.
The probability that A speaks truth is $$\frac{4}{5}$$ while this probability for B is $$\frac{3}{4}$$. The probability that they contradict each others when asked to speak ana fact is
(a) $$\frac{7}{20}$$
(b) $$\frac{1}{5}$$
(c) $$\frac{3}{20}$$
(d) $$\frac{4}{5}$$

Answer: (d) $$\frac{4}{5}$$

Question 18.
If P(a) = $$\frac{7}{10}$$ P(b) = $$\frac{7}{10}$$ and P(A∪B) = $$\frac{7}{10}$$ then P (B|A) + P(A|B) equals
(a) $$\frac{1}{4}$$
(b) $$\frac{1}{3}$$
(c) $$\frac{5}{12}$$
(d) $$\frac{7}{12}$$

Answer: (d) $$\frac{7}{12}$$

Question 19.
Two dice are tossed once. The probability of getting an even number at the first dice ora total of 8 is
(a) $$\frac{1}{36}$$
(b) $$\frac{3}{36}$$
(c) $$\frac{11}{36}$$
(d) $$\frac{5}{9}$$

Answer: (d) $$\frac{5}{9}$$

Question 20.
A pair of dice are rolled. The probability of obtaining an even prime number on each dice is
(a) $$\frac{1}{36}$$
(b) $$\frac{1}{12}$$
(c) $$\frac{1}{6}$$
(d) 0

Answer: (a) $$\frac{1}{36}$$

Question 21.
If A, B are two events associated with same random experiment such that P(a) = 0.4, P(b) = 0.8 and P(B/A) = 0.6 then P(A/B) is
(a) 0.3
(b) 0.4
(c) 0.5
(d) 0.6

Question 22.
If P(a) = $$\frac{3}{8}$$, P(b) = $$\frac{5}{8}$$, P(A∪B) = $$\frac{3}{4}$$ then p($$\frac{B}{A}$$) is
(a) $$\frac{3}{47}$$
(b) $$\frac{5}{49}$$
(c) $$\frac{2}{3}$$
(d) $$\frac{1}{4}$$

Answer: (c) $$\frac{2}{3}$$

Question 23.
If A and B are two events such that P(a) ≠ 0 and P($$\frac{B}{A}$$) = 1 then
(a) P($$\frac{A}{B}$$) = 1
(b) P($$\frac{B}{A}$$) = 1
(c) P($$\frac{A}{B}$$) = 0
(d) P($$\frac{B}{A}$$) = 0

Answer: (b) P($$\frac{B}{A}$$) = 1

Question 24.
An urn contain’s balls of which 3 are red, 4 are blue and 2 are green, 3 balls are drawn at random without replacement from the urn. The probability that the 3 balls haye different colours is
(a) $$\frac{1}{3}$$
(b) $$\frac{2}{7}$$
(c) $$\frac{1}{21}$$
(d) $$\frac{2}{23}$$

Answer: (b) $$\frac{2}{7}$$

Question 25.
An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. A consists 4 outcomes, the number of outcomes that B must have so that A and B are independent is
(a) 2, 4 or 8
(b) 36 or 9
(c) 4 or 8
(d) 5 or 10

Question 26.
The mean and the variance of binomial distribution are 4 and 2, respectively. Then the probability of 2 success
(a) $$\frac{128}{256}$$
(b) $$\frac{219}{256}$$
(c) $$\frac{7}{64}$$
(d) $$\frac{28}{256}$$

Answer: (c) $$\frac{7}{64}$$

Question 27.
If P(a) = $$\frac{4}{5}$$ and P(A∩B) = $$\frac{7}{10}$$, then P(B/A) is equal
(a) $$\frac{1}{10}$$
(b) $$\frac{1}{8}$$
(c) $$\frac{7}{8}$$
(d) $$\frac{17}{20}$$

Answer: (d) $$\frac{17}{20}$$

Question 28.
If P(A∩B) = $$\frac{7}{10}$$ and P(b) = $$\frac{17}{20}$$, then P(A|B) equals
(a) $$\frac{14}{17}$$
(b) $$\frac{17}{20}$$
(c) $$\frac{7}{8}$$
(d) $$\frac{1}{8}$$

Answer: (a) $$\frac{14}{17}$$

Question 29.
If A and B are two independent events with P(a) = $$\frac{3}{5}$$ and P (b) = $$\frac{4}{9}$$, then P (A’∩B’) equals
(a) $$\frac{4}{15}$$
(b) $$\frac{8}{15}$$
(c) $$\frac{1}{3}$$
(d) $$\frac{2}{9}$$

Answer: (d) $$\frac{2}{9}$$

Question 30.
Let A and B two event such that P(a) = $$\frac{3}{8}$$, P(b) = $$\frac{5}{8}$$ and P(A∪B) = $$\frac{3}{4}$$. Then P(A|B).P(A’|B) is equal to
(a) $$\frac{2}{5}$$
(b) $$\frac{3}{8}$$
(c) $$\frac{3}{20}$$
(d) $$\frac{6}{25}$$
Ans. (d)

Answer: (d) $$\frac{6}{25}$$

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# Stop Thinking, Just Do!

Sung-Soo Kim's Blog

# Diving into Spark Streaming’s Execution Model

With so many distributed stream processing engines available, people often ask us about the unique benefits of Spark Streaming. From early on, Apache Spark has provided an unified engine that natively supports both batch and streaming workloads. This is different from other systems that either have a processing engine designed only for streaming, or have similar batch and streaming APIs but compile internally to different engines. Spark’s single execution engine and unified programming model for batch and streaming lead to some unique benefits over other traditional streaming systems. In particular, four major aspects are:

• Fast recovery from failures and stragglers
• Better load balancing and resource usage
• Combining of streaming data with static datasets and interactive queries
• Native integration with advanced processing libraries (SQL, machine learning, graph processing)

In this post, we outline Spark Streaming’s architecture and explain how it provides the above benefits. We also discuss some of the interesting ongoing work in the project that leverages the execution model.

## Stream Processing Architectures – The Old and the New

At a high level, modern distributed stream processing pipelines execute as follows:

1. Receive streaming data from data sources (e.g. live logs, system telemetry data, IoT device data, etc.) into some data ingestion system like Apache Kafka, Amazon Kinesis, etc.
2. Process the data in parallel on a cluster. This is what stream processing engines are designed to do, as we will discuss in detail next.
3. Output the results out to downstream systems like HBase, Cassandra, Kafka, etc.

To process the data, most traditional stream processing systems are designed with a continuous operator model, which works as follows:

• There is a set of worker nodes, each of which run one or more continuous operators.
• Each continuous operator processes the streaming data one record at a time and forwards the records to other operators in the pipeline.
• There are “source” operators for receiving data from ingestion systems, and “sink” operators that output to downstream systems.

Figure 1: Architecture of traditional stream processing systems

Continuous operators are a simple and natural model. However, with today’s trend towards larger scale and more complex real-time analytics, this traditional architecture has also met some challenges. We designed Spark Streaming to satisfy the following requirements:

• Fast failure and straggler recovery – With greater scale, there is a higher likelihood of a cluster node failing or unpredictably slowing down (i.e. stragglers). The system must be able to automatically recover from failures and stragglers to provide results in real time. Unfortunately, the static allocation of continuous operators to worker nodes makes it challenging for traditional systems to recover quickly from faults and stragglers.

• Load balancing – Uneven allocation of the processing load between the workers can cause bottlenecks in a continuous operator system. This is more likely to occur in large clusters and dynamically varying workloads. The system needs to be able to dynamically adapt the resource allocation based on the workload.

• Unification of streaming, batch and interactive workloads – In many use cases, it is also attractive to query the streaming data interactively (after all, the streaming system has it all in memory), or to combine it with static datasets (e.g. pre-computed models). This is hard in continuous operator systems as they are not designed to the dynamically introduce new operators for ad-hoc queries. This requires a single engine that can combine batch, streaming and interactive queries.

• Advanced analytics like machine learning and SQL queries – More complex workloads require continuously learning and updating data models, or even querying the “latest” view of streaming data with SQL queries. Again, having a common abstraction across these analytic tasks makes the developer’s job much easier.

To address these requirements, Spark Streaming uses a new architecture called discretized streams that directly leverages the rich libraries and fault tolerance of the Spark engine.

## Architecture of Spark Streaming: Discretized Streams

Instead of processing the streaming data one record at a time, Spark Streaming discretizes the streaming data into tiny, sub-second micro-batches. In other words, Spark Streaming’s Receivers accept data in parallel and buffer it in the memory of Spark’s workers nodes. Then the latency-optimized Spark engine runs short tasks (tens of milliseconds) to process the batches and output the results to other systems. Note that unlike the traditional continuous operator model, where the computation is statically allocated to a node, Spark tasks are assigned dynamically to the workers based on the locality of the data and available resources. This enables both better load balancing and faster fault recovery, as we will illustrate next.

In addition, each batch of data is a Resilient Distributed Dataset (RDD), which is the basic abstraction of a fault-tolerant dataset in Spark. This allows the streaming data to be processed using any Spark code or library.

Figure 2: Spark Streaming Architecture

## Benefits of Discretized Stream Processing

Let’s see how this architecture allows Spark Streaming to achieve the goals we set earlier.

Dividing the data into small micro-batches allows for fine-grained allocation of computations to resources. For example, consider a simple workload where the input data stream needs to partitioned by a key and processed. In the traditional record-at-a-time approach taken by most other systems, if one of the partitions is more computationally intensive than the others, the node statically assigned to process that partition will become a bottleneck and slow down the pipeline. In Spark Streaming, the job’s tasks will be naturally load balanced across the workers — some workers will process a few longer tasks, others will process more of the shorter tasks.

### Fast failure and straggler recovery

In case of node failures, traditional systems have to restart the failed continuous operator on another node and replay some part of the data stream to recompute the lost information. Note that only one node is handling the recomputation, and the pipeline cannot proceed until the new node has caught up after the replay. In Spark, the computation is already discretized into small, deterministic tasks that can run anywhere without affecting correctness. So failed tasks can be relaunched in parallel on all the other nodes in the cluster, thus evenly distributing all the recomputations across many nodes, and recovering from the failure faster than the traditional approach.

Figure 4: Faster failure recovery with redistribution of computation

### Unification of batch, streaming and interactive analytics

The key programming abstraction in Spark Streaming is a DStream, or distributed stream. Each batch of streaming data is represented by an RDD, which is Spark’s concept for a distributed dataset. Therefore a DStream is just a series of RDDs. This common representation allows batch and streaming workloads to interoperate seamlessly. Users can apply arbitrary Spark functions on each batch of streaming data: for example, it’s easy to join a DStream with a precomputed static dataset (as an RDD).

// Create data set from Hadoop file

// Join each batch in stream with the dataset
batchRDD.join(dataset).filter(...)
}

Since the batches of streaming data are stored in the Spark’s worker memory, it can be interactively queried on demand. For example, you can expose all the streaming state through the Spark SQL JDBC server, as we will show in the next section. This kind of unification of batch, streaming and interactive workloads is very simple in Spark, but hard to achieve in systems without a common abstraction for these workloads.

### Advanced analytics like machine learning and interactive SQL

Spark interoperability extends to rich libraries like MLlib (machine learning), SQL, DataFrames, and GraphX. Let’s explore a few use cases:

#### Streaming + SQL and DataFrames

RDDs generated by DStreams can be converted to DataFrames (the programmatic interface to Spark SQL), and queried with SQL. For example, using Spark SQL’s JDBC server, you can expose the state of the stream to any external application that talks SQL.

val hiveContext = new HiveContext(sparkContext)
...
wordCountsDStream.foreachRDD { rdd =>
// Convert RDD to DataFrame and register it as a SQL table
val wordCountsDataFrame = rdd.toDF("word”, “count")
wordCountsDataFrame.registerTempTable("word_counts")
}
...
// Start the JDBC server
HiveThriftServer2.startWithContext(hiveContext)

Then you can interactively query the continuously updated “word_counts” table through the JDBC server, using the beeline client that ships with Spark, or tools like Tableau.

1: jdbc:hive2://localhost:10000> show tables;
+--------------+--------------+
|  tableName   | isTemporary  |
+--------------+--------------+
| word_counts  | true         |
+--------------+--------------+
1 row selected (0.102 seconds)
1: jdbc:hive2://localhost:10000> select * from word_counts;
+-----------+--------+
|   word    | count  |
+-----------+--------+
| 2015      | 264    |
| PDT       | 264    |
| 21:45:41  | 27     |

#### Streaming + MLlib

Machine learning models generated offline with MLlib can applied on streaming data. For example, the following code trains a KMeans clustering model with some static data and then uses the model to classify events in a Kafka data stream.

// Learn model offline
val model = KMeans.train(dataset, ...)

// Apply model online on stream
val kafkaStream = KafkaUtils.createDStream(...)
kafkaStream.map { event => model.predict(featurize(event)) }

We demonstrated this offline-learning-online-prediction at our Spark Summit 2014 Databricks demo. Since then, we have also added streaming machine learning algorithms in MLLib that can continuously train from a labelled data stream. Other Spark libraries can also easily be called from Spark Streaming.

## Performance

Given the unique design of Spark Streaming, how fast does it run? In practice, Spark Streaming’s ability to batch data and leverage the Spark engine leads to comparable or higher throughput to other streaming systems. In terms of latency, Spark Streaming can achieve latencies as low as a few hundred milliseconds. Developers sometimes ask whether the micro-batching inherently adds too much latency. In practice, batching latency is only a small component of end-to-end pipeline latency. For example, many applications compute results over a sliding window, and even in continuous operator systems, this window is only updated periodically (e.g. a 20 second window that slides every 2 seconds). Many pipelines collect records from multiple sources and wait for a short period to process delayed or out-of-order data. Finally, any automatic triggering algorithm tends to wait for some time period to fire a trigger. Therefore, compared to the end-to-end latency, batching rarely adds significant overheads. In fact, the throughput gains from DStreams often means that you need fewer machines to handle the same workload.

## Future Directions for Spark Streaming

Spark Streaming is one of the most widely used components in Spark, and there is a lot more coming for streaming users down the road. Some of the highest priority items our team is working on are discussed below. You can expect these in the next few releases of Spark:

• Backpressure – Streaming workloads can often have bursts of data (e.g. sudden spike in tweets during the Oscars) and the processing system must be able to handle them gracefully. In the upcoming Spark 1.5 release (next month), Spark will be adding better backpressure mechanisms that allow Spark Streaming dynamically control the ingestion rate for such bursts. This feature represents joint work between us at Databricks and engineers at Typesafe.

• Dynamic scaling – Controlling the ingestion rate may not be sufficient to handle longer terms variations in data rates (e.g. sustained higher tweet rate during the day than night). Such variations can be handled by dynamically scaling the cluster resource based on the processing demands. This is very easy to do within the Spark Streaming architecture — since the computation is already divided into small tasks, they can be dynamically redistributed to a larger cluster if more nodes are acquired from the cluster manager (YARN, Mesos, Amazon EC2, etc). We plan to add support for automatic dynamic scaling.

• Event time and out-of-order data– In practice, users sometimes have records that are delivered out of order, or with a timestamp that differs from the time of ingestion. Spark streaming will support “event time” by allowing user-defined time extraction function. This will include a slack duration for late or out-of-order data.

• UI enhancements – Finally, we want to make it easy for developers to debug their streaming applications. For this purpose, in Spark 1.4, we added new visualizations to the streaming Spark UI that let developers closely monitor the performance of their application. In Spark 1.5, we are further improving this by showing more input information like Kafka offsets processed in each batch.

To learn more about Spark Streaming, read the official programming guide, or the Spark Streaming research paper that introduces its execution and fault tolerance model.


text:
Math Help - Trigo- product

1. Trigo- product

Solve in R this equation :

2. $sin x cos x tan x=1$
$sin x cos x\frac{sin x}{cos x}=1$
$sin^2x=1$
$sin x=+1$ or $sin x=-1$
$x=\frac{\pi}{2}+2n\pi$ or $x=-\frac{\pi}{2}+2n\pi$

3. Originally Posted by alexmahone
$sin x cos x tan x=1$
$sin x cos x\frac{sin x}{cos x}=1$
$sin^2x=1$
$sin x=+1$ or $sin x=-1$
$x=\frac{\pi}{2}+2n\pi$ or $x=-\frac{\pi}{2}+2n\pi$
The above poster is right but bear in mind $n \in \mathbb{Z}$

(ie: n must be an integer)


text:
Lecture35 - Chapter 16 Numerical Integration b I = f x)dx a Graphical Representation of Integral Integral = area under the curve Use of a grid to

# Lecture35 - Chapter 16 Numerical Integration b I = f x)dx a...

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Chapter 16 Numerical Integration ( ) b a I f x dx =
Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral
Use of strips to approximate an integral
Numerical Integration: applications Net force against a skyscraper Cross-sectional area and volume flowrate in a river Survey of land area of an irregular lot
Integration Weighted sum of functional values at discrete points Newton-Cotes closed or open formulae -- evenly spaced points Approximate the function by Lagrange interpolation polynomial Integration of a simple interpolation polynomial Guassian Quadratures Richardson extrapolation and Romberg integration
Basic Numerical Integration Weighted sum of function values ) ( ) ( ) ( ) ( ) ( n n 1 1 0 0 i n 0 i i b a x f c x f c x f c x f c dx x f + + + = = x 0 x 1 x n x n-1 x f ( x )
0 2 4 6 8 10 12 3 5 7 9 11 13 15 Numerical Integration Idea is to do integral in small parts, like the way you first learned integration - a summation Numerical methods just try to make it faster and more accurate
Newton-Cotes formulas - based on idea dx x f dx x f I b a n b a 2245 = ) ( ) ( Approximate f ( x ) by a polynomial n n 1 n 1 n 1 0 n x a x a x a a x f + + + + = - - ) ( Numerical integration
f n ( x ) can be linear f n ( x ) can be quadratic
f n ( x ) can also be cubic or other higher-order polynomials
Polynomial can be piecewise over the data
Numerical Integration Newton-Cotes Closed Formulae -- Use both end points Trapezoidal Rule : Linear

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