In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions can also be represented as the sum of a scalar and a vector.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. They can be used alongside other methods, such as Euler angles and matrices, or as an alternative to them depending on the application.

In modern language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and thus also form a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H (for ''Hamilton''), or in blackboard bold by $\backslash mathbb\{H\}$ (Unicode U+210D, ). It can also be given by the Clifford algebra classifications ''C''ℓ_0,2(R) = ''C''ℓ⁰_3,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers.

The unit quaternions can therefore be thought of as a choice of a group structure on the 3-sphere $S^3$, the group Spin(3), the group SU(2), or the universal cover of SO(3).

History

Quaternion algebra was introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was only published in 1900.

Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and subtract triples of numbers. (This is trivial.) However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space.

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. While walking 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 formulas for the quaternions

:$i^2\; =\; j^2\; =\; k^2\; =\; ijk\; =\; -1.\; \backslash ,$

into the stone of Brougham Bridge as he paused on it.

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 the ''London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', vol. xxv (1844), pp 489–95. On the letter, Hamilton states,

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

Hamilton called a quadruple with these rules of multiplication a ''quaternion'', and he devoted most of the remainder of his life to studying and teaching them. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, ''Elements of Quaternions'', was 800 pages long and was published shortly after his death.

After Hamilton's death, his student Peter Tait continued promoting 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 of 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 prolix and opaque.

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 are not susceptible to gimbal lock. For this reason, quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude-control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.

Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including the physicists Murray Gell-Mann in 2002, Steven Weinberg in 2005, and the mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where no trace of Hamilton's carving remains, unfortunately.

Definition

As a set, the quaternions H are equal to R⁴, a four-dimensional vector space over the real numbers. H has three operations: addition, scalar multiplication, and quaternion multiplication. The sum of two elements of H is defined to be their sum as elements of R⁴. Similarly the product of an element of H by a real number is defined to be the same as the product in R⁴. To define the product of two elements in H requires a choice of basis for R⁴. The elements of this basis are customarily denoted as 1, ''i'', ''j'', and ''k''. Every element of H can be uniquely written as a linear combination of these basis elements, that is, as ''a''1 + ''bi'' + ''cj'' + ''dk'', where ''a'', ''b'', ''c'', and ''d'' are real numbers. The basis element 1 will be the identity element of H, meaning that multiplication by 1 does nothing, and for this reason, elements of H are usually written ''a'' + ''bi'' + ''cj'' + ''dk'', suppressing the basis element 1. Given this basis, associative quaternion multiplication is defined by first defining the products of basis elements and then defining all other products using the distributive law.

Multiplication of basis elements

The equations

:$i^2\; =\; j^2\; =\; k^2\; =\; i\; j\; k\; =\; -1,\backslash$

where ''i'', ''j'', and ''k'' are basis elements of H, determine all the possible products of ''i'', ''j'', and ''k''. For example, since :$-1\; =\; i\; j\; k,\backslash$ right-multiplying both sides by ''k'' gives : All the other possible products can be determined by similar methods, resulting in : which can be arranged as a table whose rows represent the left factor of the product and whose columns represent the right factor: {|class="wikitable" style="text-align:center" |+Quaternion multiplication |- !width=15|× !width=15|1 !width=15|i !width=15|j !width=15|k |- !1 |1 |''i'' |''j'' |''k'' |- !''i'' |''i'' | −1 |''k'' | −''j'' |- !''j'' |''j'' | −''k'' | −1 |''i'' |- !''k'' |''k'' |''j'' | −''i'' | −1 |}

Hamilton product

For two elements ''a''₁ + ''b''₁''i'' + ''c''₁''j'' + ''d''₁''k'' and ''a''₂ + ''b''₂''i'' + ''c''₂''j'' + ''d''₂''k'', their Hamilton product (''a''₁ + ''b''₁''i'' + ''c''₁''j'' + ''d''₁''k'')(''a''₂ + ''b''₂''i'' + ''c''₂''j'' + ''d''₂''k'') 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: :$a\_1a\_2\; +\; a\_1b\_2i\; +\; a\_1c\_2j\; +\; a\_1d\_2k$ :$\{\}+\; b\_1a\_2i\; +\; b\_1b\_2i^2\; +\; b\_1c\_2ij\; +\; b\_1d\_2ik$ :$\{\}+\; c\_1a\_2j\; +\; c\_1b\_2ji\; +\; c\_1c\_2j^2\; +\; c\_1d\_2jk$ :$\{\}+\; d\_1a\_2k\; +\; d\_1b\_2ki\; +\; d\_1c\_2kj\; +\; d\_1d\_2k^2.$ Now the basis elements can be multiplied using the rules given above to get: :$a\_1a\_2\; -\; b\_1b\_2\; -\; c\_1c\_2\; -\; d\_1d\_2$ :$\{\}+\; (a\_1b\_2\; +\; b\_1a\_2\; +\; c\_1d\_2\; -\; d\_1c\_2)i$ :$\{\}+\; (a\_1c\_2\; -\; b\_1d\_2\; +\; c\_1a\_2\; +\; d\_1b\_2)j$ :$\{\}+\; (a\_1d\_2\; +\; b\_1c\_2\; -\; c\_1b\_2\; +\; d\_1a\_2)k.$

Ordered list form

Using the basis 1, ''i'', ''j'', ''k'' of H makes it possible to write H as a set of quadruples: :$\backslash mathbf\{H\}\; =\; \backslash \{(a,\; b,\; c,\; d)\; \backslash mid\; a,\; b,\; c,\; d\; \backslash in\; \backslash mathbf\{R\}\backslash \}.$ Then the basis elements are: : and the formulas for addition and multiplication are: :$\backslash begin\{align\}\; (a\_1,\backslash \; b\_1,\backslash \; c\_1,\backslash \; d\_1)\; +\; (a\_2,\backslash \; b\_2,\backslash \; c\_2,\backslash \; d\_2)\; \backslash \backslash \; =\; (a\_1\; +\; a\_2,\backslash \; b\_1\; +\; b\_2,\backslash \; c\_1\; +\; c\_2,\backslash \; d\_1\; +\; d\_2).\; \backslash end\{align\}$

and

:

Scalar and vector parts

A number of the form a + 0i + 0j + 0k, where ''a'' is a real number, is called real, and a number of the form 0 + ''bi'' + ''cj'' + ''dk'', where ''b'', ''c'', and ''d'' are real numbers, is called pure imaginary. If ''a'' + ''bi'' + ''cj'' + ''dk'' is any quaternion, then ''a'' is called its scalar part and ''bi'' + ''cj'' + ''dk'' is called its vector part. The scalar part of a quaternion is always real, and the vector part is always pure imaginary. Even though every quaternion is a vector in a four-dimensional vector space, it is common to define a vector to mean a pure imaginary quaternion. With this convention, a vector is the same as an element of the vector space R³.

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

If a quaternion is divided up into a scalar part and a vector part, i.e.

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

then the formulas for addition and multiplication are:

:$\backslash begin\{align\}\; (r\_1,\backslash \; \backslash vec\{v\}\_1)\; +\; (r\_2,\backslash \; \backslash vec\{v\}\_2)\; \backslash \backslash \; =\; (r\_1\; +\; r\_2,\backslash \; \backslash vec\{v\}\_1+\backslash vec\{v\}\_2)\; \backslash end\{align\}$

and

:

where "" is the dot product and "" is the cross product.

Remarks

Noncommutative

Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative: For example, $ij\; =\; k$, while $ji\; =\; -k$. The noncommutativity of multiplication has some unexpected consequences, among them that polynomial equations over the quaternions can have more distinct solutions than the degree of the polynomial. The equation $z^2\; +\; 1\; =\; 0$, for instance, has infinitely many quaternion solutions $z\; =\; bi\; +\; cj\; +\; dk$ with $b^2\; +\; c^2\; +\; d^2\; =\; 1$, so that these solutions lie on the two-dimensional surface of a sphere centered on zero in the three-dimensional subspace of quaternions with zero real part. This sphere intersects the complex plane at the two poles $i$ and $-i$.

Historical impact on physics

In 1984 the ''European Journal of Physics'' (5:25–32) published P.R. Girard’s essay ''The quaternion group and modern physics''. It "shows how various physical covariance groups: SO(3), the Lorentz group, the general relativity group, the Clifford algebra SU(2), and the conformal group can be readily 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 use 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.

A more personal view of quaternions was written by Jim Lambek in 1995. In the ''Mathematical Intelligencer'' (17(4):7) he contributed "If Hamilton Had Prevailed: Quaternions in Physics" which recalled the use of biquaternions: "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."

Sums of four squares

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.

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let ''q'' = ''a'' +''bi'' +''cj'' + ''dk'' be a quaternion. The conjugate of ''q'' is the quaternion ''a'' − ''bi'' − ''cj'' − ''dk''. It is denoted by ''q''^*, $\backslash overline\; q$, ''q''^t, or $\backslash tilde\; q$. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates ''in the reverse order''. That is, if ''p'' and ''q'' are quaternions, then (''pq'')^* = ''q''^*''p''^*, not ''p''^*''q''^*.

Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition:

:$q^*\; =\; -\; \backslash frac\; 1\; 2\; (q\; +\; iqi\; +\; jqj\; +\; kqk)$

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

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 modern usage. See tensor.) It has the formula :$\backslash lVert\; q\; \backslash rVert\; =\; \backslash sqrt\{qq^*\}\; =\; \backslash sqrt\{q^*q\}\; =\; \backslash sqrt\{a^2\; +\; b^2\; +\; c^2\; +\; d^2\}.$ This is always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R⁴. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then :$\backslash lVert\backslash alpha\; q\backslash rVert\; =\; |\backslash alpha|\backslash lVert\; q\backslash rVert.$ This is a special case of the fact that the norm is ''multiplicative'', meaning that :$\backslash lVert\; pq\; \backslash rVert\; =\; \backslash lVert\; p\; \backslash rVert\backslash lVert\; q\; \backslash rVert.$ for any two quaternions ''p'' and ''q''. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively multiplicativity follows directly from the corresponding property of determinants of square matrices and the formula : where i denotes the usual imaginary unit.

This norm makes it possible to define the distance ''d''(''p'', ''q'') between ''p'' and ''q'' as the norm of their difference: :$d(p,\; q)\; =\; \backslash lVert\; p\; -\; q\; \backslash rVert.$ This makes H into a metric space. Addition and multiplication are continuous in the metric topology.

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion ''q'' by its norm produces a unit quaternion U''q'' called the versor of ''q'': :$\backslash mathbf\{U\}q\; =\; \backslash frac\{q\}\{\backslash lVert\; q\backslash rVert\}.$ Every quaternion has a polar decomposition q = ||''q''|| U''q''.

Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of $q$ and $q^*/\backslash lVert\; q\; \backslash rVert^2$ (in either order) is 1. So the reciprocal of ''q'' is defined to be :$q^\{-1\}\; =\; \backslash frac\{q^*\}\{\backslash lVert\; q\backslash rVert^2\}.$ This makes it possible to divide two quaternions ''p'' and ''q'' in two different ways. That is, their quotient can be either ''pq''⁻¹ or ''q''⁻¹''p''. The notation ''p''/''q'' is ambiguous because it does not specify whether ''q'' divides on the left or the right.

Algebraic properties

The set H of all quaternions is a vector space over the real numbers with dimension 4. (In comparison, the real numbers have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) The quaternions have a multiplication that is associative and that distributes over vector addition, but which is not commutative. Therefore the quaternions H are a non-commutative associative algebra over the real numbers. Even though 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 commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three: R, C, and H. The norm makes the quaternions into a normed algebra, and normed division algebras over the reals are also very rare: Hurwitz's theorem says that there are only four: R, C, H, and O (the octonions). The quaternions are also an example of a composition algebra and of a unital Banach algebra.

Because the product of any two basis vectors is plus or minus another basis vector, the set {±1, ±''i'', ±''j'', ±''k''} forms a group under multiplication. This group is called the quaternion group and is denoted ''Q''₈. The real group ring of ''Q''₈ is a ring R''Q''₈ which is also an eight-dimensional vector space over R. It has one basis vector for each element of ''Q''₈. The quaternions are the quotient ring of R''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 geometry of R³

Because the vector part of a quaternion is a vector in R³, the geometry of R³ is reflected in the algebraic structure of the quaternions. Many operations on vectors can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. For instance, this is true in electrodynamics and 3D computer graphics.

For the remainder of this section, ''i'', ''j'', and ''k'' will denote both imaginary basis vectors of H and a basis for R³. Notice that 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''.

Choose two imaginary quaternions ''p'' = ''b''₁''i'' + ''c''₁''j'' + ''d''₁''k'' and ''q'' = ''b''₂''i'' + ''c''₂''j'' + ''d''₂''k''. Their dot product is :$p\; \backslash cdot\; q\; =\; b\_1b\_2\; +\; c\_1c\_2\; +\; d\_1d\_2.$ This is equal to the scalar parts of ''p''^*''q'', ''qp''^*, ''pq''^*, and ''q''^*''p''. (Note that the vector parts of these four products are different.) It also has the formulas :$p\; \backslash cdot\; q\; =\; \backslash textstyle\backslash frac\{1\}\{2\}(p^*q\; +\; q^*p)\; =\; \backslash textstyle\backslash frac\{1\}\{2\}(pq^*\; +\; qp^*).$

The cross product of ''p'' and ''q'' relative to the orientation determined by the ordered basis ''i'', ''j'', and ''k'' is :$p\; \backslash times\; q\; =\; (c\_1d\_2\; -\; d\_1c\_2)i\; +\; (d\_1b\_2\; -\; b\_1d\_2)j\; +\; (b\_1c\_2\; -\; c\_1b\_2)k.$ (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 −''q''^*''p''^*. It also has the formula :$p\; \backslash times\; q\; =\; \backslash textstyle\backslash frac\{1\}\{2\}(pq\; -\; q^*p^*).$

In general, let ''p'' and ''q'' be quaternions (possibly non-imaginary), and write :$p\; =\; p\_s\; +\; \backslash vec\{p\}\_v,$ :$q\; =\; q\_s\; +\; \backslash vec\{q\}\_v,$ where ''p''_s and ''q''_s are the scalar parts of ''p'' and ''q'' and $\backslash vec\{p\}\_v$ and $\backslash vec\{q\}\_v$ are the vector parts of ''p'' and ''q''. Then we have the formula :$pq\; =\; p\_sq\_s\; -\; \backslash vec\{p\}\_v\backslash cdot\backslash vec\{q\}\_v\; +\; p\_s\backslash vec\{q\}\_v\; +\; \backslash vec\{p\}\_vq\_s\; +\; \backslash vec\{p\}\_v\; \backslash times\; \backslash vec\{q\}\_v.$ This shows that the noncommutativity of quaternion multiplication comes from the multiplication of pure imaginary quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear.

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 the terminology of abstract algebra, these are injective homomorphisms from H to the matrix rings M₂(C) and M₄(R), respectively.

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

:

This representation has the following properties:

Complex numbers (''c'' = ''d'' = 0) correspond to diagonal matrices.

The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.

The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.

Restricted to unit quaternions, this representation provides an isomorphism between ''S''³ and SU(2). The latter group is important for describing spin in quantum mechanics; see Pauli matrices.

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

:

:

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. Complex numbers are block diagonal matrices with two 2×2 blocks.

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 C² be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and ''j''. A vector in C² can be written in terms of the basis elements 1 and ''j'' as :$(a\; +\; bi)1\; +\; (c\; +\; di)j.\backslash$ If we define ''j''² = −1 and ''ij'' = −''ji'', then we can multiply two vectors using the distributive law. Writing ''k'' in place of the product ''ij'' leads to the same rules for multiplication as the usual quaternions. Therefore the above vector of complex numbers corresponds to the quaternion ''a'' + ''bi'' + ''cj'' + ''dk''. If we write the elements of C² as ordered pairs and quaternions as quadruples, then the correspondence is :$(a\; +\; bi,\backslash \; c\; +\; di)\; \backslash leftrightarrow\; (a,\; b,\; c,\; d).$

Square roots of −1

In the complex numbers, there are just two numbers, i and −i, whose square is −1 . In H there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the surface of the unit sphere in 3-space. To see this, let ''q'' = ''a'' + ''bi'' + ''cj'' + ''dk'' be a quaternion, and assume that its square is −1. In terms of ''a'', ''b'', ''c'', and ''d'', this means :$a^2\; -\; b^2\; -\; c^2\; -\; d^2\; =\; -1,$ :$2ab\; =\; 0,$ :$2ac\; =\; 0,$ :$2ad\; =\; 0.$ 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 ''a''² = −1. Therefore ''a'' = 0 and ''b''² + ''c''² + ''d''² = 1. In other words, a quaternion squares to −1 if and only if it is a vector (that is, pure imaginary) with norm 1. By definition, the set of all such vectors forms the unit sphere.

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

The identification of the square roots of minus one in H was given by Hamilton 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. Also in Conway (2003) ''On Quaternions and Octonions'' we read on page 40: "any imaginary unit may be called i, and perpendicular one j, and their product k", another statement of the sphere.

H as a union of complex planes

Each square root of −1 creates a distinct copy of the complex numbers inside the quaternions. If ''q''² = −1, then the copy is determined by the function

:$a\; +\; b\backslash sqrt\{-1\}\; \backslash mapsto\; a\; +\; bq.$

In the language of abstract algebra, each is an injective ring homomorphism from C to H. The images of the embeddings corresponding to ''q'' and ''-q'' are identical.

Every non-real quaternion lies in a unique copy of C. Write ''q'' as the sum of its scalar part and its vector part: :$q\; =\; q\_s\; +\; \backslash vec\{q\}\_v.$ Decompose the vector part further as the product of its norm and its versor: :$q\; =\; q\_s\; +\; \backslash lVert\backslash vec\{q\}\_v\backslash rVert\backslash cdot\backslash mathbf\{U\}\backslash vec\{q\}\_v.$ (Note that this is not the same as $q\_s\; +\; \backslash lVert\; q\backslash rVert\backslash cdot\backslash mathbf\{U\}q$.) The versor of the vector part of ''q'', $\backslash mathbf\{U\}\backslash vec\{q\}\_v$, is a pure imaginary unit quaternion, so its square is −1. Therefore it determines a copy of the complex numbers by the function :$a\; +\; b\backslash sqrt\{-1\}\; \backslash mapsto\; a\; +\; b\backslash mathbf\{U\}\backslash vec\{q\}\_v.$ Under this function, ''q'' is the image of the complex number $q\_s\; +\; \backslash lVert\backslash vec\{q\}\_v\backslash rVert\; i$. Thus H is the union of complex planes intersecting in a common real line, where the union is taken over the sphere of square roots of minus one.

Commutative subrings

The relationship of quaternions to each other within the complex subplanes of H can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions ''p'' and ''q'' commute (''p q'' = ''q p'') only if they lie in the same complex subplane of H, the profile of H as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. This method of commutative subrings is also used to profile the coquaternions and 2 × 2 real matrices.

Functions of a quaternion variable

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable.

Exponential, logarithm, and power

The exponential and logarithm of a quaternion are relatively inexpensive to compute, particularly compared with the cost of those operations for other charts on ''SO''(3) such as rotation matrices, which require computing the matrix exponential and matrix logarithm respectively.

Given a quaternion, :$q=a+bi+cj+dk=a+\backslash mathbf\{v\}$, the exponential is computed as :$\backslash exp(q)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{q^n\}\{n!\}=e^\{a\}\; \backslash left(\backslash cos\; \backslash |\backslash mathbf\{v\}\backslash |\; +\; \backslash frac\{\backslash mathbf\{v\}\}\{\backslash |\backslash mathbf\{v\}\backslash |\}\; \backslash sin\; \backslash |\backslash mathbf\{v\}\backslash |\backslash right)$ and :$\backslash ln(q)\; =\; \backslash ln\; \backslash |q\backslash |\; +\; \backslash frac\{\backslash mathbf\{v\}\}\{\backslash |\backslash mathbf\{v\}\backslash |\}\; \backslash cos^\{-1\}\; \backslash frac\{a\}\{\backslash |q\backslash |\}$. It follows that the polar decomposition of a quaternion may be written :$q=\backslash |q\backslash |e^\{\backslash hat\{n\}\backslash theta\},$ where the angle $\backslash theta$ and the unit vector $\backslash hat\{n\}$ are defined by: :$a=\backslash |q\backslash |\backslash cos(\backslash theta)$ and :$\backslash mathbf\{v\}=\backslash hat\{n\}\; \backslash |\backslash mathbf\{v\}\backslash |=\backslash hat\{n\}\backslash |q\backslash |\backslash sin(\backslash theta).$ Any unit quaternion may be expressed in polar form as $e^\{\backslash hat\{n\}\backslash theta\}$.

The power of a quaternion raised to an arbitrary (real) exponent is given by: :$q^\backslash alpha=\backslash |q\backslash |^\backslash alpha\; e^\{\backslash hat\{n\}\backslash alpha\backslash theta\}$

Three-dimensional and four-dimensional rotation groups

The term "conjugation", besides the meaning given above, can also mean taking an element ''a'' to ''r a r''^-1 where ''r'' is some non-zero element (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 R³ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(''θ'') is a rotation by an angle 2''θ'', the axis of the rotation being the direction of the imaginary part. The advantages of quaternions are:

# Non singular representation (compared with Euler angles for example). # More compact (and faster) than matrices. # Pairs of unit quaternions represent a rotation in 4D space (see ''Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations'').

The set of all unit quaternions (versors) forms a 3-dimensional sphere ''S''³ and a group (a Lie group) under multiplication. ''S''³ is the double cover of 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.

The image of a subgroup of ''S''³ is a point group, and conversely, the preimage of a point group is a subgroup of ''S''³. 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 group ''S''³ 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'' + ''bi'' + ''cj'' + ''dk'' where ''a'', ''b'', ''c'', and ''d'' are either all integers or all rational numbers with odd numerator and denominator 2. 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 24-cell regular polytope with Schläfli symbol {3,4,3}.

Generalizations

If ''F'' is any field with characteristic different from 2, and ''a'' and ''b'' are elements of ''F'', one may define a four-dimensional unitary associative algebra over ''F'' with basis 1, ''i'', ''j'', and ''ij'', where ''i''² = ''a'', ''j''² = ''b'' and ''ij'' = −''ji'' (so ''ij''² = −''ab''). These algebras are called ''quaternion algebras'' and 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 Cℓ_3,0(R)

The usefulness of quaternions for geometrical computations can be generalised to other dimensions, by identifying the quaternions as the even part Cℓ⁺_3,0(R) of the Clifford algebra Cℓ_3,0(R). This is an associative multivector algebra built up from fundamental basis elements σ₁, σ₂, σ₃ using the product rules :$\backslash sigma\_1^2\; =\; \backslash sigma\_2^2\; =\; \backslash sigma\_3^2\; =\; 1,$ :$\backslash sigma\_i\; \backslash sigma\_j\; =\; -\; \backslash sigma\_j\; \backslash sigma\_i\; \backslash qquad\; (j\; \backslash neq\; i).$ 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: :$r^\{\backslash prime\}\; =\; -\; w\backslash ,\; r\backslash ,\; w.$ Two reflections make a rotation by an angle twice the angle between the two reflection planes, so :$r^\{\backslash prime\backslash prime\}\; =\; \backslash sigma\_2\; \backslash sigma\_1\; \backslash ,\; r\; \backslash ,\; \backslash sigma\_1\; \backslash sigma\_2$ corresponds to a rotation of 180° in the plane containing σ₁ and σ₂. This is very similar to the corresponding quaternion formula, :$r^\{\backslash prime\backslash prime\}\; =\; -\backslash mathbf\{k\}\backslash ,\; r\backslash ,\; \backslash mathbf\{k\}.$

In fact, the two are identical, if we make the identification :$\backslash mathbf\{k\}\; =\; \backslash sigma\_2\; \backslash sigma\_1,\; \backslash mathbf\{i\}\; =\; \backslash sigma\_3\; \backslash sigma\_2,\; \backslash mathbf\{j\}\; =\; \backslash sigma\_1\; \backslash sigma\_3$ and it is straightforward to confirm that this preserves the Hamilton relations :$\backslash mathbf\{i\}^2\; =\; \backslash mathbf\{j\}^2\; =\; \backslash mathbf\{k\}^2\; =\; \backslash mathbf\{i\}\; \backslash mathbf\{j\}\; \backslash mathbf\{k\}\; =\; -1.$

In this picture, 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 Cℓ_4,0(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.

Dorst ''et al.'' identify the following advantages for placing quaternions in this wider setting:

Rotors are natural and non-mysterious in 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 required when augmenting linear algebra with quaternions).

A rotor is 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.

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 reals is Brauer equivalent to either the reals or the quaternions. Explicitly, the Brauer group of the reals consists of two classes, represented by the reals 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 reals.

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 reals (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial field extension of the reals.

Quotes

"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 (''ed''. Quoted in a letter from Tait to Cayley).

"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 (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton").

"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." — Lord Kelvin, 1892.

"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 every-day 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, preparing the second edition of his Theory of Relativity in 1924.

"… 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.

"...the thing about a Quaternion 'is' is that we're obliged to encounter it in more than one guise. As a vector quotient. As a way of plotting complex numbers along three axes instead of two. As a list of instructions for turning one vector into another..... And considered subjectively, as an act of becoming longer or shorter, while at the same time turning, among axes whose unit vector is not the familiar and comforting 'one' but the altogether disquieting square root of minus one. If you were a vector, mademoiselle, you would begin in the 'real' world, change your length, enter an 'imaginary' reference system, rotate up to three different ways, and return to 'reality' a new person. Or vector..." — Thomas Pynchon, ''Against the Day'', 2006.

See also

Notes

External articles and resources

Books and publications

Hamilton, William Rowan. On quaternions, or on a new system of imaginaries in algebra. Philosophical Magazine. Vol. 25, n 3. p. 489–495. 1844.

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.

Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0

Maxwell, James Clerk (1873), "''A Treatise on Electricity and Magnetism''". Clarendon Press, Oxford.

Tait, Peter Guthrie (1886), "''Quaternion''". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped PostScript file)

Joly, Charles Jasper (1905), "''A manual of quaternions''". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84

Macfarlane, Alexander (1906), "''Vector analysis and quaternions''", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048

1911 encyclopedia: "''Quaternions''".

Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "''Foundations of quaternion quantum mechanics''". J. Mathematical Phys. 3, pp. 207–220, MathSciNet.

Du Val, Patrick (1964), "''Homographies, quaternions, and rotations''". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81

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. (1986), "''Rotations, quaternions, and double groups''". Oxford [Oxfordshire] : Clarendon Press ; New York : Oxford University Press. LCCN 85013615 ISBN 0-19-855372-2

Altmann, Simon L. (1989), "''Hamilton, Rodrigues, and the Quaternion Scandal''". Mathematics Magazine. Vol. 62, No. 5. p. 291–308, Dec. 1989.

Adler, Stephen L. (1995), "''Quaternionic quantum mechanics and quantum fields''". New York : Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X

Trifonov, Vladimir (1995), "''A Linear Solution of the Four-Dimensionality Problem''", Europhysics Letters, 32 (8) 621–626, DOI: 10.1209/0295-5075/32/8/001

Ward, J. P. (1997), "''Quaternions and Cayley Numbers: Algebra and Applications''", Kluwer Academic Publishers. ISBN 0-7923-4513-4

Kantor, I. L. and Solodnikov, A. S. (1989), "''Hypercomplex numbers, an elementary introduction to algebras''", Springer-Verlag, New York, ISBN 0-387-96980-2

Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "''Quaternionic and Clifford calculus for physicists and engineers''". Chichester ; New York : Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 ISBN 0-471-96200-7

Kuipers, Jack (2002), "''Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality''" (reprint edition), Princeton University Press. ISBN 0-691-10298-8

Conway, John Horton, and Smith, Derek A. (2003), "''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry''", A. K. Peters, Ltd. ISBN 1-56881-134-9 (review).

Kravchenko, Vladislav (2003), "''Applied Quaternionic Analysis''", Heldermann Verlag ISBN 3-88538-228-8.

Hanson, Andrew J. (2006), "''Visualizing Quaternions''", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3

Trifonov, Vladimir (2007), "''Natural Geometry of Nonzero Quaternions''", International Journal of Theoretical Physics, 46 (2) 251–257, DOI: 10.1007/s10773-006-9234-9

Ernst Binz & Sonja Pods (2008) ''Geometry of Heisenberg Groups'' American Mathematical Society, Chapter 1: "The Skew Field of Quaternions" (23 pages) ISBN 978-0-8218-4495-3.

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 D.J. Evans, (1977), "On the Representation of Orientation Space", Mol. Phys., vol 34, p 317.

Links and monographs

Matrix and Quaternion FAQ v1.21 Frequently Asked Questions

"Geometric Tools documentation" (frame; body) includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.

Patrick-Gilles Maillot Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.

"Geometric Tools source code" (frame; body) includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.

Doug Sweetser, Doing Physics with Quaternions

Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)

The Physical Heritage of Sir W. R. Hamilton (PDF)

D. R. Wilkins, Hamilton’s Research on Quaternions

Quaternion Julia Fractals 3D Raytraced Quaternion Julia Fractals by David J. Grossman

Quaternion Math and Conversions Great page explaining basic math with links to straight forward rotation conversion formulae.

John H. Mathews, Bibliography for Quaternions.

Quaternion powers on GameDev.net

Andrew Hanson, Visualizing Quaternions home page.

Representing Attitude with Euler Angles and Quaternions: A Reference, Technical report and Matlab toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.

Charles F. F. Karney, ''Quaternions in molecular modeling'', J. Mol. Graph. Mod. 25(5), 595–604 (Jan. 2007); DOI: 10.1016/j.jmgm.2006.04.002; E-print arxiv:0506177.

Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations., ''arXiv General Mathematics'' 2005.

Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., ''arXiv General Mathematics'' 2007.

NUI Maynooth Department of Mathematics, Hamilton Walk.

OpenGL:Tutorials:Using Quaternions to represent rotation

David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. Drdc-rddc.gc.ca

Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",Utexas.edu

D. Stahlke, Quaternions in Classical Mechanics Stahlke.org (PDF)

Morier-Genoud, Sophie, and Valentin Ovsienko. "Well, Papa, can you multiply triplets?", arxiv.org describes how the quaternions can be made into a skew-commutative algebra graded by Z/2×Z/2×Z/2.

Curious Quaternions by Helen Joyce hosted by John Baez.

Software

Quaternion Calculator [javascript], bluetulip.org

Quaternion Calculator [Java], theworld.com

Quaternion Toolbox for Matlab, qtfm.sourceforge.net

Boost library support for Quaternions in C++, boost.org

Mathematics of flight simulation >Turbo-PASCAL software for quaternions, Euler angles and Extended Euler angles, xs4all.nl

Category:Rotational symmetry

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Coordinates	40°37′29″N73°57′8″N
name	For Real
background	group_or_band
genre	R&B;, Pop, Dance
years active	1993–1999
label	Arista RecordsA&M; Records, Rowdy Records
past members	Wendi WilliamsLatanyia BaldwinJosina Elder Necia Bray-Gates
notable instruments	}}

For Real was an American R&B; and soul quartet, that formed in 1993. In the latter part of that decade they were nominated for a Grammy, Billboard Music Award, and Soul Train Music Award.

Biography

For Real secured their recording contract by accident. "We were picking up our manager from the airport. So we decided to greet him with an acappella song. Someone from A&M; records just happened to be in the airport and heard them perform. Not long after, they were signed," said Latanyia Baldwin.

The band released their debut album, ''It's a Natural Thang'', with production from Brian McKnight on A&M; Records in 1994, and it became a critical success, including a rare four stars from Rolling Stone Magazine. Their first single, "You Don't Wanna Miss" hit #28 on the ''Billboard'' chart, courtesy of a danceable new jack swing remix by Steve "Silk" Hurley, which was featured in the song's video. A second single, "Easy To Love," peaked at #65 on the Billboard Hot 100 chart. The album scored another chart hit with the a cappella love song, "You Don't Know Nothin'," which was written and produced by Mervyn Warren, and which peaked at #27. The song peaked at #54 in the UK Singles Chart in July 1995. The album sold over one million copies worldwide, and peaked at #80 on the Billboard 200.

In 1995, the band appeared in Italian ''Vogue'', modeling mens suits. They toured alongside Stevie Wonder, and lent their voices to the Martin Scorsese film project ''Grace of My Heart.'' The foursome also forayed into acting with appearances in the film ''Shake, Rattle and Rock!'' starring Renée Zellweger and Howie Mandel on ''Showtime''.

They also hit the US Top 20 with the single "Freedom (Theme from Panther)" featuring Aaliyah, TLC, En Vogue, BlackGirl, SWV, and Vanessa Williams from the movie ''Panther''. They also recorded a duet with Stevie Wonder called "Stubborn Kind Of Fellow", which appeared on the Marvin Gaye tribute album, ''Inner City Blues: The Music Of Marvin Gaye.'' For Real also recorded songs for the soundtracks of the films ''Waiting to Exhale'' and ''Fled'' respectively.

After a label change from A&M; to Dallas Austin's Arista imprint Rowdy Records in 1996, For Real struck again with the Austin-produced, 60s-leaning single "Like I Do," from their second album ''Free''. "Like I Do" hit #10 on the Billboard singles chart, the video also earned a Billboard nomination for Best R&B; Clip, the album made the Top 100, and nearly 400,000 copies in the US, the album's next and last single was the Diane Warren-penned pop ballad "The Saddest Song I Ever Heard" which hit #65 on the Top 100 singles chart. The band performed "Saddest Song" on the TV show All That. ''Free'' included production from top R&B;/pop producers like Babyface, Mario Winans, and Jon B.

The band also recorded three songs ("I Do", "Born to Love That Boy" and "Unwanted Number") and appeared in the critically acclaimed film ''Grace of My Heart''. The films soundstrack won a Satellite Award in 1997.

In January 1997, the band won an American Music Award for their work on the ''Waiting to Exhale'' soundtrack, in Summer 1997, the group earned a Soul Train Award nomination for Album Of The Year (By A Group, Band Or Duo) for "Free".

In 1999, Wendi Williams lent her voice to the Emmy winning film ''Introducing Dorothy Dandridge'', with Halle Berry, in which Berry plays a singer but Williams is behind her singing voice.

In 2002, their song "Love Will Be Waiting At Home", appeared in the movie, ''Two Can Play That Game'', that starred Vivica A. Fox and Morris Chestnut.

In 2009, CoryLavel began to work with Josina Elder on her new album.

Discography

''It's a Natural Thang'' (1994)

''Free'' (1996)

Awards and nominations

Year	Result	Award	Category	Work
	Nominated	Grammy Award	Album of the Year	Waiting to Exhale
	Nominated	Billboard Music Award	Best Clip - R&B;	"Like I Do"
	Nominated	Soul Train Music Award	Album of the Year (By a Group, Band or Duo)	''Free''

References

External links

SingingFool.com Streaming music videos for "Like I Do" and "The Saddest Song I Ever Heard"

Allmusic Discography, biographies, and pictures

IUMA Group photos, lyrics, message board, audio streams, and bio.

Google Album info, photos, and lyrics

Category:American pop music groups Category:American rhythm and blues musical groups Category:African American musical groups Category:American girl groups Category:American soul musical groups Category:Musical groups established in 1992 Category:Musical groups disestablished in 1999

Coordinates	40°37′29″N73°57′8″N
name	Donna Summer
background	solo_singer
background	solo_singer
alias	Donna Gaines
birth name	LaDonna Adrian Gaines
born	December 31, 1948 Boston, Massachusetts, USA
origin	Dorchester, Massachusetts, USA
genre	Dance-pop, disco, pop, rock, new wave
occupation	Singer-songwriter
years active	1968–present
instrument	Vocals, piano
label	Oasis RecordsCasablanca (1975–1980)Geffen (1980–1988)Atlantic (1988–1991)Mercury (1994–1996) Warner-Elektra-Atlantic (Outside of U.S. 1980–1991)Epic (1999–2001)Burgundy (2006–present)
associated acts	Giorgio Moroder, Brooklyn Dreams }}

LaDonna Adrian Gaines (born December 31, 1948), known by her stage name, Donna Summer, is an American singer/songwriter who gained prominence during the disco era of the 1970s. She has a mezzo-soprano vocal range. Summer is a five-time Grammy winner and was the first artist to have three consecutive double albums reach number one on the US ''Billboard'' chart. She also charted four number-one singles in the US within a thirteen-month period.

Early life and career

Donna Summer was born on New Year's Eve 1948 in the Dorchester neighborhood of Boston, Massachusetts, she was one of seven children raised by devout Christian parents. Influenced by Mahalia Jackson, Summer began singing in the church at a young age. In her teens, she formed several musical groups including one with her sister and a cousin, imitating Motown girl groups such as The Supremes and Martha and the Vandellas in Boston.

In the late 1960s, Summer was influenced by Janis Joplin after listening to her albums as member of Big Brother and the Holding Company, and joined the psychedelic rock group the Crow as lead singer. Beforehand, Summer dropped out of school convinced that music was her way out of Boston, where she had always felt herself to be an outsider, even among her own family who ridiculed her for her voice and her looks. The group was short-lived, as they split upon their arrival in New York. In 1968, Summer auditioned for a role in the Broadway musical, ''Hair''. She lost the part of Sheila to Melba Moore. When the musical moved to Europe, Summer was offered the role. She took it and moved to Germany for several years. While in Germany, where she learned to speak German fluently, she participated in the musicals ''Godspell'' and ''Show Boat''. After settling in Munich, she began performing in several ensembles including the Viennese Folk Opera and even sang as a member of the pop group FamilyTree – "invented" and created by the German music producer Guenter "Yogi" Lauke & the Munich Machine. She came to the group in 1973 and toured with the 11-people pop group throughout Europe. She also sang as a studio session singer and in theaters. In 1971, while still using her birth name Donna Gaines, she released her first single, a cover of "Sally Go 'Round the Roses", though it was not a hit. In 1972, she married Austrian actor Helmuth Sommer and gave birth to their daughter Mimi Sommer in 1973. Citing marital problems caused by his frequent absences, she divorced him but kept his last name, changing the "o" to a "u".

1970s

It was while singing background for the hit-making 1970s trio Three Dog Night that Summer met producers Giorgio Moroder and Pete Bellotte. She eventually made a deal with the European label Groovy Records and issued her first album, ''Lady of the Night'', in 1974. The album was not released in America, but found some limited European success on the strength of the song "The Hostage", which reached number one in Belgium and number two in the Netherlands. Summer's early material consisted of pop rock and folk rock material. She stated years later that had she not recorded disco she would have been a black rock singer, but considering there was not a market for black rock singers, Summer thought it would be hard to get promoted as such.

In 1975, Summer approached Moroder with an idea for a song he and Bellotte were working on for another singer. She had come up with the lyrics "love to love you, baby". Moroder was interested in developing the new sound that was becoming popular and used Summer's lyric to develop the song. Moroder persuaded Summer to record what was to be a demo track for another performer. She later said that she had thought of how the song might sound if Marilyn Monroe had sung it and began cooing the lyrics. To get into the mood of recording the song, she requested the producers turn off the lights while she sat on a sofa inducing moans and groans. After hearing playback of the song, Moroder felt Summer's version should actually be distributed. Released as "Love to Love You" in Europe, some radio stations refused to play it, but the song found modest chart success in several countries there. The song was a real piece of work with sensual elegance.

The song was then sent to America and arrived in the office of Casablanca Records president Neil Bogart in hopes of getting an American release. Casablanca was known around the industry for throwing lavish parties. At one of these parties, Bogart, still undecided about releasing the song on his label, had "Love To Love You" played so he could gauge the reaction of people on the dancefloor. The crowd took to the song (which was less than five minutes) so strongly that they kept asking for it to be played over and over consecutively so they could continue dancing in the same groove. Soon after that night, Bogart informed Summer and Moroder he would release the song but requested that Moroder produce a longer version, about 15 to 20 minutes in length. Moroder, Bellotte, and Summer returned with a 17 minute version that included a soulful chorus and an instrumental break where Summer invoked even more moans. Bogart stated the name would be slightly changed to "Love to Love You Baby" for the American release. Casablanca signed Summer and it released the single in November 1975. The shorter version of the single was promoted to radio stations while clubs received the 17 minute version (the longer version would also appear on the album). When Casablanca released the 17 minute version in its entirety as a single, it became one of the first record labels, to help make popular a format that would later be known as the 12 inch. By early 1976, "Love To Love You Baby" had reached #2 on the US Billboard Hot 100, while the parent album of the same name sold over a million copies. The song generated controversy for Summer's moans and groans and some American radio stations, like several in Europe, refused to play it. ''Time magazine'' would report that 22 orgasms were simulated in the making of the song. Other upcoming singles included "Try Me, I Know We can Make It", US #80; "Could It Be Magic", US #52; "Spring Affair", US #58; and "Winter Melody", US #43. The subsequent albums ''Love Trilogy'' and ''Four Seasons of Love'' both went gold in the US.

In 1977, Summer released the concept album ''I Remember Yesterday''. This album included her second top ten single, "I Feel Love", which reached number six in the US and number one in the UK. These US Hot 100 entries on the singles chart would help get Summer deemed in the press as "The First Lady of Love", a title which she was not totally comfortable with.

Another concept album, also released in 1977, was the double album, ''Once Upon a Time'', which told of a modern-day Cinderella "rags to riches" story through the elements of orchestral disco and ballads. This album would also attain gold status. In 1978, Summer released her version of the Richard Harris ballad, "MacArthur Park", which became her first number one US hit. The song was featured on Summer's first live album, ''Live and More'', which also became her first album to hit number one on the US Billboard 200 chart, and went platinum selling over a million copies. Other studio tracks included the top ten hit, "Heaven Knows", which featured the group Brooklyn Dreams accompanying her on background and Joe "Bean" Esposito singing alongside her on the verses. Summer would later be involved romantically with Brooklyn Dreams singer Bruce Sudano and the couple married two years after the song's release. Also in 1978, Summer acted in the film, ''Thank God It's Friday'', playing a singer determined to perform at a hot disco club. The film met modest success, but a song from the film, entitled "Last Dance", reached number three on the Hot 100 and resulted in Summer winning her first Grammy Award. Its writer, Paul Jabara, won an Academy Award for the composition. Despite her musical success, Summer was struggling with anxiety and depression and fell into a prescription drug addiction for several years.

In 1979, Summer was a performer on the world-televised Music for UNICEF Concert. The United Nations organization Unicef had declared 1979 as the Year of the Child. Summer joined contemporaries like Abba, Olivia Newton-John, the Bee Gees, Andy Gibb, Rod Stewart, John Denver, Earth, Wind and Fire, Rita Coolidge and Kris Kristofferson for an hour's TV special that raised funds and awareness for the world's children. Artists donated royalties of certain songs, some in perpetuity, to benefit the cause.

''Bad Girls'' and the break from disco

Following her recovery from prescription drug addiction, Summer worked on her next album with Moroder and Bellotte. The result was ''Bad Girls'', an album that had been in production for nearly two years. Summer based the concept of the album on a prostitute, as was made clear in the lyrics. The album became a success, spawning the number one hits "Hot Stuff" and Bad Girls, and the number two ballad "Dim All the Lights" With ''MacArthur Park'', ''Hot Stuff'', ''Bad Girls'', and the Barbra Streisand duet "No More Tears (Enough is Enough)" (which appeared on Streisand's album ''Wet''), Summer achieved four number-one hits within a thirteen month period. Those aforementioned songs, along with ''Heaven Knows'', ''Last Dance'', ''Dim All The Lights'', and ''On the Radio'' (from her upcoming double-album) would give her eight US Top 5 singles within a two year period. "Hot Stuff" later won her a second Grammy in the Best Female Rock Vocal Performance, which was the first time that category was ever brought to the award's show. That year, Summer played eight sold-out nights at the Universal Amphitheater in Los Angeles.

Summer released her first greatest hits set in 1979, a double-album entitled ''On The Radio: Greatest Hits Volumes 1 & 2''. The album reached number one in the US, becoming her third consecutive number one album. A new song from the compilation, "On the Radio", reached the US top five.

1980s

After the release of the greatest hits album, Summer wanted to branch out with other musical styles in addition to disco. This led to tensions between her and Casablanca Records. Sensing that they could no longer come to terms, Summer and the label parted ways in 1980, and she signed with Geffen Records, the new label started by David Geffen.

Summer's first release on Geffen Records was ''The Wanderer''; it replaced the disco sound of Summer's previous releases with more of the burgeoning new wave sound and elements of rock, such as the material being recorded at this time by Pat Benatar. The first single, the title track, became a hit and peaked at #3 in the US, subsequent singles were moderate hits. The album achieved gold status in the US, but met limited success on the UK charts.

Summer's projected second Geffen release, entitled ''I'm a Rainbow'', was shelved by Geffen Records (though two of the album's songs would surface in soundtracks of the 1980s films ''Fast Times at Ridgemont High'' and ''Flashdance''). Summer reluctantly parted company with Moroder after seven years working together as Geffen had recruited Quincy Jones to produce her next album. The result was the 1982 album ''Donna Summer''. The album had taken a lengthy six months to record. The album's first single, the dance track "Love Is in Control (Finger on the Trigger)", became an American top ten hit on the Hot 100, followed by more moderate hits "State of Independence"(#41 pop) and "The Woman In Me"(#33 pop). Problems then increased between Summer and Geffen Records after they were notified by Polygram Records, the parent company of Summer's former label Casablanca, that she needed to deliver them one more album to fulfill her agreement with them. Summer delivered the album ''She Works Hard for the Money'' and Polygram released it on its Mercury imprint in 1983. The title song became a hit reaching number three on the US Hot 100, and would provide Summer with a Grammy nomination. The album also featured the reggae-flavored top 20 UK hit "Unconditional Love", which featured the group Musical Youth who were riding high from the success of their single "Pass the Dutchie". The third US single, "Love Has A Mind of Its Own", reached the top forty of the Billboard R&B; chart. The album was certified gold. During the '80's Donna Summers stage and wardrobe designer was James Hopkins of Jopkin Design designing costumes and wardrobe that help tell the story of her music.

In late 1984, with no more albums due to Polygram, Summer returned on Geffen Records with her next release. Geffen, wanting to keep the momentum going, enlisted ''She Works Hard For the Money'''s producer Michael Omartian to produce ''Cats Without Claws''. The album, however, was not as successful as ''She Works Hard For the Money'' and failed to attain gold status of 500,000 copies sold in the US, becoming her first album since her 1974 debut not to do so. It did include a moderate hit in "There Goes My Baby", which peaked at #21.

Controversy

In the mid 1980s, Summer was in the midst of a controversy. She had allegedly made anti-gay remarks regarding the then-relatively new disease AIDS, which as a result had a significantly negative impact on her career and saw thousands of her records being returned to her record company by angered fans. Summer, by this time a born-again Christian, was alleged to have said that AIDS was a punishment from God for the immoral lifestyles of homosexuals. However, she later denied that she had ever made any such comment and, in a letter to the AIDS campaign group ACT UP in 1989, she claimed that it was a ''"terrible misunderstanding. I was unknowingly protected by those around me from the bad press and hate letters....If I have caused you pain, forgive me."'' She went on to apologise for the delay in refuting the rumors and closed her letter with quotes from Chapter 13 of 1 Corinthians from the Bible.

Also in 1989, Summer told ''The Advocate'' magazine that ''"A couple of the people I write with are gay, and they have been ever since I met them. What people want to do with their bodies is their personal preference."''. A couple of years later she filed a lawsuit against ''New York'' magazine when it reprinted the rumors as fact just as she was about to release her album ''Mistaken Identity'' in 1991. According to an ''A&E; Biography'' program dedicated to Summer in which she participated in 1995, the lawsuit was settled out of court though neither side were able to divulge any details.

European success

In 1987, Summer returned with the album ''All Systems Go'', which did not sell well, becoming her second consecutive album not to achieve gold status. It featured the single "Dinner with Gershwin" (written by Brenda Russell), which was only a minor US hit, though it peaked at #13 in the UK. The album's title track, "All Systems Go", was released only in the UK where it peaked at #54.

For Summer's next album, Geffen Records hired the hit production team of Stock Aitken Waterman (or ''SAW''), who had experienced incredible success by writing and producing for such acts as Kylie Minogue, Dead or Alive, Bananarama, and Rick Astley among others. However, Geffen decided not to release the album, entitled ''Another Place and Time'', and Summer and Geffen Records parted ways in 1988. The album was released in Europe in March 1989 on Warner Bros. Records, which had been Summer's label in Europe since 1982. The single "This Time I Know It's For Real" had become a top ten hit in several countries in Europe, prompting the Warner Bros. subsidiary company Atlantic Records to sign Summer in the US and pick up the album for a North American release soon after. The single peaked at #7 on the Hot 100 in the US, and became her twelfth gold single there. It was also Summer's final Top 40 hit on the American pop charts, though she scored two more UK hits from the album, "I Don't Wanna Get Hurt" (UK #7) and "Love's About To Change My Heart" (UK #20).

1990s

In 1990, ''The Best of Donna Summer'' was released on Warner Bros. Records. It featured some of Summer's biggest hits from this period (with the exception of any material from the Mercury album, as this was not a division of Warner Bros.) plus a small selection of her 1970s disco hits licensed from PolyGram.

In 1991, Summer released the new jack swing style album ''Mistaken Identity''. It did not sell well, but did contain the #18 R&B; hit "When Love Cries".

In 1993, Polygram Records released an extended greatest hits collection entitled ''The Donna Summer Anthology''. It included 34 songs totally over two and a half hours of music. It not only included songs from the Polygram-owned labels of Casablanca and Mercury, but also material from Atlantic and Geffen Records as well.

In 1994, Summer return with a new album on Mercury/Polygram, a gospel-influenced Christmas album entitled ''Christmas Spirit''. It included classic Christmas songs such as "O Holy Night", "Joy To The World", and "O Come All Ye Faithful" as well as some original songs.

Some of Summer's dance releases including "Carry On" (her first collaboration with Moroder in a decade) and "Melody of Love (Wanna Be Loved)" charted on the US Dance Chart, with "Melody of Love" reaching number one on that chart and also reaching number 21 on the UK Singles Chart

Also in 1994, Polygram would release yet another Summer album (Polygram's third Summer album within a two year period); a collection called "Endless Summer: Greatest Hits". The differences between this greatest hits album and the ''Anthology'' collection would be that this package contained 18 songs, while ''Anthology'' contained 34; and the songs here were mainly the radio versions heard at the time of their release, while ''Anthology'' contained somewhat longer versions of the songs.

While touring, Summer was offered a role guest-starring on the sitcom ''Family Matters'' as Steve Urkel's (Jaleel White) Aunt Oona in 1994, and again in 1997. In 1998, Summer received a Grammy Award for Best Dance Recording, being the first to do so, after a remixed version of her 1992 collaboration with Giorgio Moroder, "Carry On", was released in 1997. In 1999, Summer taped a live television special for VH1 titled ''Donna Summer – Live and More Encore'', producing the second highest ratings that year for the network, after their annual ''Divas'' special. A CD of the event was released by Epic Records and featured two studio recordings, "I Will Go with You (Con te partirò)" and "Love Is the Healer" reached number one on the Billboard Dance Charts.

2000s

Summer scored top ten hits on Billboard's Dance Chart in the beginning of the new millennium. In 2000, she also appeared on the third annual ''Divas'' special, dedicated to Diana Ross, though Summer sang her own material for the show. In 2004, Summer was inducted to the Dance Music Hall of Fame alongside The Bee Gees and Barry Gibb as an artist. Her classic song, "I Feel Love", was also inducted that night. Also, that year, she became one of the first acts, as both an Artist Inductee and a Record Inductee (for 1977’s “I Feel Love”) into the Dance Music Hall of Fame in New York City.

In 2008, Summer released her first studio album of original music in 14 years since 1994 '' (Christmas Spirit)''; years entitled ''Crayons'', which peaked at #17 on the US Top 200 Album Chart, and achieved modest international success. The album was released on the Sony BMG label Burgundy Records. The songs "I'm A Fire", "Stamp Your Feet", and "Fame (The Game)" reached number one on the US Billboard Dance Chart. The ballad "Sand on My Feet" was released to adult contemporary stations and reached number thirty on that chart.

While commenting on the album, Summer said “I wanted this album to have a lot of different directions on it. I did not want it to be any one baby. I just wanted it to be a sampler of flavors and influences from all over the world. There’s a touch of this, a little smidgeon of that, a dash of something else…like when you’re cooking.” On the song “The Queen Is Back”, Summer reveals her wry and witty self-awareness of her musical legacy and her public persona. “I’m making fun of myself,” she admits. “There’s irony, it’s poking fun at the idea of being called a queen. That’s a title that has followed me, followed me, and followed me. We were sitting and writing and that title kept popping up in my mind and I’m thinking, ‘Am I supposed to write this? Is this too arrogant to write?’ But people call me ‘the queen,’ so I guess it's ok to refer to myself as what everybody else refers to me as. We started writing the song and thought it was kind of cute and funny.” Summer wrote “The Queen Is Back” and “Mr. Music” with J.R. Rotem and Evan Bogart, the son of Neil Bogart, Casablanca Records founder. Neil Bogart died from cancer at the age of 39. He signed Summer to his Casablanca Records label in 1975 and released most of her biggest records during the 1970s. “I adored him and would have given up everything for him to be alive,” says Summer, remembering a time backstage long ago “when the nail person didn’t show up and Neil got on his knees and did my toenails. In many ways he was my mentor and I didn’t get to say goodbye to him.” When Summer met Evan Bogart, she was struck by his uncanny resemblance to his father. “It’s almost like they chiseled him out of his father,” Summer observed. “I’m in the studio looking at him and I get tears in my eyes, he has no idea why. I just wanted to hug him because it’s like I’m seeing someone I haven’t seen since his father passed away. It’s almost like Neil is looking at me through him. Evan and I hit it off immediately; there was a synergy that happened really quickly.”

On December 11, 2009, Summer performed at the Nobel Peace Prize Concert in Oslo, Norway, in honor of US President Barack Obama. She was backed by the Norwegian Radio Orchestra.

2010s

In August 2010, she released the single "To Paris With Love", co-written with Bruce Roberts and produced by Peter Stengaard. In October 2010, the single reached #1 on the US Billboard Dance Chart. Also that month, Summer appeared in the PBS Television Special ''Hitman Returns: David Foster and Friends''. In it, Summer performed with Seal on a medley of the songs "Unbreak My Heart / Crazy / On The Radio", before Summer closed the show with "Last Dance".

On July 29, 2010, Summer gave an interview with allvoices.com where she was asked if she would consider doing an album of standards. She replied:

:"I actually am, probably in September. I will begin work on a standards album. I will probably do an all-out dance album and a standards album. I'm gonna do both, and we will release them however were gonna release them. We are not sure which is going first."

On September 15, 2010, Summer appeared as a guest celebrity singing alongside rising star Prince Poppycock on the television show ''America's Got Talent''.

On October 16, 2010, she performed at a benefit concert at the Phoenix Symphony.

On June 6, 2011, Summer was a guest judge on the show, ''Platinum Hit'' in week two titled, Dance Floor Royalty. ''Platinum Hit'' is a reality competition series on Bravo launched in 2011 in which 12 singer-songwriters compete through innovative songwriting challenges that will test their creativity, patience and drive. Every episode features a different topic from a dance track to a love ballad, that require the contestants to write and perform lyrics from a multiple of genres, for a cash prize of $100,000, a publishing deal with songwriting collective The Writing Camp, and a recording deal with RCA/Jive label

In July, 2011 Summer was working at Paramount Recording Studios in Los Angeles with her nephew, the rapper and producer O'Mega Red. Together they worked on a track entitled "Angel".

Awards and recognition

One NAACP Image Award.

One time Juno Award nomination for Best Selling International Single,"I feel Love".

Three Multi-Platinum albums in the US.

Eleven of her albums went Gold in the US.

Twelve Gold singles.

Six American Music Awards.

She was the first female African American to receive an MTV Video Music Awards nomination. ("Best Female Video" and "Best Choreography" for "She Works Hard For The Money")

Academy Award for Best Original Song "Last Dance".

Two Golden Globe Award nominations (one win for "Last Dance" Song and one nominated for "The Deep" John Barry's Song).

Summer has received five Grammy Awards.

*1979 – Best R&B; Vocal Performance (Female), ''Last Dance''

*1980 – Best Rock Vocal Performance (Female), ''Hot Stuff''

*1984 – Best Inspirational Performance, ''He's a Rebel''

*1985 – Best Inspirational Performance, ''Forgive Me''

*1998 – Best Dance Recording, ''Carry On''

Twelve Grammy Award nominations (total of seventeen).

*1979 – Best Pop Vocal Performance (Female), ''MacArthur Park''

*1980 – Album of the Year, ''Bad Girls''

*1980 – Best Pop Vocal Performance (Female), ''Bad Girls''

*1980 – Best R&B; Vocal Performance (Female), ''Dim All the Lights''

*1980 – Best Disco Recording, ''Bad Girls''

*1981 – Best Pop Vocal Performance (Female), ''On the Radio''

*1982 – Best Rock Vocal Performance (Female), ''Cold Love''

*1982 – Best Inspirational Performance, ''I Believe in Jesus''

*1983 – Best Rock Vocal Performance (Female), ''Protection''

*1983 – Best R&B; Vocal Performance (Female), ''Love is in Control (Finger on the Trigger)''

*1984 – Best Pop Vocal Performance (Female), ''She Works Hard for the Money''

*2000 – Best Dance Recording, ''I Will Go with You (Con te Partiro)''

Summer placed a Top Forty hit on the ''Billboard'' Hot 100 in every year from 1976 ("Love to Love You Baby") to 1984 ("There Goes My Baby").

Summer was the first artist to score three consecutive number-one double albums.

Summer was twice honored by the Dance Music Hall of Fame; once with her induction as a recording artist and again with the induction for her influential single "I Feel Love". Summer's music career has landed her as the eighth most successful female recording artist in history according to ''Billboard'' .

Summer's career span of ''Billboard'' number-one Disco/Club Play hits spans from 1975's "Love to Love You Baby" through 2010's "[To Paris With Love]".

Summer was nominated for 2010 induction into the Rock and Roll Hall of Fame but was not chosen.

Cover versions by other artists

"I Feel Love" has been sampled several times. The song was sampled by Madonna, Whitney Houston, Bette Midler, Diana Ross, Moloko, Britney Spears, Robbie Williams, Darren Hayes, Mylo, David Guetta, Stuart Price, and Moby among others. "I Feel Love" was recorded by classical pop musician Vanessa-Mae for her 1998 album ''Storm''. Italian company Gucci used a special version of it in the "Flora" perfume advertising, filmed by Chris Cunningham. "I Feel Love" has been covered onstage by Madonna, the Red Hot Chili Peppers' John Frusciante, Kylie Minogue, Blondie, Basement Jaxx, and Finnish progressive rock band Kingston Wall. Venus Hum with Blue Man Group also performed this song with Japanese singer Kumi Koda for their album ''The Complex'', and in 2006, Tracy Bonham stood in for Hum on the Blue Man Group tour, performing the song. Bronski Beat and Marc Almond released the track as a duet with an added bridge section and titled it "I Feel Love/Johnny Remember Me", reaching number 3 in the UK charts in April 1985. In 1992 U.K. alterna-pop group Curve recorded a version for the NME's 40th anniversary compilation ''Ruby Trax'', which became an instant underground classic.

"Love to Love You Baby" was sampled in Beyoncé Knowles's "Naughty Girl" and by TLC on their original version of "I'm Good at Being Bad", but was removed by request of Summer on later editions. The song has been covered in portions onstage by Dionne Warwick.

"I Feel Love" was covered by French Canadian artist Jorane for her 2004 album ''The You and the Now''. LA based band Dengue Fever covered this song on their album ''Radio Dance Floor''. Kylie Minogue's team also sampled this for her 2002 KylieFever. Kylie performed "Light Years" which contains elements of "I Feel Love".

Madonna combined her song "Future Lovers" from her hit album ''Confessions on a Dancefloor'' with "I Feel Love" as the opening song on her Confessions Tour in 2006.

Summer's "Starting Over Again" was a number one hit on the Hot Country Songs chart as a single for Dolly Parton in 1980. It also was a top-forty hit for Parton on the Billboard Hot 100. Reba McEntire named her 1995 album after the song, and McEntire's version hit no. 17 on the country singles chart in 1996. McEntire stated in the album's liner notes that her recording of the song was intended as something of a tribute to Summer and Parton, both artists whom she admired.

Summer's "On the Radio" was covered by country artist Emmylou Harris for her 1983 album ''White Shoes''. British singer and actress Martine McCutcheon recorded a version that reached number 7 in the UK charts in February 2001. An acoustic rendition was released by alt. rock band Hypnogaja in January 2010.

Summer's "Sunset People" was covered by E.G. Daily on her ''Wild Child'' album (1985).

Summer's "Dim All the Lights" was a Top 40 Dance hit for Laura Branigan in 1995, appearing on her ''The Best of Branigan'' album.

Summer's "Last Dance" and "On the Radio" were covered by Tejano/pop singer Selena, most famously at one of her last shows at the Houston Astrodome, on February 26, 1995.

Summer's "All Through the Night" was covered in 1982 by KC and The Sunshine Band on their Painter album, and in 1995 by supermodel Naomi Campbell for her album ''Baby Woman'', featuring Luther Vandross on backing vocals.

Summer's "Bad Girls" was recorded by British Jazz and US pop singer Juliet Roberts in 1998, and again in 2000 by Cheryl Chase for the Nick film ''Rugrats in Paris: The Movie''.

Summer's "Love Is in Control" was covered by pop singer Sheena Easton on her 2000 disco album ''Fabulous''.

In 1985 Dusty Springfield reached number 83 on the UK charts with a cover of "Sometimes like Butterflies".

Summer's "Heaven Knows" was recorded by Faith Evans for inclusion on the soundtrack for the film, The Fighting Temptations.

Summer's "The Woman In Me" was recorded by Heart in the early 1990s.

Summer's "She Works Hard For The Money" was often performed by Millie Jackson during her 1980s concerts. "She Works Hard For The Money" was covered by Kris Allen during an American Idol Season 8 episode and recorded by Allen for the American Idol Season 8 CD.

Summer's "All Through The Night" from the album "Bad Girls", is recorded by model Naomi Campbell in 1994 included in her album "BabyWoman".

Jeffree Star covered her song "Bad Girls" for the upcoming movie called "Turn The Beat Around" for MTV.

Monica Naranjo Fuego de pasion "Love's About To Change My Heart".

Mexican singer Lorena Herrera samples "Try Me ,I Know We Can Make It" in her song Dame Amor from the album Desnudame el Alma.

In 2010, Keri Hilson portrayed Summer in her music video for her hit-single, Pretty Girl Rock.

In 2010 – Rhythm Masters and MYNC Project released a remake of "I Feel Love" on CR2 Records in the UK, with vocals by American Electro/Pop/Dance Artist Wynter Gordon.

Donna Summer concert tours

''Once Upon A Time Tour'' (1977–1978)

''Bad Girls Tour'' (1979)

''The Wanderer Tour'' (1981)

''Hard For The Money Tour'' (1983)

''The Rainbow Tour'' (1984)

''Silver Girl Tour'' (1986)

''All Systems Go Tour'' (1987)

''Endless Summer Tour'' (1995)

''Mid Summer Nights Dream Tour'' ( 1996–1997)

''Live & More Encore Tour'' (1999)

''Greatest Hits Tour'' (2005)

''Crayons Tour'' (2008)

Discography

1974: ''Lady of the Night''

1975: ''Love to Love You Baby''

1976: ''A Love Trilogy''

1976: ''Four Seasons of Love''

1977: ''I Remember Yesterday''

1977: ''Once Upon a Time''

1979: ''Bad Girls''

1980: ''The Wanderer''

1981: ''I'm a Rainbow''

1982: ''Donna Summer''

1983: ''She Works Hard for the Money''

1984: ''Cats Without Claws''

1987: ''All Systems Go''

1989: ''Another Place and Time''

1991: ''Mistaken Identity''

1994: ''Christmas Spirit''

2008: ''Crayons''

See also

Brooklyn Dreams

Best selling music artists

References

External links

Official website

A hit search page

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