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Definition

A quaternion is an expression of the form

a + b i + c j + d k , {\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 q = a + b i + c j + d k {\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }, as consisting of a scalar part and a vector part. The quaternion b i + c j + d k {\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.) A quaternion that equals its vector part is called a vector quaternion.

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

(a 1 + b 1 i + c 1 j + d 1 k) + (a 2 + b 2 i + c 2 j + d 2 k) = (a 1 + a 2) + (b 1 + b 2) i + (c 1 + c 2) j + (d 1 + d 2) k, {\displaystyle {\begin{aligned}&(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})\\&\qquad =(a_{1}+a_{2})+(b_{1}+b_{2})\,\mathbf {i} +(c_{1}+c_{2})\,\mathbf {j} +(d_{1}+d_{2})\,\mathbf {k},\end{aligned}}}

and the component-wise scalar multiplication

λ (a + b i + c j + d k) = λ a + (λ b) i + (λ c) j + (λ d) k. {\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:

Thus the quaternions form a division algebra.