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{{Template:Math-h}}
In [[mathematics]], a '''quaternion''' is a four-dimensional [[object]] important in [[group theory]] and [[geometry]]. As with the complex numbers, the quaternions can be viewed as an extension of the [[real number]] line. Unlike the complex numbers, however, the quaternions are not a [[field]], since multiplication is not commutative. Instead, the quaternions are a [[skew field]].
In [[mathematics]], a '''quaternion''' is a four-dimensional [[object]] important in [[group theory]] and [[geometry]]. As with the complex numbers, the quaternions can be viewed as an extension of the [[real number]] line. Unlike the complex numbers, however, the quaternions are not a [[field]], since multiplication is not commutative. Instead, the quaternions are a [[skew field]].
==Operations==
==Operations==
The quaternions obey all the usual arithmetic operations. Quaternions may be [[addition|added]], [[subtraction|subtracted]], [[multiplication|multiplied]], and [[division|divided]]. Addition is [[associative]] and [[commutative]], while multiplication is only associative. Moreover, addition distributes over multiplication and nonzero quaternions have multiplicative inverses, and so the quarternions are termed a skew field.
The quaternions obey all the usual arithmetic operations. Quaternions may be [[addition|added]], [[subtraction|subtracted]], [[multiplication|multiplied]], and [[division|divided]]. Addition is [[associative]] and [[commutative]], while multiplication is only associative. Moreover, addition distributes over multiplication and nonzero quaternions have multiplicative inverses, and so the quarternions are termed a skew field.
==Properties==
Quaternions are basically an extension of the complex plane to four dimensions. In addition to an imaginary unit <math>i</math> quaternions also have j and k, which all follow the relation:
<math>i^2 = j^2 = k^2 = ijk = -1</math>.
Hamilton made this discovery while on a walk and wrote it down on the Brougham Bridge.
A quaternion, <math>H</math> can be written as
<math>p= a(1) + bi + cj + dk</math>.
If <math>p= a_1(1) + b_1j + c_1i + d_1k</math> and <math>q = a_2(1) + b_2i + c_2j + d_2k</math> are two quaternions then their sum is <math>p + q = (a_1 + a_2)(1) + (b_1 + b_2)i + (c_1 + c_2 )j + (d_1 + d_2)k</math>. Multiplicaton is more complicated. To find a find a product one makes repeated use of the relations given of i, j and k. It turns out that quanternion multiplication is messy to write. As mentioned above, quanternion multiplication is noncommutative.
==References==
Weisstein, Eric W. "Quaternion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Quaternion.html
[[Category:Mathematics]]
[[Category:Mathematics]]