# Difference between revisions of "Quaternion"

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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]]. | ||

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==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. | ||

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+ | ==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: | ||

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+ | <math>i^2 = j^2 = k^2 = ijk = -1</math>. | ||

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+ | Hamilton made this discovery while on a walk and wrote it down on the Brougham Bridge. | ||

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+ | A quaternion, <math>H</math> can be written as | ||

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+ | <math>p= a(1) + bi + cj + dk</math>. | ||

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+ | 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== | ||

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+ | Weisstein, Eric W. "Quaternion." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Quaternion.html | ||

[[Category:Mathematics]] | [[Category:Mathematics]] |

## Revision as of 23:46, June 16, 2010

This article/section deals with mathematical concepts appropriate for late high school or early college. |

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.

Quaternions were invented by Irish mathematician William Rowan Hamilton in the 1840s. Quaternions have proved useful in describing the mechanics of rotation.

## Operations

The quaternions obey all the usual arithmetic operations. Quaternions may be added, subtracted, multiplied, and 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 quaternions also have j and k, which all follow the relation:

.

Hamilton made this discovery while on a walk and wrote it down on the Brougham Bridge.

A quaternion, can be written as

.

If and are two quaternions then their sum is . 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