# Quaternion definition

*a*+

*bi*+

*cj*+

*dk*where

*a, b, c,*and

*d*are real numbers,

*ij*=

*k, i*

^{2}=

*j*

^{2}= −1, and

*ij*= −

*ji.*Under addition and multiplication, quaternions have all the properties of a field, except that multiplication is not commutative.

*a*+

*bi*+

*cj*+

*dk*where

*a, b, c,*and

*d*are real numbers,

*ij*=

*k, i*

^{2}=

*j*

^{2}= −1, and

*ij*= −

*ji.*Under addition and multiplication, quaternions have all the properties of a field, except multiplication is not commutative.

## Other Word Forms

### Noun

## Origin of quaternion

- Middle English
*quaternioun**from*Late Latin*quaterniō**quaterniōn-**from*Latin*quaternī**by fours**from**quater**four times*kwetwer- in Indo-European roots### From American Heritage Dictionary of the English Language, 5th Edition

- From Middle English
*quaternioun*, from Late Latin*quaterniō*, from*quaternÄ«*(ablative form of*quater*, “four times") +*-iōn*, “-ion".### From Wiktionary