Two conceptualizations of four directions in mathematics are Complex numbers and Quaternions. For complex numbers, two of the directions are the opposite or negative of the other two. Complex numbers are like Cartesian coordinates in that they combine an (x,y) coordinate pair of real numbers into one complex number x + yi. Plus, complex numbers can by manipulated by extensions of arithmetic operations, like addition, subtraction, multiplication, and division.

Quaternions are an extension of the complex numbers where the directions are all different, and each direction is perpendicular to the other three. Like complex numbers are a notion of numbers that cover a plane, quaternions are a notion of number that fill four dimensional space. Thus they combine a 4-tuple (w,x,y,z) into one quaternion number w + xi + yj + zk. Like complex numbers, quaternions can be manipulated by further extensions of arithmetic operations.

http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Quaternion

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September 1, 2014 at 10:57 AM |

[…] easily construct multidimensional spaces. One representation of four dimensional space is by using quaternions, which have four dimensions to the complex numbers’ two. Tuples of real numbers or even […]

February 22, 2019 at 9:50 AM |

[…] rotations in 3D space, instead of using matrices. Above is a pitiful diagram (although better than my last one) of the Quaternion units 1, i, j, and k used in the typical representation a + b i + c j + d k. […]