QuaternionsOverview on Hamilton Quaternions
A Hamilton Quaternion is a hypercomplex number with one real part (the scalar) and three imaginary parts (the vector).
This is an extension of the concept of numbers. We have found that a real number is a one-part number that can be represented on a number line and a complex number is a two-part number that can be represented on a plane. Extending that logic, we have also found that we can produce more numbers by adding more parts.
Quaternion --> a + b*i + c*j + d*k, where the coefficients a, b, c, d are elements of the reals
The hypercomplex number-quaternion-is a non-commutative division ring. This is what we call a four-dimensional number. Here is an example of a quaternion: 5 + 2i + 3j + 4k. The first term is called the scalar term; it is simply a real number. The other terms consisting of, i, j, k. are called the imaginary terms. As a group, they are called the vector of the quaternion. Vector algebra uses the same name vector as the quaternion number, but with a different meaning. Although vector algebra is an offspring of quaternions, vectors are not numbers. Starting with complex numbers, we lost the permanence of trichotomy. The permanencies we lose with quaternion numbers are the trichotomy property and the commutative property under multiplication. In addition, the imaginary units are anti-commutative under multiplication. Anti-commutative means the sign of the imaginary unit changes when we transpose the two operands under multiplication, e.g., i*j = -(j*i). The imaginary elements, i, j, and k, give cyclic permutations with each other (see Behaviors of Quaternions);.
If we set the coefficients of the imaginary elements j and k to zero, the quaternion number becomes an ordinary complex number. We can deduce all of the other algebraic numbers and/or the transcendental numbers from the quaternions simply by setting all the coefficients of the imaginary elements to zero.
This is an update from an earlier version that was updated in March 2005.