Quaternions
An associative, noncommutative algebra based on four linearly independent units or basal elements. Quaternions were originated in 1843, by W. R. Hamilton.
The four linearly independent units in quaternion algebra are commonly denoted by 1, i, j, k, where 1 commutes with i, j, k and is called the principal unit or modulus. These four units are assumed to have the following multiplication table: 
The i, j, k do not commute with each other in multiplication, that is, ij ≠ ji, jk ≠ kj, ik ≠ ki, etc. But all real and complex numbers do commute with i, j, k, thus if c is a real number, then ic = ci, jc = ci, and kc = ck. On multiplying ijk = −1 on the left by i, so that iijk = i(−1) = −i, it is found, since i2 = −1, that jk = i. Similarly jjk = ji = −k; when exhausted, this process leads to all the simple noncommutative relations for i, j, k, namely, 
More complicated products, for example, jikjk = −kki = i, are evaluated by substituting for any adjoined pair the value given in the preceding series of relations and then proceeding similarly to any other adjoined pair in the new product, and so on until the product is reduced to ±1, ±i, ±j, or ±k. Multiplication on the right is also permissible; thus from ij = k, one has ijj = kj, or −i = kj. Products such as jj and jjj may be written j2 and j3.
All the laws and operations of ordinary algebra are assumed to be valid in the definition of quaternion algebra, except the commutative law of multiplication for the units i, j, k. Thus the associative and distributive laws of addition and multiplication apply Without restriction throughout. Addition is also commutative, for example, i + j = j + i.