Quaternions

• Quaternions are a type of mathematical object that extends complex numbers.
• Notation: A quaternion $q$ can be written as $q=a+bi+cj+dk$, where $a$, $b$, $c$, and $d$ are real numbers, and $i$, $j$, and $k$ are the fundamental quaternion units.

Properties of Quaternions

1. Non-commutativity:
   • The multiplication of quaternions is not commutative. For example, $ij\ne ji$; instead, $ij=k$ and $ji=-k$.
2. Associativity:
   • Quaternion addition and multiplication are associative.
3. Norm:
   • The norm of a quaternion $q=a+bi+cj+dk$ is defined as $\sqrt{a^2+b^2+c^2+d^2}$.
4. Conjugate:
   • The conjugate of a quaternion $q=a+bi+cj+dk$ is $q^{\ast }=a-bi-cj-dk$.