A 4 dimensional extension of [[Complex Numbers]] whose multiplication is [[Commutative Property|non-commutative]]. More formally, the [[Quaternions]] are a [[Number System]] composing of [[Linear Combination|linear combinations]] of 4 axis', $1,i,j,k$. $i$ denotes the [[Imaginary Number|square root of negative 1]], and $j, k$ represents different roots of $-1$. $\huge \H = \setbuild{ a+bi+cj+dk}{a,b,c,d\in\R}$ The set of all quaternions is denoted as $\H$. $\huge\begin{align} i^2=j^2&=k^2=-1 \\ \end{align}$ Famously, multiplication of these new constants are [[Commutative Property|non-commutative]] with each other. $\huge \begin{array} iij =k & jk = i & ki=j \\ ji=-k & kj=-i & ik = -j \end{array}$ $\huge ijk = -1 $ Note that for $i,j,k$, multiplication by the [[Real Numbers]] is still [[Associative Property|associative]] and [[Distributive Property|distributive]], and *only* [[Commutative Property|commutative]] on $+$. Although the [[Quaternions|quaternion axis']] are [[Commutative Property|non-commutative]], $\H$ constitutes a [[Vector Space]] as there is still a [[Field]] $\R$ that satisfies the [[Axiom|axioms]] to be a scalar. $\H$ constitutes a [[Vector Space]] over $\R$ with 4 dimensions and a [[Orthonormal Basis]] $\set{1,i,j,k}$. >[!example]- >$\huge \begin{align} >(1+2i+3j+4k)(2-i+2j+5k) >\end{align}$ > >>[!multi-column] >>| | $1$ | $2i$ | $3j$ | $4k$ | >>| ---- | ---- | ----- | ----- | ----- | >>| $2$ | $2$ | 4i | 6j | 8k | >>| $-i$ | $-i$ | $2$ | $k$ | -j | >>| $2j$ | $2j$ | $4k$ | $-6$ | $8j$ | >>| $5k$ | $5k$ | $10k$ | $15i$ | $-20$ | >$\begin{align} >&= 2 +4i+6j+8k + -i + 2 + k + -j + 2j + 4k -6 + 8j + 5k + 10k + 15i -20\\ >&= \boxed{-22 + 10i - 6j - 20k} >\end{align}$ ### Multiplication Multiplication works the same as if $i,j,k$ were any other variables, with the added constraint that you cant just switch their orderings around as their multiplication is [[Commutative Property|non-commutative]] . ### Division Division between two quaternions $q,s\in \H$ is defined as: $\huge \frac{s}{q} = s \frac{1}{q} = s q^{-1} $ Where $q^{-1}$ denotes the [[Quaternion Inverse]] of $q$.