To add and take scalar multiples of quaternions just treat them like 4-dimensional vectors, for example:

${\displaystyle (a_1+b_{1}i+c_{1}j+d_{1}k)+(a_2+b_{2}i+c_{2}j+d_{2}k)=(a_1+a_2)+(b_1+b_2)i+(c_1+c_2)j+(d_1+d_2)k).}$

Multiplication is defined by the rules of ordinary algebra where

${\displaystyle i^2=j^2=k^2=-1,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}ij=k=-ji,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}jk=i=-kj,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}ki=j=-ik.}$

For example

${\displaystyle (2+3i+4j+5k)(6+7i+8j+9k)=(12-21-32-45)+(36-40)i+(35-27)j+(21-28)k=-86-4i+8j-7k.}$

To read about number systems, where quaternions fit in, why there are no three-dimensional numbers and numbers in higher dimensions, see the NRICH article What Are Numbers?