The octonions form an 8dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers. Every octonion is a real linear combination of the unit octonions 1, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6} and e_{7}, the multiplication table for which looks as follows.
·  1  e_{1}  e_{2}  e_{3}  e_{4}  e_{5}  e_{6}  e_{7} 
1  1  e_{1}  e_{2}  e_{3}  e_{4}  e_{5}  e_{6}  e_{7} 
e_{1}  e_{1}  1  e_{4}  e_{7}  e_{2}  e_{6}  e_{5}  e_{3} 
e_{2}  e_{2}  e_{4}  1  e_{5}  e_{1}  e_{3}  e_{7}  e_{6} 
e_{3}  e_{3}  e_{7}  e_{5}  1  e_{6}  e_{2}  e_{4}  e_{1} 
e_{4}  e_{4}  e_{2}  e_{1}  e_{6}  1  e_{7}  e_{3}  e_{5} 
e_{5}  e_{5}  e_{6}  e_{3}  e_{2}  e_{7}  1  e_{1}  e_{4} 
e_{6}  e_{6}  e_{5}  e_{7}  e_{4}  e_{3}  e_{1}  1  e_{2} 
e_{7}  e_{7}  e_{3}  e_{6}  e_{1}  e_{5}  e_{4}  e_{2}  1 
See also Hypercomplex numbers.
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