6 NUMBER. [INT. I. 



is defined as 



a + b = (!+ /3 l }e l + (a,, + &)e 2 ...... + (<* n + n ) e n . 



.With respect to multiplication, the units are commutative with 

 respect to real numbers : 



abe r ce s = ae r bce s = abce r e s = e r e s abc, etc. 

 The associative and distributive principles also hold, so that 



(ae r ) e s a (e r e s ) and either a (e r + e s ) = ae r + ae 8 , 

 or e r (e s + e t ) = e r e s + e r e t . 



We may accordingly define the multiplication of any two com- 

 plex numbers 



ab = (otA + 2 e 2 ...... + a n e) (&*! 



It will be convenient to consider a system of units of such a 

 nature that instead of the commutative property with respect to 

 multiplication we have e r e s e s e r where r and s are different, and 

 for any r, e? = 1. 



If we consider a set of three units, each possessing the above 

 properties, and in addition the property that the product of any 

 two taken in cyclic order is equal to the third, we have the system 

 proposed by Hamilton, and denoted by him by the letters i, j, k. 

 Accordingly by definition 



ij = ji = k, jk = kj = i, ki = ik =j. 



Multiplying each equation by the first unit appearing in it, and 

 observing the associative law, we have 



iij = fij = iji = (ij) i = ki = j, 

 jjk = fk = -jkj = - (jk)j = -ij=-k, 

 kki = k?i = kik (ki) k jk = i, 



necessitating 



= ja = # = _!. 



Even powers of the three units are real, and equal even powers 

 are equal, while odd powers of any unit are equal to real multiples 

 of itself, and equal odd powers of different units are not equal. 

 The product of two threefold complex numbers of this system, a 

 and b, 



