4, 5] NUMBER. 



(ai + 13 j + yk) (vti+ffj + yk) = aaV + /3/8'j 2 + yy'k* + affij + fa'ji 

 + y@'kj + ya'ki + ay'ik = - (act + 00' -f yy') 



is accordingly equal to a real number plus a complex number of 

 the same system, and this may be considered as a fourfold complex 

 number compounded of the units 1, i,j t k. Such a fourfold number 

 was called by Hamilton .a Quaternion. We shall in this book 

 seldom need the fourfold number, but shall frequently use the 

 threefold one. 



5. Geometrical Representation of Numbers. The 



natural numbers may be represented by an unlimited series of 

 points laid off at equal distances along a straight line. If we take 

 a certain point to represent zero, the positive integers will lie on 

 one side of it and the negative on the other. Points between the 

 integer points will represent fractions and irrationals, and to every 

 real number will correspond a point. For any rational number we 

 may find others lying as near it as we please, and as we have 

 already stated, for any irrational we may find rational numbers 

 lying as near it as we please. It may be shown, however, that 

 between any two rational numbers, however close together, there 

 can always be found an irrational, consequently the rational 

 numbers do not form a continuous series. It may be shown that 

 every point on the line corresponds to either a rational or an 

 irrational number, so that the whole series of real numbers is 

 continuous. Quantities which, like the real numbers, require for 

 their specification but a single given quantity, which may take 

 any of an unlimited series of values, are said to have one degree 

 of freedom. It is also said that there is a single infinity of such 

 quantities. 



Complex quantities in the narrow sense, involving two 

 different units, 1 and i, cannot be represented by points on a line. 

 If however we lay off the real numbers on a straight line, we may 

 lay off the pure imaginary numbers on a line at right angles with 

 it through the point representing zero. The point i is to be taken 

 at the same distance from zero on this line that the point 1 is on 

 the other LLae. The two lines are called respectively the axes of 

 reals and of pure imaginaries, or the axes of X and Y. Any 

 complex number a = a + /3i may now be represented by a point in 

 the plane whose rectangular x and y coordinates are respectively a 



