20 Proceedings of the Roj/al Irish Academy. 



Note. — Were I to substitute the words " 31 miles in the same direction " 

 for "31 miles in a forward direction," the problem could not be stated 

 algebraically, and with the number chosen would not have any solution. 



Addition and Multiplication of generalised or real numbers. — (x u x z , x 3 , etc.) 

 or (#i + x 2 + x s + etc.) has the following definite meaning as an operator on 

 any quantity tea, namely, it means that we multiply wa first by x u then by 

 x~, and so on, and add the results. 



Now if x x a + x z a + x 3 a + etc. = xa, we may multiply as we saw in the 

 above problem by any real number w, so that 



XiWa + j\iv a + x 3 v:a + etc. = xwa : 

 (a - , + x 2 + x 3 + etc.) ica = xwa. 



Thus if (Xi + .r 2 + r 3 + etc.) a = xa. the operator .i\ + x„ + ,r 3 + etc., on any 

 quantity > r a. produces the same result as the operator x, and so in this sense 

 is equal to it. where x is derived from .c, + x 2 + x s + etc., by combining the 

 numbers in the ?ame way as they would combine should we operate with it 

 on a. This is precisely the same as what we mean in arithmetic by such a 

 statement as 3+4 = 7, which asserts not only that 3 units + 4 units = 7 units, 

 but also that if we multiply any quantity say 13 units by 3 and then by 4 

 and add the results, we get the same quantity as if we multiplied 13 units 

 by 7. 



Similarly if x-x,xji = xa. multiply each side by V) (as we saw we could 

 do), so that xix..r 3 u a = xwa. Hence if we combine the real numbers in the 

 operator r x x^t 3 in the same way as they would combine were we to operate 

 with it on a, we get an operator x which produces the same effect as X&&* 

 when we operate with it on any quantity. In this sense and according to 

 this rule x x t~t z = x. We have thus shown how to add and multiply real 

 numbers, and we note in such operations the numbers are commutative and 

 associative. 



Xow. (x + y) ( x - y' I a = x < -•' + >f) a + y {■•■' + >f) a 



= (x + f).ra + {x' + y')ya 



= ,<•'.'« - xy'a + x'ija + yy'a, 



(■'• - //)(«' + y') = "' '//' +x'y + yy. 



Replacing < ' by x and //' by //. and using the notation xx = x z . 



< /• + y )- = x 2 + 2xy + if. 

 Replacing y by - ,y, 



{x - yf = 3?- 2xi/ + f. 



Replacing x by x and y by - y, 



