F K Y — Real and Complex Numbers as Adjectives or Operators. 21 



Fractions. — "What we mean by - or x/y is a real number such that when 



we multiply it by y, or y by it, we get .r. Such a number exists, for, denoting 

 the numerical part of a real number x by \x\, its numerical part is \a>\ / \y\ ; 

 and, in addition, we have to prefix the sign + or - as required. 



Now if - a = to, multiplying each side by y and by any real number vj 

 V 

 as we saw we can do, we get 



xwa = ywza ; 



* xvj 



y ~ " ~ yw y 



or a fraction can be multiplied above and below by any real number without 

 altering its value. 



Addition of Fractions. — 



x x' x" 



- + '— + — - is a real number %, given by 



y y y 



r rr\ 



XX X \ 



- + — + — a = Sa. 



y y y J 



Midtiply each side by yy'y" ; 



••■ ( y y'y" + *yy" + x "yy') « = yy'y" za ; 



xy'y" + x'yy" + x'y y 



yy'y" 



Quadratic Equations. — In problems, such arise in the form 



(ic 2 + 2ax ±b)a = 0. 

 Adding (+■ « 2 - a 2 ) a, if « 2 - b is +, we get 



((./• ± a) 2 - (a 2 ±b)}a = (it ± a + </ar ± b) (x ± a - y/a- ± b) a = 0. 

 This equals 0, when and only when, either 



x ± a + y/a' ± b = or x ± a - ^/a'- ±5 = 0. 



Thus, the mathematical game played with the units a, (3 is incomplete, as 

 we cannot find a real number x such that x 2 a = aQ = - aa ; nor can we 

 change x 2 + a' into the form (x + w) (x + w') ; nor can we alter of ± 2ax + b 

 into the form (x + w) (x + vj), in other words, find its factors when a 2 - b is -. 



Example of a Problem. — What are eggs a dozen, if two more in a shilling's 

 worth lowers the price a penny a dozen ? 



Here two sets of units occur. Let a denote an egg I am to receive, |3 one 

 I am to give away, a a penny I am to pay, /3' a penny I am to receive. If 



