20 • Algebra. 



We obtain the following equations : 



If we attempt to go one step further by the same rule, viz : sub- 

 tract one from the exponent, we get 



Now, «° is a symbol that has not been used before, and indeed 

 one that has no meaning according to the definition already given 

 of a power of a number. But we know that a-^a=\, and if we 

 agree that this new symbol <2° shall be i, (^ being any number 

 whatever (not zero), then we may carry our process of successive 

 division one step further than we could without this agreement. 



More than this, it may easily be seen that by giving this mean- 

 ing to ^° each of our formulas (a)^ (b), (c), (d) is slightly more 

 general than it was before. Let us examine these formulas sep- 

 arately. 



11. First, a"a''=a"-^\ 

 If we here make 7^=o, we get 



a°a''=a'', 

 and this is true if a°=i. 



Again, if we make r=o, we get 



a"a''=a'% 

 and this is true if a°= i . 



12. Second, (a''y=.a"''. (d) 

 If we here make n=o, we get 



which is true if a°=i, since i''==i ; and if we make r=o, we get 



and this, too, is true if a?z-y quantity affected with a zero exponent 

 equals one. 



* The student may think that such an equation as 



4°=2° 

 involves the absiardity that 4=2, but it does not. No one thinks that the equation 



4 2 



j=.— involves any absurdity, and so if we look upon a quantity affected with an expon- 

 ent zero as only another way of writing a quantity divided by itself, there is no con- 

 fusion. 



