2 . , NUMBER. [INT. I. 



integers,, which we can v add, multiply, and subtract according to the 

 same laws as v the natural numbers. 



Defining the operation of division as the inverse of multiplica- 

 tion, so that a divided by b is c, if c multiplied by b will give a, 

 we find that the operation can be performed only when b is a factor 

 of a. We are thus again led to extend our definition of numbers 

 so as to call that which, being multiplied by b, will give a, even 

 when b is not a factor of a, a number. We are thus led to the con- 

 ception of fractions, which may be operated upon like the positive 

 and negative integers. Every fraction is of the form m/n where 

 m and n are positive or negative integers. This system of 

 numbers suffices for all the ordinary operations of arithmetic, 

 including the solution of equations of the first degree. 



Any number may be raised to any power, the process being 

 known as involution. If we define evolution, or the operation of 

 taking a root, as the inverse of involution, so that the 6th root of 

 a is c when c b = a, we find that the operation can be performed 

 only when a is one of the series of numbers, 



c, c 2 , c 3 , ....;. 



If we further extend our definition of number, so that that 

 which raised to the 6th power will give a, even in the contrary 

 case, we are led to the conception of irrational numbers. No 

 irrational number can be expressed as the quotient of two integers, 

 though for any given irrational a we can always find two integers 

 such that their quotient differs from a by an amount that is as 

 small as we please. In symbols, if e is any given positive number 

 as small as we please, we can always find m and n so that 



< e. 



By | a | is meant the absolute, or arithmetical value of a, irre- 

 spective of sign. E.g., the square root of 2 is an irrational, but 

 the rules for the extraction of square roots enable us to find a 

 value that differs from it by as little as we please. The ordinary 

 theory of enumeration shows that we can express any rational 

 number in terms of any integer, 6, called the base, as the sum of a 

 definite number of terms, each of which is the product of some 

 integer less than b by some power of b with positive or negative 

 exponent, or else as the limit of a sum of such terms, where as the 



