PureMath.pdf

# In other cases φ n may be defined by a formula such

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In other cases φ ( n ) may be defined by a formula, such as ( - 1) n , which ceases to define for some values of x (as here in the case of fractional values of x with even denominators, or irrational values). But it may be possible to transform the formula in such a way that it does define for all values of x . In this case, for example, ( - 1) n = cos nπ, if n is an integer, and the problem of interpolation is solved by the func- tion cos . In other cases φ ( x ) may be defined for some values of x other than positive integers, but not for all. Thus from y = n n we are led to y = x x . This expression has a meaning for some only of the remaining values of x . If for simplicity we confine ourselves to positive values of x , then x x has a meaning for all rational values of x , in virtue of the definitions of fractional powers adopted in elementary algebra. But when x is irrational x x has (so far as we are in a position to say at the present moment) no meaning at all. Thus in this case the problem of interpolation at once leads us to consider the question of extending our definitions in such a way that x x shall have a meaning even when x is irrational. We shall see later on how the desired extension may be effected. Again, consider the case in which y = 1 · 2 . . . n = n ! . In this case there is no obvious formula in x which reduces to n ! for x = n , as x ! means nothing for values of x other than the positive integers. This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides. 52. Finite and infinite classes. Before we proceed further it is necessary to make a few remarks about certain ideas of an abstract and logical nature which are of constant occurrence in Pure Mathematics. In the first place, the reader is probably familiar with the notion of a class . It is unnecessary to discuss here any logical difficulties which may be involved in the notion of a ‘class’: roughly speaking we may say that

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[IV : 53] POSITIVE INTEGRAL VARIABLE 131 a class is the aggregate or collection of all the entities or objects which possess a certain property, simple or complex. Thus we have the class of British subjects, or members of Parliament, or positive integers, or real numbers. Moreover, the reader has probably an idea of what is meant by a finite or infinite class. Thus the class of British subjects is a finite class: the aggregate of all British subjects, past, present, and future, has a finite number n , though of course we cannot tell at present the actual value of n . The class of present British subjects , on the other hand, has a number n which could be ascertained by counting, were the methods of the census effective enough.
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