PRELIMINARY ALGEBRA
be obvious, but they are both formed in the same way in terms of recurrence
relations. Whatever the sign ofn, the series of coefficientsnCkcan be generated
by starting withnC 0 = 1 and using the recurrence relation
nC
k+1=n−k
k+1nC
k. (1.59)The difference is that for positive integernthe series terminates whenk=n,
whereas for negativenthere is no such termination – in line with the infinite
series of terms in the corresponding expansion.
Finally we note that, in fact, equation (1.59) generates the appropriate coef-ficients for all values ofn, positive or negative, integer or non-integer, with the
obvious exception of the case in whichx=−yandnis negative. For non-integer
nthe expansion does not terminate, even ifnis positive.
1.7 Some particular methods of proofMuch of the mathematics used by physicists and engineers is concerned with
obtaining a particular value, formula or function from a given set of data and
stated conditions. However, just as it is essential in physics to formulate the basic
laws and so be able to set boundaries on what can or cannot happen, so it
is important in mathematics to be able to state general propositions about the
outcomes that are or are not possible. To this end one attempts to establish
theorems that state in as general a way as possible mathematical results that
apply to particular types of situation. We conclude this introductory chapter by
describing two methods that can sometimes be used to prove particular classes
of theorems.
The two general methods of proof are known as proof by induction (whichhas already been met in this chapter) and proof by contradiction. They share
the common characteristic that at an early stage in the proof an assumption
is made that a particular (unproven) statement is true; the consequences of
that assumption are then explored. In an inductive proof the conclusion is
reached that the assumption is self-consistent and has other equally consistent
but broader implications, which are then applied to establish the general validity
of the assumption. A proof by contradiction, however, establishes an internal
inconsistency and thus shows that the assumption is unsustainable; the natural
consequence of this is that the negative of the assumption is established as true.
Later in this book use will be made of these methods of proof to explore newterritory, e.g. to examine the properties of vector spaces, matrices and groups.
However, at this stage we will draw our illustrative and test examples from earlier
sections of this chapter and other topics in elementary algebra and number theory.