Rational Expressions

  Rational Numbers:

 Loosely speaking,  a rational number is an expression of the form p/q where and q are integers and q 0. What is ``loose'' about this  definition is that two distinct expressions p/q  and r/s may represent the same rational number.   1/For example,   1/4 and 2/8 represent the same rational  number.  This equivalence is reflected by the fact  if we only think about the quantity involved, having 1/4 of a pie is the same as having 2/8 of a pie.

Expressions of the form p/q where p and q are integers and q 0 are frequently called fractions.  What we really want to say is that rational numbers are the abstract quantities that are represented by fractions.

The rule is that the fractions  p/q and r/s represent the same  rational number  (and we write p/s = r/q)  if, and only if,  ps = rq as integers (that is, the product of the integers  p and s equals the product of the integers r and q).   The operations of  addition, subtraction and multiplication of integers are used to define corresponding operations on rational numbers.   Thus we define:
 

  where  on the left side of the equations the +,- and × denote the operation on  the rationals while on the right side of the equations they denote the operations  on integers.

We can also define an operation of addition for rational numbers for rational expressions  p/q and  r/s where r0.

An important property of the rational numbers is that any  expression built up  from rational  numbers using addition,  subtraction , multiplication and division  can be reduced  (simplified) to a rational number.

Given any  integer p we can regard it as being the same as the rational number  p/1. Furthermore,  the definitions of addition, subtraction and multiplication of rational numbers are such that they agree  with the definitions for those operations on the integers, that is
 

Because of this it is common to speak of the integers as if they were rationals and to write  rather than p/1 for the rational number corresponding to  an integer p. Technically this is incorrect, but  it is certainly convenient  and  while it may lead to conceptual errors it does not lead to arithmetic errors.

Rational Expressions

An important generalization of a rational number  is the concept of a rational expression.  A  rational expression  is an expression of  form p/q  where p and q are polynomials and  q\[NotEqual] 0. Broadly speaking,  everything we said above
concerning integers and rational numbers can be carried over to polynomials and rational expressions. In particular,  we can use exactly the same rules for defining equality,  addition,  subtraction, multiplication and division of  rational expressions as we did for rational numbers,  but using the underlying operatiions of addtion,  subtraction,  and multiplication of polynomials in place of these operations on integers. It is worthwhile stating  the definition of equality of rational expressions in detail:

Given two rational expressions  p/q  and r/s we say they are equal,  and write p/q = r/s, if , and only if,  ps = rq  as polynomials.
 
 

 Interpretation of Rational expressions:

Any rational expression can be interpreted as a partial function taking integers to rational numbers.   Thus,  x/y  can be interpreted as the partial function taking  each pair of integers, <p,q> , to the rational number  p/q. This function is partial because if  q=0  then p/q does not represent a rational number!    Let's look at some slightly more complicated cases. For
example, the rational expressions 1/(x-1) and x/(x-5)  correspond to partial functions,  this is because  1/(x-1) is not defined when x=1  and  x/(x-5) is not defined when x=5 .  In many cases the existence of an undefined point is obvious from the graph of the function corresponding to the rational expression.  For example the graph of 1/(x-1) is as follows


However,  there are some very important cases where the partiality of the
function is not obvious from the graph. An example would be the rational
expression (x2-5x)/(x-5).  The function corresponding to this expression is not defined at  x=5.   In
particular,  its corresponding function is not the same as that for the
polynomial x  even though it might appear that we could cancel out the  two
occurences of (x-5). However the only difference between the two functions is
at the point x=5.  Both  graphs "look like"

but the graph for  (x2-5x)/(x-5) has one point missing!.

What  this example shows is that two rational expressions may be "equal" without this meaning that their corresponding functions are identical.   As we  will see later,  it follows from this that two rational expressions may be equal  but still not have the same set of solutions  (values of the variables for which the corresponding function takes the value 0). However, it will be the case that if two rational expressions are equal and their corresponding functions are both defined for a given value of the variables then  the corresponding functions will also be equal for those arguments.

 It is very easy to draw incorrect conclusions if one allows  division by zero   in  algebraic manipulations.  Here is an  (amusing ?) example

  "Claim":  All numbers are equal.

   "Proof":  Let  A and B be two arbitrarily chosen numbers. Let C= -B and let  D = A+C.  Now multiply both sides of that equation by (A+C) getting


 Rearranging the terms we get
 


 Now (A + C + D)  is a common factor of both sides, giving us

 from which it follows that  A = -C.  But then, since C=-B, we get A=B.  Thus,
 because A and B were chosen arbitrarily, it  follows  that all numbers are
equal.

 Exercise:  Find the error in the above "proof".