Loosely speaking, a rational number is an expression of the
form p/q where
p and q are integers
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
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
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
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.
p/q + r/s = (ps+ rq)/
p/q - r/s = (ps-rq)/ qs
r/s = (pr)/(qs)
We can also define an operation of addition for rational numbers for
rational expressions p/q and
r/s = (ps)/(qr)
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 p 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.
p/1 + q/1 = (p+q)/1
p/1 - q/1 = (p-q)/1
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
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
example, the rational expressions 1/(x-1)
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)
However, there are some very important cases where the partiality
function is not obvious from the graph. An example would be the rational
The function corresponding to this expression is not defined at
particular, its corresponding function is not the same as that
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)
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
Rearranging the terms we get
+ AC -AD = -AC -C2 +CD
Now (A + C + D) is
a common factor of both sides, giving us
+ C - D) = -C(A + C - D)
from which it follows that A = -C.
But then, since C=-B, we get A=B.
because A and B
were chosen arbitrarily, it follows that all numbers are
Exercise: Find the error in the above "proof".