The Rational Numbers Page 2
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3Exercises
3. Show that addition, multiplication, order, negation, and taking reciprocals of
rational numbers is well-defined. That is, show that these “definitions” do not
depend on the particular choice of representatives.
As was the case for the integers, there is a list of basic properties of rational numbers
from which all other properties can be deduced. Here is that list:
[Keep in mind that every rational number a
I Q is to be represented by an equivalence
class a = p/q = [(p, q)] with p, q
Z Z and q = 0.]
Exercises
4. Let a, b, c
I Q. Show that
(i) a + b = b + a
(ii) (a + b) + c = a + (b + c)
(iii) a b = b a
(iv) (a b) c = a (b c)
(v) (a + b) c = (a c) + (b c)
5. a + (0/1) = a for all a
I Q.
6. a + ( a) = 0/1 for all a
I Q.
7. a (1/1) = a for all a
I Q.
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8. a a
= 1/1 for all a
I Q with a = 0/1.
9. For all a, b
I Q, a < b if and only if 0/1 < b
a.
Here, we write b
a for b + ( a) as we did before.
10. For each a
I Q exactly one of the following is true: either 0/1 < a, or 0/1 <
a,
or 0/1 = a.
11. If a, b
I Q with 0/1 < a and 0/1 < b, then 0/1 < a + b and 0/1 < a b.
If you tried to solve Exercises 11 through 18 on the integers without directly referring
to equivalence classes, then you already have a proof of most of the following exercises.
Exercises
12. a (0/1) = 0/1 for all a
I Q.
13.
( a) = a for all a
I Q.
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14. (a
)
= a for all a
I Q with a = 0/1.
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