The Rational Numbers Page 3
ADVERTISEMENT
1
2
315. ( 1/1) a =
a for all a
I Q.
16. If a
I Q with a = 0/1, then 0/1 < a a.
17. If a, b, c
I Q with a < b, then
(i) a + c < b + c
(ii) a c < b c, provided 0/1 < c
(iii) b c < a c, provided c < 0/1
18.
is a linear order on I Q (where a
b stands for “a < b or a = b”).
19. If a, b, c
I Q with a + c = b + c, then a = b.
20. If a, b, c
I Q with a c = b c and c = 0/1, then a = b.
Just like the natural numbers I N can be regarded as a subset of the integers Z Z, the
integers Z Z can be regarded as a subset of the rational numbers I Q, namely the set
p/1 p
Z Z . This is a consequence of the following theorem. We will therefore no
longer distinguish between the equivalence class p/1 = [(p, 1)] (which is a set of pairs
of integers) and the integer p.
Theorem 2
The function f : Z Z
I Q given by f (p) = p/1 has the following properties:
(i) If p
, p
Z Z with p
< p
, then f (p
) < f (p
). In particular, f is injective.
1
2
1
2
1
2
(ii) f (p
+ p
) = f (p
) + f (p
) for all p
, p
I Q.
1
2
1
2
1
2
(iii) f (p
p
) = f (p
) f (p
) for all p
, p
I Q.
1
2
1
2
1
2
In short, the function f is a bijection between the set Z Z of integers and the subset
p/1 p
Z Z of I Q which preserves all relevant properties.
Exercises
21. Prove Theorem 2.
1
22. Writing a
b for a (b
) when a, b
I Q and b = 0/1, show that we have
p/q = (p/1)
(q/1) for all p, q
Z Z with q = 0.
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education