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WEEK 1: THE AXIOMS OF AN ORDERED FIELD
Lemma (1.1). If a, b, c are numbers and a
ba
c, then b c.
Corollary (1.2, Uniqueness of additive identity). Suppose there exist numbers x, a
such that a x a. Then x 0.
Corollary (1.3, Uniqueness of additive inverses). Let a be a number. There exists
a unique number a such that a paq 0.
paq a.
Lemma (1.5). If a, b, c are numbers, a 0 and a.b a.c, then b c.
Lemma (1.6, Zero multiplication). If a is a number, then a.0 0.
Lemma (1.7, No zero divisors). Suppose a, b are numbers and a.b 0. Then a 0
or b 0.
Corollary (1.8). For a, b numbers, a.b 0 if and only if a 0 or b 0.
Corollary (1.4). For any number a one has
Lemma (1.9). Let a, b be numbers.
i) paq.b pa.bq
ii) paq.pbq a.b
Lemma (2.2, Trichotomy law for
following holds:
i) a b,
ii) a ¡ b,
iii) a b.
).
If a, b are numbers, then precisely one of the
Lemma (2.3, Properties of ). If a, b, c are numbers, then the following hold.
i) If a b, then a c a b.
ii) If a b and b c, then a c (the relation is transitive).
iii) If a, b ¡ 0 or a, b 0, then a.b ¡ 0.
Corollary (2.4). If a is a number and a 0, then a2
¡ 0. In particular, 1 ¡ 0.
Theorem (2.6, Triangle inequality). If a, b are numbers, then
|a
b| ¤ |a|
| b|
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]
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