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1.5 The Archimedean Property 31

1.5 The Archimedean Property

In this section, we discuss a property so natural that it may seem hard to imag-
ine an ordered field without it. Indeed, an ordered field without this property
would not be acceptable as a number system. We would like to use our number
system to record measurements, for example, in geometry. We often let the
number 1 stand for a unit of measurement. Then, given any positive quantity
x, we expect to be able to reach beyond x by "counting off" sufficiently many
l 's. An ordered field with this property will be called an Archimedean ordered
field. Not every ordered field has this property.

Definition 1.5.1 An ordered field F is Archimedean if it satisfies the

Archimedean property: Vx E F, :Jn EN 3 n > x.

That is, for every element x in F , there is a natural number larger than x. The
field of rational numbers is an example of an Archimedean ordered field (see
Exercise 1). The next theorem shows other, equivalent, forms of this property.

Theorem 1.5.2 Let F be an ordered field. The following properties are equiv-
alent to the Archimedean property^7 in F:

(a) Vx > 0, :Jn E N 3 n > x.

(b) If a> 0, then Vx E F, :Jn E N 3 na > x.

1

"'(c) Ve> 0, :Jn EN 3 - < e.

n

Proof. Our strategy will be to prove (a) => Archimedean prop. => (b) =>

(c) => (a).

(1) (a) => Archimedean property. (Exercise 2)

(2) Archimedean property=> (b). (Exercise 3)

. 1
(3) To prove (b) => (c), suppose (b) 1s true. Let e > 0. Then - > 0. Hence, by
e
1 1 1
(b) with a = 1 and x = -, :Jn E N 3 n · 1 > -. Then ne > 1, so - < e.
e e n
Therefore, Ve > 0, :Jn E N 3 ~ < e. That is, ( c) is true.
n
(4) (c) => (a). (Exercise 4) •
7. Since these three properties are equivalent to the Archimedean property, any one of them
may be called the Archimedean property.