Axiom of Archimedes
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The Axiom of Archimedes can be stated in modern notation as follows:
Let x be any real number. Then there exists a natural number n such that n > x.
In field theory this statement is called the Axiom of Archimedes. The same name is also applied to similar statements about other fields or other systems of magnitudes; chiefly as one of David Hilbert's axioms for geometry.
In modern Real Analysis, it is not an axiom. It is rather a consequence of the completeness of the real numbers. For this reason it is often referred to as the Archimedean property of the reals instead.
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[edit] Formal Statement and Proof
Formally, the Archimedean Property can be stated as follows:
Let c, ε be in R, the real numbers. Then the following two properties hold:
(i) For any positive c, there exists a natural number n such that n > c.
(ii) For any positive ε, there exists a natural number n, such that 1/n < ε.
The proof follows from the completeness of the real numbers:
First, observe that (i) and (ii) are equivalent. Indeed, if ε = 1/c, then (i) follows from (ii), and (ii) follows from (i), so the two are equivalent.
We will prove (i) (thus also proving (ii)), using a proof by contradiction. Suppose there is a positive number c such that there is no natural number n greater than c. Clearly, n ≤ c for every n. This implies that the set of natural numbers, N, is bounded above by c. Thus, the completeness axiom of R asserts that N has a least upper bound, which we will call b.
Since b is the least upper bound of N, b - 1/2 is not an upper bound of N. Hence, we can choose an n in N such that n > b - 1/2. This implies that n + 1 > b - 1/2 + 1 > b. Therefore, n + 1 is a natural number that is larger than b. This contradicts b being an upper bound of N. This is a contradiction, which implies there is no upper bound for N.
[edit] Interpretation
In simple terms, the Archimedean Property can be thought of as either of the following two statements:
(1) Given any number, you can always pick another number that is larger than the original number.
(2) Given any positive number, you can always pick another positive number that is less than the original number.
These two statements correspond to (i) and (ii), respectively. To a mathematician, (1) and (2) indicate that the Archimedean property is capturing an important intuitive property of the real numbers.
[edit] Example Usage
One of the most important uses of the Archimedean property in analysis is proving the important result:
as 
This is the statement that the sequence of unit fractions {1 / n} converges to 0.
Proof:
Let ε > 0. We need to find an N such that 1 / n < ε for all n ≥ N.
Using the Archimedean property, we can choose N such that 1 / N < ε. Thus 1 / n ≤ 1 / N < ε for all n ≥ N.
This statement is vital in establishing many of the properties of sequences of real numbers.
[edit] History
The first known statement of what is now called Archimedes Axiom is found in the writing of Eudoxus. The term itself was first used by the Austrian mathematician Otto Stolz in 1883. An equivalent statement was also used by David Hilbert as one of his axioms of modern Euclidean geometry. See Hilbert's Axioms.
[edit] References
- ^ Fitzpatrick, Patrick M. (2006) Advanced Calculus (2nd ed.). Belmont, CA: Thompson Brooks/Cole. ISBN 0-534-37603-7.
- ^ Eric W. Weisstein. "Archimedes' Axiom." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedesAxiom.html, Retrieved February 28, 2006.
