Archimedean property

The Archimedean property is a key idea in abstract algebra and analysis, named after the ancient Greek mathematician Archimedes of Syracuse. It describes a special feature found in certain mathematical structures, like ordered groups and fields. In simple terms, this property tells us that for any two positive numbers, no matter how small or large, we can always find a whole number that, when multiplied by the smaller number, becomes bigger than the larger number.

This property is important because it means there are no "infinitely large" or "infinitely small" elements in these structures. It was Otto Stolz who named this concept after Archimedes, as it appears in Archimedes’ work On the Sphere and Cylinder.

The Archimedean property comes from ancient Greek ideas about magnitudes and is still used today in modern mathematics. For example, it is used in David Hilbert's axioms for geometry and in the study of ordered groups, ordered fields, and local fields. In fields like the real numbers, this property holds true, but it does not apply to all mathematical structures, such as rational functions with real coefficients.

History and origin of the name of the Archimedean property

The Archimedean property was named by Otto Stolz in the 1880s after the ancient Greek mathematician Archimedes of Syracuse. This idea appears in Book V of Euclid's Elements as Definition 4, which talks about how magnitudes can have a ratio to one another.

Archimedes often used infinitesimals in his thinking, even though he did not consider them to be formal mathematical proofs. Because Archimedes credited this idea to Eudoxus of Cnidus, it is sometimes called the "Theorem of Eudoxus" or the Eudoxus axiom.

Definition for linearly ordered groups

Main article: Archimedean group

In mathematics, the Archimedean property helps us understand how numbers compare to each other. Imagine you have two positive numbers, x and y. The Archimedean property tells us that you can always find a whole number, n, so that when you add x to itself n times, the total becomes bigger than y. This shows that the natural numbers, like 1, 2, 3, and so on, can grow without stopping.

This idea is important in studying ordered groups and fields, where it helps distinguish between different types of number systems. Some systems follow this property, while others, called non-Archimedean, have different rules for how numbers add up.

Examples and non-examples

Archimedean property of the real numbers

The real numbers follow the Archimedean property. This means that for any two positive numbers, no matter how small one is compared to the other, you can always multiply the smaller number by a whole number (like 1, 2, 3, and so on) enough times to exceed the larger number. For example, if you have a very tiny number like 0.001 and a larger number like 10, multiplying 0.001 by 10,001 will give you 10.001, which is just a bit more than 10.

Real numbers also do not have "infinitesimally small" numbers — numbers so small that they are practically zero but still positive. This is one reason why the real numbers are considered Archimedean.

Non-Archimedean ordered field

There are other number systems that do not follow the Archimedean property. For example, consider functions made by dividing one polynomial by another, where the coefficients (numbers used in the polynomials) are real numbers. We can give these functions an order, similar to how we order regular numbers.

In this system, there exists a function like 1/x, which is always positive but gets smaller as x gets larger. No matter how many times you add 1/x to itself, it will never exceed 1. This means 1/x behaves like an "infinitesimally small" number, and this system is therefore non-Archimedean.

Equivalent definitions of Archimedean ordered field

An Archimedean field is one where the natural numbers (1, 2, 3, ...) grow without bound. In such a field, every number is smaller than some natural number. Another way to think about it is that between any two different numbers, no matter how close they are, there is always a fraction (like 1/2, 1/3, etc.) that fits in between them. This makes the rational numbers "dense" within the field.