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 in some math 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. When we multiply the smaller number by this whole number, it 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 sizes and amounts. It is still used today in modern math. 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 is true, but it does not apply to all math 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. It talks about how sizes can compare to each other.

Archimedes used very small numbers in his thinking, but he did not use them as official math rules. Because Archimedes gave credit to Eudoxus of Cnidus, this idea is sometimes called the "Theorem of Eudoxus" or the Eudoxus axiom.

Definition for linearly ordered groups

Main article: Archimedean group

In math, the Archimedean property helps us see how numbers compare. Imagine you have two positive numbers, x and y. The Archimedean property says you can always find a whole number, n. If you add x to itself n times, the total will be bigger than y.

This shows that numbers like 1, 2, 3, and so on can keep growing forever. This idea is useful when we study groups and fields of numbers. It helps us tell apart different kinds of number systems. Some systems follow this rule, while others, called non-Archimedean, work in different ways.

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, you can always multiply the smaller number by a whole number enough times to get past 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. These are numbers so small that they are almost zero but still positive. This is one reason why the real numbers are 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 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.