Superreal number

The superreal numbers are a special kind of number that builds on the real numbers we use every day. They were created by two mathematicians, H. Garth Dales and W. Hugh Woodin, to help with advanced areas of math like non-standard analysis, model theory, and the study of Banach algebras. These numbers are like an expanded version of the hyperreal numbers, which themselves are useful for understanding very small or very large values.

The superreal numbers fit inside an even bigger number system called the surreal numbers. This means they share some properties with surreal numbers but also have their own unique features that make them interesting to mathematicians.

There is also another idea called "super-real numbers" created by David O. Tall. These are different from Dales and Woodin's superreals. Tall's version uses special kinds of fractions and power series, which are arranged in a specific order called lexicographical order.

Overall, superreal numbers help mathematicians explore new ideas and solve problems that are difficult with regular real numbers. They show how numbers can be stretched and expanded to open up new areas of mathematical discovery.

Formal definition

In mathematics, a superreal field is a special kind of number system that includes all the normal real numbers, but also has extra numbers. To build this system, mathematicians start with continuous functions on a certain space and then use special rules to create a new field of numbers.

When these rules use a maximal ideal, the resulting field is related to hyperreal numbers, which are used in non-standard analysis. This helps mathematicians study complex problems in different areas of math.