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.

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.

There is also another idea called "super-real numbers" created by David O. Tall. These are different from Dales and Woodin's superreals.

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. It includes all the normal real numbers and also has extra numbers.

Mathematicians build this system by starting with continuous functions and using special rules to create a new field of numbers.

When these rules use a maximal ideal, the resulting field is related to hyperreal numbers. Hyperreal numbers are used in non-standard analysis to help study complex math problems.