Surreal Numbers DavidFern´andez–Bret´ on Abstract for 01 December 2016 world where expressi Imagine a world where expressions such as dy/dx really represent a quotient. In such a world, dx and dy would need to be quite peculiar entities, capa- ble of somehow interacting with real numbers but not being real numbers themselves: they would be smaller than every positive real number, yet nonzero. These peculiar entities, infinitely smaller than real numbers, are called “infinitesimals.” The field of Surreal Numbers arises as the result of tak- ing this idea to the extreme: starting with the field of Real Numbers, attaching infinitesimals to it, and then attaching “second order infinitesimals” (quantities that are infinitely smaller than infinites- imals themselves), and afterwards attaching also “third order infinitesimals,” and so on... trans- finitely many times.
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
The title is ”Surreal Numbers”
Abstract: Imagine a world where expressions such as dy/dx really repre-sent a quotient. In such a world, dx and dy would need to be quite pecu-liar entities, capable of somehow interacting with real numbers but notbeing real numbers themselves –them being smaller than every positivereal number, yet nonzero–. These peculiar entities, infinitely smaller thanreal numbers, are called ”infinitesimals”. The field of Surreal Numbersarises as the result of taking this idea to the extreme: starting with thefield of Real Numbers, attaching infinitesimals to it, and then attaching”second order infinitesimals” (quantities that are infinitely smaller thaninfinitesimals themselves), and afterwards attaching also ”third order in-finitesimals”, and so on... transfinitely many times.
Surreal Numbers
David Fernandez–Breton
Abstract for 01 December 2016
Imagine a world where expressions such as dy/dx really represent a quo-tient. In such a world, dx and dy would need to be quite peculiar en-tities, capable of somehow interacting with real numbers but not beingreal numbers themselves: they would be smaller than every positive realnumber, yet nonzero. These peculiar entities, infinitely smaller thanreal numbers, are called “infinitesimals.” The field of Surreal Numbersarises as the result of taking this idea to the extreme: starting with thefield of Real Numbers, attaching infinitesimals to it, and then attaching“second order infinitesimals” (quantities that are infinitely smaller thaninfinitesimals themselves), and afterwards attaching also “third order in-finitesimals,” and so on... transfinitely many times.
1
The title is ”Surreal Numbers”
Abstract: Imagine a world where expressions such as dy/dx really repre-sent a quotient. In such a world, dx and dy would need to be quite pecu-liar entities, capable of somehow interacting with real numbers but notbeing real numbers themselves –them being smaller than every positivereal number, yet nonzero–. These peculiar entities, infinitely smaller thanreal numbers, are called ”infinitesimals”. The field of Surreal Numbersarises as the result of taking this idea to the extreme: starting with thefield of Real Numbers, attaching infinitesimals to it, and then attaching”second order infinitesimals” (quantities that are infinitely smaller thaninfinitesimals themselves), and afterwards attaching also ”third order in-finitesimals”, and so on... transfinitely many times.
Surreal Numbers
David Fernandez–Breton
Abstract for 01 December 2016
Imagine a world where expressions such as dy/dx really represent a quo-tient. In such a world, dx and dy would need to be quite peculiar en-tities, capable of somehow interacting with real numbers but not beingreal numbers themselves: they would be smaller than every positive realnumber, yet nonzero. These peculiar entities, infinitely smaller thanreal numbers, are called “infinitesimals.” The field of Surreal Numbersarises as the result of taking this idea to the extreme: starting with thefield of Real Numbers, attaching infinitesimals to it, and then attaching“second order infinitesimals” (quantities that are infinitely smaller thaninfinitesimals themselves), and afterwards attaching also “third order in-finitesimals,” and so on... transfinitely many times.
1
Surreal Numbers
David Fernandez–Breton
Abstract for01 December 2016
Imagine a world where expressions suchas dy/dx really represent a quotient. Insuch a world, dx and dy would needto be quite peculiar entities, capable ofsomehow interacting with real numbersbut not being real numbers themselves:they would be smaller than every pos-itive real number, yet nonzero. Thesepeculiar entities, infinitely smaller thanreal numbers, are called “infinitesimals.”The field of Surreal Numbers arises asthe result of taking this idea to the ex-treme: starting with the field of RealNumbers, attaching infinitesimals to it,and then attaching “second order in-finitesimals” (quantities that are infin-itely smaller than infinitesimals them-selves), and afterwards attaching also“third order infinitesimals,” and so on...transfinitely many times.
1
Surreal Numbers
David Fernandez–Breton
Abstract for01 December 2016
Imagine a world where expressions such as dy/dxreally represent a quotient. In such a world, dx anddy would need to be quite peculiar entities, capa-ble of somehow interacting with real numbers butnot being real numbers themselves: they wouldbe smaller than every positive real number, yetnonzero. These peculiar entities, infinitely smallerthan real numbers, are called “infinitesimals.” Thefield of Surreal Numbers arises as the result of tak-ing this idea to the extreme: starting with the fieldof Real Numbers, attaching infinitesimals to it,and then attaching “second order infinitesimals”(quantities that are infinitely smaller than infinites-imals themselves), and afterwards attaching also“third order infinitesimals,” and so on... trans-finitely many times.
1
Related Documents
