Hackenbush and

Surreal Numbers

Christopher CunninghamElgin Community College

Abstract for 24 October 2019

The game of Blue-Red Hackenbush is simple to play, but answering the question “howmuch am I winning by?” turns out to be complicated. We will delve into the game tosee that it is possible to be ahead by any integer number of moves, and then any rationalnumber of moves. When we try to find a game that puts us ahead by any real number ofmoves, we will overshoot a bit and accidentally invent the surreal numbers, which includenumbers like ! (larger than any real number), !+1 (one more than that), and !�1 (justa bit less than infinity). Oh, and don’t forget 1/! (which is infinitesimally small). Whilewe are here, we might as well build a game where we are winning by exactly !�⇡ moves.This talk is accessible to anyone willing to play some games, work hard to understandthem, and think about infinity.

z = 0

H := {c 2 C | fc has an attracting cycle}

1

Hackenbush and

Surreal Numbers

Christopher CunninghamElgin Community College

Abstract for 24 October 2019

The game of Blue-Red Hackenbush is simple to play, but answering the question “howmuch am I winning by?” turns out to be complicated. We will delve into the game tosee that it is possible to be ahead by any integer number of moves, and then any rationalnumber of moves. When we try to find a game that puts us ahead by any real number ofmoves, we will overshoot a bit and accidentally invent the surreal numbers, which includenumbers like ! (larger than any real number), !+1 (one more than that), and !�1 (justa bit less than infinity). Oh, and don’t forget 1/! (which is infinitesimally small). Whilewe are here, we might as well build a game where we are winning by exactly !�⇡ moves.This talk is accessible to anyone willing to play some games, work hard to understandthem, and think about infinity.

z = 0

H := {c 2 C | fc has an attracting cycle}

1

Hackenbush and

Surreal Numbers

Christopher CunninghamElgin Community College

Abstract for 24 October 2019

The game of Blue-Red Hackenbush is simple to play, but answering the question “howmuch am I winning by?” turns out to be complicated. We will delve into the game tosee that it is possible to be ahead by any integer number of moves, and then any rationalnumber of moves. When we try to find a game that puts us ahead by any real number ofmoves, we will overshoot a bit and accidentally invent the surreal numbers, which includenumbers like ! (larger than any real number), !+1 (one more than that), and !�1 (justa bit less than infinity). Oh, and don’t forget 1/! (which is infinitesimally small). Whilewe are here, we might as well build a game where we are winning by exactly !�⇡ moves.This talk is accessible to anyone willing to play some games, work hard to understandthem, and think about infinity.

z = 0

H := {c 2 C | fc has an attracting cycle}

1

Hackenbush and

Surreal Numbers

Christopher CunninghamElgin Community College

Abstract for 24 October 2019

The game of Blue-Red Hackenbush is simple to play, but answering the question“how much am I winning by?” turns out to be complicated. We will delve intothe game to see that it is possible to be ahead by any integer number of moves,and then any rational number of moves. When we try to find a game that puts usahead by any real number of moves, we will overshoot a bit and accidentally inventthe surreal numbers, which include numbers like ! (larger than any real number),! + 1 (one more than that), and !� 1 (just a bit less than infinity). Oh, and don’tforget 1/! (which is infinitesimally small). While we are here, we might as wellbuild a game where we are winning by exactly ! � ⇡ moves. This talk is accessibleto anyone willing to play some games, work hard to understand them, and thinkabout infinity.

z = 0

H := {c 2 C | fc has an attracting cycle}

1