Why Does -1/12 Keep Appearing Everywhere In Physics?
From PBS Space Time.
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Can adding every positive whole number really give you -1/12? We explore how physicists extract meaningful finite values from infinite sums, and how this strange mathematical technique shows up in the measurable Casimir effect and even helps explain why bosonic string theory requires 26 dimensions.
.
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Hosted by Matt O'Dowd
Written by Richard Dyer & Matt O'Dowd
Post Production by Leonardo Scholzer
Directed by Andrew Kornhaber
Associate Producer: Bahar Gholipour
Executive Producer: Andrew Kornhaber
Executive in Charge for PBS: Maribel Lopez
Director of Programming for PBS: Gabrielle Ewing
Assistant Director of Programming for PBS: Mike Martin
Spacetime is a production of Kornhaber Brown for PBS Digital Studios.
This program is produced by Kornhaber Brown, which is solely responsible for its content.
© 2026 PBS. All rights reserved.
End Credits Music by J.R.S. Schattenberg: https://www.youtube.com/user/MultiDroideka
Space Time Was Made Possible In Part By:
Big Bang
Alexander Tamas
Filip Rolenec
Juan Benet
Kenneth See
Mark Rosenthal
MATTHEW OCKO
Morgan Hough
Peter Barrett
Vinnie Falco
Supernova
Ethan Cohen
Grace Biaelcki
Justin Lloyd
Mark Heising
Rad Antonov
Stephen Wilcox
Tristan Lucian Claudius Aurelius Tyacke
Hypernova
Alex Kern
Ben Delo
chuck zegar
Dean Galvin
Donal Botkin
D. Rollere
Gregory Forfa
Jeff White
John R. Slavik
Paul C. Pedersen
Pala Massimiliano
Patrick O’hara
Scott Gorlick
Scott Gray
Spencer Jones
Vlad Shipulin
Zachary Haberman
Антон Кочков
Gamma Ray Burst
Alex Gan
Aaron Pinto
Almog Cohen
Arko Provo Mukherjee
Austin Richards
Ayden Miller
Bert Whetstone
Brandon Lattin
Brandon R
Cal Harrington
Cal Stephens
Chris Liao
Collin Dutrow
Craig Falls
Craig Stonaha
Dan Warren
Daniel Donahue
Daniel Jennings
Darrell Stewart
David Giltinan
David Johnston
Diana S Polijar
Donovan King
Doug Richardson
Doyle Vann
Eric Kiebler
Eric Raschke
Eric Schrenker
Faraz Khan
Frederic Simon
Harsh Khandhadia
Isaac Suttell
James Trimmier
Jason Holtkamp
Jeb Campbell
Jeff Harris
Jerry Thomas
Jim Bartosh
John Anderson
John De Witt
John Hanavan
John Mieras USN Ret
Joseph Salomone
Junaid Ali
Justin Zheng
Kacper Cieśla
Kent Durham
Koen Wilde
Kyle Atkinson
Lori Ferris
Marcelo Garcia
Mark Daniel Cohen
Mark Delagasse
Mathew Ray
Matt Kaprocki
Matthew Johnson
Matthew Twyman
Michael Barton
Michael Clark
Michael Lev
Michael Purcell
Michael Spallino
Mikk Mihkel Nurges
Nicolas Katsantonis
Nils Anderson
Paul Wood
Reuben Brewer
Robert DeChellis
Ross Kennedy
Ross Story
Russell Moore
SamSword
Sandhya Devi
Sean Owen
Shane Calimlim
Terje Vold
Terrance Cennon
Thomas Dougherty
Tybie Fitzhugh
WILLIAM HAY III
Zac Sweers
00:00 The Impossible Equation: 1 + 2 + 3 + … = -1/12?
01:59 The Riemann Zeta Function
02:46 How Analytic Continuation Makes -1/12 Possible
03:28 The Casimir Effect: A Force From Empty Space
05:28 How Physicists Tame Infinity
08:16 What Does This Have to Do With String Theory?
11:59 Why String Theory Needs 26 Dimensions
13:16 What Does -1/12 Actually Mean?
Can adding every positive whole number really give you -1/12? We explore how physicists extract meaningful finite values from infinite sums, and how this strange mathematical technique shows up in the measurable Casimir effect and even helps explain why bosonic string theory requires 26 dimensions.
.
Sign Up on Patreon to get access to the Space Time Discord!
https://www.patreon.com/pbsspacetime
Check out the Space Time Merch Store
https://www.pbsspacetime.com/shop
Sign up for the mailing list to get episode notifications and hear special announcements!
https://mailchi.mp/1a6eb8f2717d/spacetime
Search the Entire Space Time Library Here: https://search.pbsspacetime.com/
Hosted by Matt O'Dowd
Written by Richard Dyer & Matt O'Dowd
Post Production by Leonardo Scholzer
Directed by Andrew Kornhaber
Associate Producer: Bahar Gholipour
Executive Producer: Andrew Kornhaber
Executive in Charge for PBS: Maribel Lopez
Director of Programming for PBS: Gabrielle Ewing
Assistant Director of Programming for PBS: Mike Martin
Spacetime is a production of Kornhaber Brown for PBS Digital Studios.
This program is produced by Kornhaber Brown, which is solely responsible for its content.
© 2026 PBS. All rights reserved.
End Credits Music by J.R.S. Schattenberg: https://www.youtube.com/user/MultiDroideka
Space Time Was Made Possible In Part By:
Big Bang
Alexander Tamas
Filip Rolenec
Juan Benet
Kenneth See
Mark Rosenthal
MATTHEW OCKO
Morgan Hough
Peter Barrett
Vinnie Falco
Supernova
Ethan Cohen
Grace Biaelcki
Justin Lloyd
Mark Heising
Rad Antonov
Stephen Wilcox
Tristan Lucian Claudius Aurelius Tyacke
Hypernova
Alex Kern
Ben Delo
chuck zegar
Dean Galvin
Donal Botkin
D. Rollere
Gregory Forfa
Jeff White
John R. Slavik
Paul C. Pedersen
Pala Massimiliano
Patrick O’hara
Scott Gorlick
Scott Gray
Spencer Jones
Vlad Shipulin
Zachary Haberman
Антон Кочков
Gamma Ray Burst
Alex Gan
Aaron Pinto
Almog Cohen
Arko Provo Mukherjee
Austin Richards
Ayden Miller
Bert Whetstone
Brandon Lattin
Brandon R
Cal Harrington
Cal Stephens
Chris Liao
Collin Dutrow
Craig Falls
Craig Stonaha
Dan Warren
Daniel Donahue
Daniel Jennings
Darrell Stewart
David Giltinan
David Johnston
Diana S Polijar
Donovan King
Doug Richardson
Doyle Vann
Eric Kiebler
Eric Raschke
Eric Schrenker
Faraz Khan
Frederic Simon
Harsh Khandhadia
Isaac Suttell
James Trimmier
Jason Holtkamp
Jeb Campbell
Jeff Harris
Jerry Thomas
Jim Bartosh
John Anderson
John De Witt
John Hanavan
John Mieras USN Ret
Joseph Salomone
Junaid Ali
Justin Zheng
Kacper Cieśla
Kent Durham
Koen Wilde
Kyle Atkinson
Lori Ferris
Marcelo Garcia
Mark Daniel Cohen
Mark Delagasse
Mathew Ray
Matt Kaprocki
Matthew Johnson
Matthew Twyman
Michael Barton
Michael Clark
Michael Lev
Michael Purcell
Michael Spallino
Mikk Mihkel Nurges
Nicolas Katsantonis
Nils Anderson
Paul Wood
Reuben Brewer
Robert DeChellis
Ross Kennedy
Ross Story
Russell Moore
SamSword
Sandhya Devi
Sean Owen
Shane Calimlim
Terje Vold
Terrance Cennon
Thomas Dougherty
Tybie Fitzhugh
WILLIAM HAY III
Zac Sweers
00:00 The Impossible Equation: 1 + 2 + 3 + … = -1/12?
01:59 The Riemann Zeta Function
02:46 How Analytic Continuation Makes -1/12 Possible
03:28 The Casimir Effect: A Force From Empty Space
05:28 How Physicists Tame Infinity
08:16 What Does This Have to Do With String Theory?
11:59 Why String Theory Needs 26 Dimensions
13:16 What Does -1/12 Actually Mean?
