8/9/2019 At the Very End–It All Adds Up to Minus One–Twelfth by Dennis Overbye From the NY Times Dated 04 February 2014

http://slidepdf.com/reader/full/at-the-very-endit-all-adds-up-to-minus-onetwelfth-by-dennis-overbye-from 1/3

 At The Very End–It All Adds Up To Minus One–Twelfth ByDennis Overbye From The NY Times Dated 04 February 2014

This is what happens when you mess with infnity. You might thin that i! you simp"y started adding the natura" numbers# 1 p"us 2 p"us $ and so on a"" theway to infnity# you wou"d get a pretty big number. %t "east & a"ways did.

'o it (ame as a sho( to a "ot o! peop"e when# in a re(ent  video# a pair o!

physi(ists purported to prove that this infnite series a(tua""y adds up to ...minus1)12. To date some 1.* mi""ion peop"e have viewed this (a"(u"ation# whi(h p"ays aey ro"e in modern physi(s and +uantum theory, the answer# as absurd as it maysound# has been verifed to many de(ima" p"a(es in "ab e-periments. %!terwat(hing the video myse"!# & (he(ed to mae sure & sti"" had my wa""et and mywat(h.

ven the maers o! the video# Brady /aran# a ourna"ist# and d ope"andand %ntonio adi""a# physi(ists at the 3niversity o! ottingham in ng"and# admitthere is a (ertain amount o! 5ho(us6po(us#7 or what some mathemati(ians have(a""ed dirty tri(s# in their presentation. 8hi(h has "ed to some on"ine grumb"ing.

 ASTOUNDING: 1 ! " # $ %%% & '1(1! Video )y Nu*)erphileBut there is broad agreement that a more rigorous approa(h to the

prob"em gives the same resu"t# as shown by a !ormu"a in 9oseph o"(hinsi:s two6 vo"ume te-tboo 5'tring Theory.7 'o what is it that is going on with infnity;

5This (a"(u"ation is one o! the best6ept se(rets in math#7 said dwardFrene"# a mathemati(s pro!essor at the 3niversity o! a"i!ornia# Bere"ey# and

8/9/2019 At the Very End–It All Adds Up to Minus One–Twelfth by Dennis Overbye From the NY Times Dated 04 February 2014

http://slidepdf.com/reader/full/at-the-very-endit-all-adds-up-to-minus-onetwelfth-by-dennis-overbye-from 2/3

author o! 5<ove and =ath> The /eart o! /idden ?ea"ity#7 @Basi( Boos# 201$A#who was in town re(ent"y promoting his boo and a(ting as an ambassador !orbetter math edu(ation. 5o one on the outside nows about it.7

The great 1th6(entury mathemati(ian <eonhard u"er# who was born in'witCer"and but did most o! his wor in Ber"in and 't. etersburg# ?ussia# wasthe frst one down this road. u"er wanted to now i! you (ou"d fnd an answer toend"ess sums o! numbers "ie 1 p"us 1)2 p"us 1)$ p"us 1)4 on up to infnity# or thes+uares o! those !ra(tions..

These are a"" dierent versions o! what has be(ome nown as the ?iemannCeta !un(tion# a!ter Bernhard ?iemann# who (ame a"ong about a (entury a!teru"er. The Ceta !un(tion is one o! the more mysterious and (e"ebrated sube(ts inmathemati(s# important in the theory o! prime numbers# among other things. &twas one o! the p"ot threads# !or e-amp"e# in Thomas yn(hon:s 200E nove"#5 %gainst the Day.7

&n 14G# u"er used a bag o! mathemati(a" tri(s to so"ve the prob"em o!adding the natura" numbers !rom 1 to infnity# a so6(a""ed divergent seriesbe(ause the terms eep growing without "imit as you go a"ong. "ear"y# i! you stopadding anywhere a"ong the way H at a +uinti""ion @1 with 1 Ceros a!ter itA# say#or a googo"p"e- @10I100 Ceros A H the sum wi"" be enormous. The prob"em with

infnity is that you (an:t stop. You never get there. &t:s more o! a ourney than adestination. %s Dr. adi""a says to =r. /aran at the end o! their video# 5You haveto !a(e infnity# Brady.7

The method in the video is essentia""y the same as u"er:s. &t invo"vesnothing more (omp"i(ated than addition and subtra(tion @a"though the thingsbeing added and subtra(ted were more infnite seriesA and a sma"" pie(e o!a"gebra that my si-th6grade daughter wou"d breeCe through. You are not a"one inwondering how this (an mae sense. The orwegian mathemati(ian ie"s /enri 

 %be"# whose notion o! an %be" sum p"ays a ro"e here# on(e wrote#5The divergent series are the invention o! the devi"# and it is a shame tobase on them any demonstration whatsoever.7

&n modern terms# Dr. Frene" e-p"ained# the gist o! the (a"(u"ations (an be

interpreted as saying that the infnite sum has three separate parts> one o! whi(hb"ows up when you go to infnity# one o! whi(h goes to Cero# and minus 1)12. Theinfnite term# he said# ust gets thrown away.

 %nd it wors. % hundred years "ater# ?iemann used a more advan(ed andrigorous method# invo"ving imaginary as we"" as rea" numbers# to (a"(u"ate theCeta !un(tion and got the same answer> minus 1)12.

5'o u"er guessed it right#7 Dr. Frene" said. Those o! us who are notmathemati(ians probab"y wou"d not (are so mu(h about infnity e-(ept that it(rops up again and again in (a"(u"ations o! things# "ie the energy o! the e"e(tron#that we now are fnite# or in string theory# whi(h physi(ists wou"d "ie to hope isfnite. &n this (ase# our (urrent understanding o! the very so"idity o! rea"ity

depends on (oming up with a (onsistent way to assign va"ues to infnite sums.&n the pro(ess nown as regu"ariCation# whi(h is a part o! many(a"(u"ations in +uantum theory# physi(ists do something simi"ar to what u"er did#arriving at a rea" number that (orresponds to the +uantity they want to now andan infnite term# whi(h they throw away. The pro(ess wors so we"" thattheoreti(a" predi(tions in +uantum e"e(trodynami(s# the !an(y version o! the!ami"iar !or(e o! e"e(tromagnetism# agree with e-periments to a pre(ision o! onepart in a tri""ion.

8hi(h is remarab"e given that infnite +uantities have been thrown away#or 5swept under the rug#7 in the words o! the a"i!ornia &nstitute o! Te(hno"ogyphysi(ist ?i(hard Feynman# who he"ped invent a "ot o! this stu but thought itwas more than !aint"y s(anda"ous.

<iewise# it is no surprise that the !a(tor 1)12 shows up a "ot in stringtheory e+uations# Dr. Frene" said. 58hy it a"" wors is sti"" a mystery. Juantumphysi(s needs its own ?iemann to (ome and give a rigorous e-p"anation o! thesemysteries.7

8/9/2019 At the Very End–It All Adds Up to Minus One–Twelfth by Dennis Overbye From the NY Times Dated 04 February 2014

http://slidepdf.com/reader/full/at-the-very-endit-all-adds-up-to-minus-onetwelfth-by-dennis-overbye-from 3/3

To him and others# this is ust another e-amp"e o! what the eminentphysi(ist ugene 8igner (a""ed the 5unreasonab"e ee(tiveness o! mathemati(s.78hy shou"d su(h woo""y and abstra(t (on(epts as Ceta !un(tions or imaginarynumbers# the produ(ts o! a (hess game in our minds# have su(h re"evan(e indes(ribing the wor"d;

?iemann:s e-p"orations o! the geometry o! (urved spa(es in 1*4 "aid the!oundation !or instein:s theory o! gravity# genera" re"ativity# ha"! a (entury "ater.There were mathemati(ians and phi"osophers who were ready to ump out thewindow "ater in the 100s when Keorg antor# a ?ussian6born mathemati(ian#set out to ("assi!y the inds o! infnity. &n a spee(h in 1G0# the Fren(hmathemati(ian /enri oin(arL (ompared 5antorism#7 as he (a""ed it# to adisease.

=athemati(ians today agree that there is an infnite number o! natura"numbers @1# 2# $ and so onA on the bottom rung o! infnity. %bove that# however# isanother rung o! so6(a""ed rea" numbers# whi(h is bigger in the sense that there isan un(ountab"e number o! them !or every natura" number. %nd so it goes.

osmo"ogists do not now i! the universe is physi(a""y infnite in eitherspa(e or time or what it means i! it is or is not. Or i! these are even sensib"e+uestions. They do not now whether someday they wi"" fnd that higher orders o! 

infnity are unreasonab"y ee(tive in understanding e-isten(e# whatever that is./ere is where we sprain our imaginations and perhaps (he( to see that we sti""have our wa""ets. M$0