["< sod > later work showed that in fact, any finite - dimensional real division algebra must be of dimension 1, 2, 4, or 8. this was independently proved again by [ [ michel kervaire ] ] and [ [ john milnor ] ] in 1958, again using techniques of [ [ algebraic topology ] ], in particular [ [ k - theory ] ]. [ [ adolf hurwitz ] ] had shown in 1898 correctly that the identity < math > q \\ overline { q } = \\ textrm { sum \\ of \\ squares } < / math > held only for dimensions 1, 2, 4 and 8. < ref > { { cite book | title = [ [ the road to reality ] ] | authorlink = roger penrose | non author = roger penrose | year = sept 2005 | publisher = vintage | isbn = 0 - 09 - 944068 - 7 } } < / ref > ( see [ [ hurwitz's theorem ( normed division algebras ) | hurwitz's theorem ] ]. ) < eod > < sod > later work showed that in fact, any finite - dimensional real division algebra must be of dimension of 1, 2, 4, or 8. this fact was independently proved by [ [ michel kervaire ] ] x and [ [ john milnor ] ] in 1958, again using techniques of [ [ algebraic topology ] ], in particular [ [ k - theory ] ]. [ [ adolf hurwitz ] ] had shown in 1898 that the identity < math > q \\ overline { q } = \\ textrm { sum \\ of \\ squares } < / math > held only for dimensions 1, 2, 4 and 8. < ref > { { cite book | title = [ [ the road to reality ] ] | authorlink = roger penrose | author = roger penrose | year = dec 2005 | x publisher = vintage | isbn = 0 - 09 - 05 944068 - 7 } }, p. 202 < / ref > ( see [ [ hurwitz's theorem ( normed division algebras ) | hurwitz's theorem ] ]. ) < eod >"]