▁this ▁regard . ▁ ▁In ▁the ▁ 1 7 th ▁century , ▁Des cart es ▁introduced ▁the ▁term ▁" real " ▁to ▁describe ▁roots ▁of ▁a ▁polynomial , ▁distingu ishing ▁them ▁from ▁" im ag inary " ▁ones . ▁ ▁In ▁the ▁ 1 8 th ▁and ▁ 1 9 th ▁centuries , ▁there ▁was ▁much ▁work ▁on ▁ir r ational ▁and ▁trans c endent al ▁numbers . ▁Johann ▁Heinrich ▁Lam bert ▁( 1 7 6 1 ) ▁gave ▁the ▁first ▁fla wed ▁proof ▁that ▁ ▁cannot ▁be ▁rational ; ▁Ad rien - Marie ▁Leg endre ▁( 1 7 9 4 ) ▁completed ▁the ▁proof , ▁and ▁showed ▁that ▁ ▁is ▁not ▁the ▁square ▁root ▁of ▁a ▁rational ▁number . ▁Paolo ▁R uff ini ▁( 1 7 9 9 ) ▁and ▁N iels ▁Hen rik ▁Ab el ▁( 1 8 4 2 ) ▁both ▁constructed ▁proofs ▁of ▁the ▁Ab el – R uff ini ▁theorem : ▁that ▁the ▁general ▁qu int ic ▁or ▁higher ▁equations ▁cannot ▁be ▁solved ▁by ▁a ▁general ▁formula ▁involving ▁only ▁ar ithmet ical ▁operations ▁and ▁roots . ▁ ▁É var iste ▁Gal ois ▁( 1 8 3 2 ) ▁developed ▁techniques ▁for ▁determ ining ▁whether ▁a ▁given ▁equation ▁could ▁be ▁solved ▁by ▁radical s , ▁which ▁gave ▁rise ▁to ▁the ▁field ▁of ▁Gal ois ▁theory . ▁Joseph ▁Li ou ville ▁( 1 8 4 0 ) ▁showed ▁that ▁neither ▁e ▁nor ▁e 2 ▁can ▁be ▁a ▁root ▁of ▁an ▁integer ▁quadratic ▁equation , ▁and ▁then ▁established ▁the ▁existence ▁of