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Old English stæger "flight of steps," also "a single step," from Proto-Germanic *staigri (cf. Old Norse and Old Frisian stiga, Middle Dutch stighen, Old High German stigan, German steigen, Gothic steigan "to go up, ascend;" Old English stigan "to climb, go;" German Steig "path," Old English stig "narrow path"), from PIE *steigh- "go, rise, stride, step, walk" (cf. Greek steikhein "to go, march in order," stikhos "row, line, rank, verse;" Sanskrit stighnoti "mounts, rises, steps;" Old Church Slavonic stignati "to overtake," stigna "place;" Lithuanian staiga "suddenly;" Old Irish tiagaim "I walk;" Welsh taith "going, walk, way").
Originally also a collective plural; stairs developed by late 14c. OED says stair still is ordinary in Scotland where flight of stairs would be used elsewhere.