I assume that Narsil, precious hierloom of Arnor, would normally be held in Elrond's safe-keeping, along with the Sceptre of the Northern Kingdom, but in his mission to bring Frodo and the Ring safely to Rivendell, Aragorn knew that he might have to face the Nazgul. No ordinary sword would have been much use but perhaps Aragorn felt that the Black Riders might hesitate when they came up against someone wielding the sword that had once defeated their master, however many pieces it was in, buying time for an escape.
As it turned out, he didn't have to tackle all Nine at the same time; at Weathertop fire was sufficient to drive them away.
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