Nut size (A.K.A. diameter) is the size bolt that the nut is used with. The same is true of the thread count. Thus a 1/4"-20 nut fits a 1/4"-20 bolt.
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A long nut used to connect pieces of threaded rod or other male threaded fasteners.
A long nut used to connect two different diameters of threaded rod or other male threaded fasteners.