Swirling flows are of considerable practical importance. They are used for example to increase heat transfer in pipes or to stabilize flames at a distance from the injector unit by means of a central reverse flow that establishes an inner recirculation zone of hot combustion products. The level of swirl is governed by a dimensionless parameter designated as the swirl number, which essentially quantifies the ratio of the flow rate of the angular momentum axial component to the flow rate of axial momentum. This ratio controls to a great extent the structure of the swirling flow and its value determines whether an inner recirculation zone is established and whether a precessing vortex core (PVC) is formed in the flow. However, a major difficulty resides in calculating the swirl number from experimental measurements and over the last 50 years, several simplified formulas have been proposed to overcome this difficulty. The present study is aimed at using velocity and pressure profiles obtained by a large eddy simulation in a generic configuration to examine these simplified expressions and determine the conditions under which they may be applicable. The geometry comprises a cylindrical swirling injector, flush mounted in the back plane of a cylindrical cavity. Although the swirl number is in principle constant when the flow is established in a duct with a constant cross-section, provided that viscous forces at the wall are negligible, one finds that this quantity varies substantially if inadequate approximations are made. Among the many possibilities one concludes that two swirl numbers should be distinguished. The first corresponding to the original definition features conservation properties, but is difficult to properly calculate from experimental data. The second is a highly simplified formulation that is commonly used today but does not share the conservation properties of the first formulation. Recommended practices are provided on how each of these swirl numbers should be calculated. It is also shown that the other formulations yield values that notably differ from those provided by the original definition.