39The "number of field lines" definition comes from the result of dot product of magnetic field and area vectors. The dot product is a number, it has no direction. But we also have to see WHICH quantities we are trying to find the dot product of.
Multiplying magnetic field with area gives you a quantity which does have dimensions. What early scientists observed was that the number of magnetic field lines corresponds to the scalar number when we write it without the units. Thus it became easier to find the magnetic flux if we knew how many field lines were passing through a particular area. However flux still has its units.
So overall, it is the scalar number (without the Wb/m² part) that is the number of field lines. It is simply a correlation, and not the exact magnetic flux, because magnetic flux has dimensions.
The weber does not signify the number of field lines, but the scalar number 'x' with which we write 'x Weber' corresponds to the number of magnetic field lines passing through that particular area.
A useful link for finding SI dimensions and units of quantities -: http://www.ebyte.it/library/educards/sidimensions/SiDimensionsAlfaList.html
