The development of 20th Century physics was intimately interconnected with the ramifications of what constitutes a physical constant or fundamental parameter. Which was first immortalized in Einsteins special relativity as an invariant constant c = the speed of light. As velocity measures length L divided by time T, c represents a metric or dimensional constant, as is mass M or any combination of these quantities. Yet when the SI metric system was formally adopted (firmly replacing the CGS system), c had acquired new meaning as a defined constant equal to a precise integer 299,792,458 represented in meters per second. However, as explained by Wikipedia (browse physical constant), the distinction between a pure or dimensionless number in mathematics and physics is that the speed of light had to be measured since no theory can predict it; for even then the value would have to be empirically confirmed. .
And though hardly an integer nor yet a defined parameter by NIST, a much purer number as a dimensionless fundamental physical constant is distinctly embodied by the inverse of the electromagnetic coupling constant alpha or fine-structure (fs-) number a ~ 137.036. Which...