You&#x27;ve got the right intuition here. The unitless part is key for several reasons. It means that the number itself is meaningful. In your example, the ratio of 1.7 long&#x2F;wide would be the same no matter what units used to make the measurement. (Obviously, the units DO have to actually cancel cm&#x2F;inches doesn&#x27;t work). The number for the speed of light (299 792 458 m&#x2F;s) is a defined property, so that number doesn&#x27;t <i>mean</i> anything deeply. However, 1&#x2F;137... itself is directly meaningful. This is why people who work in fundamental constant research use dimensionless ratios.<p>Now for a piece that&#x27;s more interesting. The fine-structure constant (alpha) is the coupling constant that sets the strength of electromagnetism. This means that its value is the thing that matters in the equation: e^2&#x2F;\hbar c. Each of the other values is a derived quantity. Further, to speak a bit loosely, only changes in alpha matter -- in the sense that if the speed of light (c) changes, but the other constants (e and \hbar) change in a way that keeps alpha the same, then you wouldn&#x27;t be able to tell with an experiment that anything has changed.<p>Contrast this situation w&#x2F; a change in alpha -- a table-top experiment would be able to detect the change (given that it&#x27;s large enough, and we have methods of measuring changes on year-time scales that are a few parts in ~10^-18 (Rosenband, 2008)), as it would mean that physics has changed in a fundamental way.<p><a href="http:&#x2F;&#x2F;phys.columbia.edu&#x2F;~millis&#x2F;1900&#x2F;readings&#x2F;Science-2008-Rosenband-1808-12.pdf" rel="nofollow">http:&#x2F;&#x2F;phys.columbia.edu&#x2F;~millis&#x2F;1900&#x2F;readings&#x2F;Science-2008-...</a>