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This is a question of basic maths. And I am sorry if this is a stupid question, but I am teaching it and there is something I remember seeing once and ignoring it when I was a child. I understand it is a consequence of a bad procedure in the use of units, but would like to confirm with your answers and thoughts.

The case is that, for example, when I have radians and degrees, if I express the equality $\pi radians=180 degrees$ I am able to construct the neutral fraction $\frac{\pi radians}{180 degrees}=1$ with which perform cancellations and change units. The problem is that this student had the interpretation of units as variables, so he proceeded to state from the first equation $radians=\frac{180}{\pi } degrees$ and so concluding the exact inverse ratio. Though it is a correct equation, he used it as a functional relationship. The proportion that does hold is $\frac{180}{\pi }=\frac{degrees}{radians }$ taking now degrees and radians as variable quantities. How can one justify as clearly as possible this false duality?

EDIT: Thanks for the answers. I was going through the same mental process of clarifying the difference of units (using the example one of you gave: $s$ is different if it is used as the unit or the quantity one second), but he made me acknowledge something: if it is not possible to handle it as such, why can we cancel them (the units)?

Curious student
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    But it is true that $1\operatorname{rad}=\frac{180^\circ}\pi$ ... –  Sep 08 '21 at 14:54

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Your student is interpreting the statement, "$\pi$ radians equals $180$ degrees" as if it meant, "$pi$ times the number of radians equals $180$ times the number of degrees," which of course, it doesn't.

It's as if we took the statement "$1$ dollar equal $100$ cents" to mean, "The number of dollars equals $100$ times the number of cents."

saulspatz
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  • It's an insidious ambiguity of language. Thinking of "100 cents" as a mathematical expression where "cents" is a variable seems pretty reasonable, especially when we start abbreviating units. Without foreknowledge, how can one tell the difference between $2s$ and $2~\textrm{s}$, unless you're explicitly told that $s$ represents the amount of time in seconds and $\textrm{s}$ represents the quantity "1 second"? This comes up pretty regularly in my non-major intro physics courses. @Curiousstudent – march Sep 08 '21 at 15:37
  • Personally, I prefer things like "There are $12$ inches to a foot," rather than $1ft = 12in$, which seems to say the wrong thing. More generally, I don't think we're really ever without foreknowledge. There's always some context. We know what the variables mean. Usually, we start with some problem and translate it into symbols, though this isn't always true of textbook exercises. – saulspatz Sep 08 '21 at 15:46