Mathematicians' way:

    Let a = distance in km;
        b = distance in mi; and
        φ = the conversion factor from mi to km.

    Then, a = φb.

Ha, I saw a downvote there!

I'm fine with that - on the condition that you never, ever treat a differential operator like a fraction again ;)

Physicists have their own battles, like electrical engineers using Ohm's Law as a definition of impedance.

What's the battle there? How do physicists define impedance?

Ohm's law is an empirical law that only holds in certain circumstances. A classic exercise is measuring the current and voltage across a lightbulb, plotting it, and measuring the slope of the line. The slope is the impedance. Then you turn up the voltage and watch the line turn into a curve, which is where the law breaks down and doesn't apply anymore. The engineers treat it like a definition and assume linearity over all voltage.

Wouldn't you start off using spherical cows?