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Water is a precious resource. So protest erupted when, last September, India’s Supreme Court ordered Karnataka state to release 15,000 cusecs (cubic feet per second) of the stuff per day to its downriver neighbour, Tamil Nadu. But what irked keen-eyed scientists was the nonsensical nature of the order. Cusec is a rate of flow, analogous to speed, not something that can be done “per day”. Taken literally, the order was meaningless.
Dimensional analysis is supposed to prevent such errors. This involves calculating units in their own equation, for example dividing metres by seconds to get the unit for speed: metres per second. It’s a handy way to sanity-check an answer — and shows that the quantity in the above court order is akin to acceleration, rather than rate or volume.
But such an analysis can come a cropper in the face of quantities that do not have dimensions such as length or time — including radians (the ratio of the length of an arc of a circle to its radius) and anything countable, such as a number of atoms. The issue arises because the International System of Units (SI) allows combinations of only seven basic dimensions and their units (such as length in metres), and allocates quantities with no extent in these dimensions a unit of ‘1’. That makes life simple, but hides crucial information. For example, a turning force, torque, is often measured in joules per radian. In dimensional analysis, that confusingly becomes joules, the same unit as energy. Hertz — cycles per second — reduces to ‘per second’, just like the frequency of non-periodic events.
Informally, physicists get around this by explicitly including the extra information. But software struggles to do that in a consistent way, and the inability to deal with this quirk formally is irksome.
Solutions exist. For example, radians could be made a new SI unit, and the unit 1 could be formally coupled with notation that includes the type of quantity that it represents. The SI system is nifty, but its real beauty is its coherence. Avoiding nonsense may require forgoing brevity for clarity.
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