As water encompasses wetness, so a quantity encompasses its units. I noticed troubles over this seemingly obvious point when I taught in England, perhaps because England, in science teaching and research, is a metric nation. My students, all physics majors, would calculate their kinetic energy during sprinting as 3000

KE=(12)(60)(100)=3000.
(1)

I didn't object to their fast sprint (who doesn't dream?) but did ask, “3000 what?” They then attached units to the end

KE=(12)(60)(100)=3000J.
(2)

This unit band-aid ruined Eq. (1)'s only valid equality! Although the equation's first equality, between kinetic energy and a purely numerical expression, was dimensionally bogus, at least its second equality, reflecting the numerical calculation, was valid. But with the joules [Eq. (2)], the second...

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