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Different units for the same quantity usually only differ by a multiplicative constant, such as inch and meter, but °F and K differ by both a multiplicative constant and an additive one. Does that make them so different they're really measuring different quantities?

If they're the same quantity, then where's the limit of what operations you can do before it becomes something different? Is 1/s still a unit of time or is m²/s² a unit of velocity? In these cases, instead of multiplying and adding a constant, we're inverting the value or squaring it.

I've heard them distinguished as "thermodynamic temperature" and "temperature" in SI but that seems to be uncommon and it's hard to find justification for it.

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Fahrenheit and Centigrade scale are related via a linear transformation: $$ T_F = a T_C + b \Leftrightarrow T_C=\alpha T_F + \beta, $$ so they are really the: once we know one, we automatically know the other. Note that this would be also true for more general functions (e.g., if we had two thermometers based in substances that expand with temperature in a non-uniform way), as long as there is a unique (one-to-one) correspondence between the two temperatures: $$ T_1 = f(T_2) \Leftrightarrow T_2=f^{-1}(T_1). $$

Note that we are talking here about temperature scales, which consist of units and the origin of the scale - the units (a degree of Fahrenheit and a degree of Centigrade) are related via a proportionality relation, like meters and inches. The origins of the two scales are not the same - the situation that can be easily transposed to distance measurements as well. A good example is the Greenwich vs. Paris meridian (although in this case we talk about the angular distance).

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  • $\begingroup$ So is 1/s a unit of time, m^2/s^2 velocity, etc.? The relationship isn't linear but as you say, it doesn't have to be. $\endgroup$
    – user1318499
    Commented Apr 12, 2021 at 9:16
  • $\begingroup$ If you want an example with time, you may think of the military time (e.g., it is 13:30) versus am/pm (it is one and a half pm) - the reference point is different,a nd the units might be different, but an hour is still an hour. $\endgroup$
    – Roger V.
    Commented Apr 12, 2021 at 9:21
  • $\begingroup$ I'm looking for examples that are more strange, like 1/s for time, to find the boundary of what counts as units of the same quantity. Doesn't matter if it's not commonly used. $\endgroup$
    – user1318499
    Commented Apr 12, 2021 at 9:24
  • $\begingroup$ People are more comfortable with linear units. But technically we are not prevented from using non-linear scales. E.g., charge on a battery can be used to measure time - even though it changes exponentially. The bars on the computer showing how much of the program installed are notoriously inconsistent with time, but based on hard measures of computational operations. Noise level is measured on a logarithmic scale (in dB). $\endgroup$
    – Roger V.
    Commented Apr 12, 2021 at 9:27
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Here is one way of looking at this which might help you.

Imagine an old-school bulb-type thermometer. You stick the bulb into the water bath, the red alcohol meniscus inside the thermometer takes up a position along the length of the thermometer, and then you read the temperature of the bath off the scale point nearest the meniscus position.

Now imagine the thermometer has not one but two scales on it, say degrees F and degrees K. You stick the bulb into the water bath, the red alcohol meniscus inside the thermometer takes up a position along the length of the thermometer, and then you read the temperature(s) off the scale(s).

There is ONE and ONLY ONE position of the meniscus, but TWO possibilities for the "temperature reading" depending on which scale you choose. In this sense, the two scales are still measuring the same thing.

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  • $\begingroup$ So we can also say 1/s measures time if a clock has those units on the dial? $\endgroup$
    – user1318499
    Commented Apr 12, 2021 at 5:16
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    $\begingroup$ @user1318499 I suppose, but you'll find it painful to work with (just like Celsius and Fahrenheit, really). If it takes me 0.55555 millihertz to get from home to work and then 3.3333 millihertz to get to my desk, then it takes 0.47619 millihertz to get from home to my desk. $\endgroup$
    – Criticize SE actions means ban
    Commented Apr 12, 2021 at 9:38
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I think K and °F measure different quantities since there are formulas like

P=σAT4

which only allow absolute temperature. You might imagine that it does work with °F, and with P being in some weird units. But that's not possible because positive and negative °F temperatures would give the same value for P even though the actual power is different. This weird P unit is therefore not a unit of power at all, so °F also isn't a unit of the same quantity that K is.

This problem also happens with 1/s for time. You can use the formula $v=x/t$ and choose two pairs of values for $x$ and $t$ which give the same $v$ in normal units but two different values of $v$ in 1/s units. Eg:

$x=2$m, $t=2$s, $v=1$ m/s or $4$ m.s in 1/s consistent units

$x=3$m, $t=3$s, $v=1$ m/s or $9$ m.s in 1/s consistent units

$4\not=9$

Again, there's no 1-1 mapping between the two units for $v$, so they aren't measuring the same quantity, which means 1/s and s aren't either.

We can allow these different units if we also change the formulas to accommodate them (P=σAT4 becomes P=σA(T+T0)4 and $v=x/t$ becomes $v=xt$). But that's a qualitatively different change compared to units which only differ by a multiplicative constant where we don't have to change any formulas.

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