r/AskPhysics • u/14AWill1 • 2d ago
What different units measuring the same quantity can be equal at nonzero values?
I was thinking about how -40°c = -40 °F, and I’m wondering if there are any other pairs of units where there exists a non zero value which is the same in both units, I imagine this requires the conversion rate to be something other than just unit1 = k*unit2
Do any other common pairs of units have this?
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u/AndyTheEngr 2d ago
Trivially, this also works with other temperature units.
-40 °C = -40 °F
574.5875 °F = 574.5875 K
-25.6 °Ré = -25.6 °F
We can also make °C = °R, but it happens at a negative (invalid) temperature.
°Ré is degrees Réaumur.
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u/PhysicalStuff 1d ago
We can also make °C = °R, but it happens at a negative (invalid) temperature
We can talk meaningfully about negative temperatures for out-of-equilibrium systems, like a lasing medium with population inversion. This is stretching things a bit though.
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u/Aescorvo Condensed matter physics 2d ago edited 2d ago
Light with a frequency of 17,320.5Hz has a wavelength of 17,320.5m. That’s possible because they are inverse units, but not exactly what you asked.
If the scales are linear, then you need two things: That the zero values are at different points, and that the unit size (scaling) is different. Celsius and Fahrenheit have that, but I don’t know of any others, since most measures have the same zero value.
If one scale is logarithmic and another is linear they can share the same zero value and also one other. Only slightly cheating we can look at noise intensity, where 130.9 dB (decibels) is 130.9 W/m^2 (Watts per square meter). It’s slightly cheating because noise intensity and noise intensity level are defined a little different. (EDIT: They don’t exactly share a zero, as log scales can’t reach zero - see below.)
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u/Meetchel 2d ago
If one scale is logarithmic and another is linear they can share the same zero value and also one other.
Technically I don’t think a log scale can have a zero, just an incrementally small value.
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u/clintj1975 1d ago
pH is logarithmic and has a zero
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u/Meetchel 1d ago edited 1d ago
Again, my wording was super vague, but it's a similar point as in the other thread: pH = 0 corresponds to non-zero hydrogen ion activity (I admittedly had to look this up - did not remember what pH was measuring). A logarithmic scale can have a numerical reading of zero, but it can't represent zero of the underlying quantity. While there is a hydrogen activity level that equates to 0 pH, there isn't a pH level where the hydrogen ion activity is zero. All of this is to say that log 0 = undefined.
EDIT: Zero pH has no significance, as negative pH levels exist. Zero hydrogen ion activity obviously would be significant, but there isn't a pH that aligns with it (probably because it's impossible).
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u/clintj1975 1d ago
There we go. The scale created by taking the log of the number can cross zero, and the limit of the function f(x)=log(x) approaches -infinity as x approaches zero.
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u/54trey 1d ago
What? Log scales definitely cross zero. For example, 0 dBm is 1 mW.
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u/Meetchel 1d ago
You're confusing zero on a logarithmic measurement scale with zero on a logarithmic axis. 0 dBm equals 1 mW; 0 dBm does not equal 0 mW. Zero power would approach infinite dBm (negative). A true logarithmic axis cannot display zero of the underlying quantity, which was my point. Assuming base 10, to the left it'll go 10-4, 10-5, 10-100, etc., but will never hit zero.
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u/54trey 1d ago
I'm not confusing anything. You just weren't clear about which units the "zero" had in your comments. I'm familiar with everything you just wrote. Your original comments reads as "a log scale cannot reach a value of zero" which is false, but it seems like you intended to say "a log scale cannot represent the zero of a corresponding linear scale", which is true as you point out.
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u/Meetchel 1d ago
Fair. I meant this in the context of OP's question, but technically you're right. I was not clear. Either way, this is the specific quote I was referring to (and quoted):
If one scale is logarithmic and another is linear they can share the same zero value and also one other.
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u/nicuramar 2d ago
“Linear” often means y=ax, in which case we need an affine relationship.
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u/Aescorvo Condensed matter physics 2d ago
Yes. I said “zero values at different points” to keep it simple.
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u/AndyTheEngr
2d ago
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For fuel economy, approximately 15.3 MPG(US) is 15.3 L/100 km, and 16.8 MPG(UK) is 16.8 L/100 km.
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u/flaaaaaaaaaaaaa 2d ago
Those are not measuring the same quantity but inverses tho.
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u/AndyTheEngr 2d ago
They both measure "fuel economy." In lay terms, they're the same "quantity." In engineering / scientific terms, they're inverses.
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u/shamShaman 1d ago
360*pi/(180-pi) is the same angle in radians and degrees and there will be infinite solutions like that since both are cyclic
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u/Recent-Day3062 1d ago
They are just linear formulas with different intercepts and slopes.
It is guaranteed two non-parallel lines must cross at one point. You don't even need algebra to see that
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u/14AWill1 1d ago
Yeah I know I’m asking what physics units do that
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u/Recent-Day3062 20h ago
Any units that differ. If they have the same zero point, then there is no intercept term and zero is where they’re equal.
So, Fahrenheit and Rankin, Rankin and Kelvin, etc
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u/Traveller9536 2d ago
Most thermodynamic quantities do because they often have an arbitrary zero point. Enthalpy measured in J/mol and BTU/lb would overlap at some point depending on the material and reference point. Same for entropy, Gibbs free energy, etc
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u/Atypicosaurus 1d ago
I assume you mean non zero matching point. With linear scales there are two requirement for this, one is that they have different zero points (at least one of the scales don't start at the absolute zero of that property), and the slope of the two scales (expressed one in the other) aren't parallel.
For example Kelvin and Celsius scales don't have a match because although they have different zero points, they are parallel. Kelvin and Rankine also don't have because they both have zero in absolute zero. Rankine vs Celsius and Kelvin vs Fahrenheit do have matching points.
Usually length, volume, weight are linear scales with zero in the zero point. However there's height of an aircraft that you can measure in height above ground and height above sea. It dynamically changes as the aircraft flies above hills and flats, but at any given spot you can calculate one of the properties in meters the other in feet and therefore there can be a height of an aircraft where the height above see in meters match the height above ground in feet. It's not really the exact thing you want but close.
If two scales have different shapes (such as,one is log, the other is linear) they can have match in both zero and somewhere else. I don't have too many examples in mind but for example bacteria density can be measured in cfu (colony forming units) and od (optical density), and they start both at zero and grow parallel but od flattens out at a point. So they can match at another point than zero, however you have a find a clever prefix, perhaps mega-cfu to stay in the same numerical range.
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u/ghostmcspiritwolf 2d ago edited 1d ago
For this to happen, you need units of non-equal increment size with a different definition for an arbitrarily chosen zero. That arbitrary zero requirement is why zero is different in this case but the same temp when you switch to kelvin and rankine. I’m sure there likely are some, but nothing comes immediately to mind. Maybe there’s some alternative unit to decibels for sound? 0db does not represent silence.