4 ms·

It's not a linear function for what it's worth. It breaks the f(x + y) = f(x) + f(y) part of linearity. It is a linear equation, confusingly enough.

kazinator 6y ago
That's linear in the sense of "linear operator" or "linear map". I've never heard "nonlinear" as the opposite of that due to to a constant offset. "Linear map" and "linear operator" are used in contexts where homogeneity is a given. A scaling plus displacement is linear in the sense of "linear equation" because, for instance, if you scale a signal and add DC offset, there is no distortion; no new harmonics are present in the output signal. ax + by + c is called linear because the plot of points which satisfy the equation form a straight line; there is nothing confusing about it whatsoever. Temperature delta conversions between C and F are certainly a linear map/operator. If it gets 5°C hotter, and then 3°C hotter, making 8°C hotter, that's exactly like 9°F hotter plus 5.4°F hotter making 14.4°F hotter. Here is a test for linearity for a real-valued one-argument function which is not confused by an offset: x + y f(x) + f(y) f( ------ ) = ----------- 2 2 Works just fine for °C/°F. There is another criterion for a linear map that hasn't been mentioned, though: f(ax) = af(x).
- contravariant 6y ago
  Well one way or another it's not a linear function, so calling it a non-linear conversion is technically not incorrect. That said I completely agree that it makes way more sense to just call it a linear conversion and to pick an example that's truly nonlinear. Not that there are that many non-linear units, but something like decibel or beaufort might work.
  - kazinator 6y ago
    Insistence that "linear function" is defined one way is not supported by facts: https://en.wikipedia.org/wiki/Linear_function https://en.wikipedia.org/wiki/Linear_function A "linear function" refers to a "linear map" only in certain contexts. The word "affine" can be used for a mapping that fails to meet the definition of a linear map operator due to offsets. That is much better than "nonlinear".
- adrianmariano 6y ago
  GNU units supports normal unit conversion, such as meters to feet. These conversions all obey f(ax)=af(x), that is, if you have zero meters you have zero feet. If you want to convert something that breaks that rule then you have to give a definition in the database as a general functional form or as a lookup table. And you have to use functional syntax to perform the conversion. Some examples in the database include dB, wire gauge, shoe size, grit size for sandpaper, ring size, musical intervals, and sugar concentration. I call units defined using a lookup table or general functional form "nonlinear units". Zero degrees Fahrenheit is of course not equal to 0 K, but conversion of absolute temperature is not presented as the best example of this class of unit conversion. It's highlighted because everybody asks about it. Temperature conversion was for years, before I figured out a way to support it elegantly, the most requested missing feature. Yes, it's a little strange to call affine transformations "nonlinear" but there is no special "affine unit" feature in the program.
  - kazinator 6y ago
    Speaking of K, Celsius to K is also "nonlinear" math: you add or subtract a constant, around 272-something.

kazinator 6y ago

That's linear in the sense of "linear operator" or "linear map". I've never heard "nonlinear" as the opposite of that due to to a constant offset. "Linear map" and "linear operator" are used in contexts where homogeneity is a given. A scaling plus displacement is linear in the sense of "linear equation" because, for instance, if you scale a signal and add DC offset, there is no distortion; no new harmonics are present in the output signal. ax + by + c is called linear because the plot of points which satisfy the equation form a straight line; there is nothing confusing about it whatsoever. Temperature delta conversions between C and F are certainly a linear map/operator. If it gets 5°C hotter, and then 3°C hotter, making 8°C hotter, that's exactly like 9°F hotter plus 5.4°F hotter making 14.4°F hotter. Here is a test for linearity for a real-valued one-argument function which is not confused by an offset: x + y f(x) + f(y) f( ------ ) = ----------- 2 2 Works just fine for °C/°F. There is another criterion for a linear map that hasn't been mentioned, though: f(ax) = af(x).

contravariant 6y ago
Well one way or another it's not a linear function, so calling it a non-linear conversion is technically not incorrect. That said I completely agree that it makes way more sense to just call it a linear conversion and to pick an example that's truly nonlinear. Not that there are that many non-linear units, but something like decibel or beaufort might work.
- kazinator 6y ago
  Insistence that "linear function" is defined one way is not supported by facts: https://en.wikipedia.org/wiki/Linear_function https://en.wikipedia.org/wiki/Linear_function A "linear function" refers to a "linear map" only in certain contexts. The word "affine" can be used for a mapping that fails to meet the definition of a linear map operator due to offsets. That is much better than "nonlinear".
adrianmariano 6y ago
GNU units supports normal unit conversion, such as meters to feet. These conversions all obey f(ax)=af(x), that is, if you have zero meters you have zero feet. If you want to convert something that breaks that rule then you have to give a definition in the database as a general functional form or as a lookup table. And you have to use functional syntax to perform the conversion. Some examples in the database include dB, wire gauge, shoe size, grit size for sandpaper, ring size, musical intervals, and sugar concentration. I call units defined using a lookup table or general functional form "nonlinear units". Zero degrees Fahrenheit is of course not equal to 0 K, but conversion of absolute temperature is not presented as the best example of this class of unit conversion. It's highlighted because everybody asks about it. Temperature conversion was for years, before I figured out a way to support it elegantly, the most requested missing feature. Yes, it's a little strange to call affine transformations "nonlinear" but there is no special "affine unit" feature in the program.
- kazinator 6y ago
  Speaking of K, Celsius to K is also "nonlinear" math: you add or subtract a constant, around 272-something.