Here you aren’t cancelling out any units, you’re just neglecting to write them down, and scaling things so that outcomes of interest happen to land at 0 and 1. Expecting special insight to come out of that operation is numerology.
Hmm. You are right, and I should fix that. When we did that trick in school, we always called it “dimensionless”, but you are right it’s distinct from the pi-theorem stuff (reynolds number, etc). I’ll rethink it.
Edit: Wait a minute, on closer inspection, your criticism seems to apply to radians (why radius?) and reynolds number (characteristic length and velocity are rather arbitrary in some problems).
Why are some unit systems “dimensionless”, and others not? More relevently, taboo “dimensionless”, why are radians better (as they clearly are) than degrees or grads or arc-minutes? Why is it useful to pick the obvious characteristic lengths and velocities for Re, as opposed to something else.
For radians, it seems to be something to do with euler’s identity and the mathematical foundations of sin and cos, but I don’t know how arbitrary those are, off the top of my head.
For Re, I’m pretty sure it’s exactly so that you can do numerology by comparing your reynolds number to reynolds numbers in other problems where you used the same charcteristic length (if you used D for your L in both cases, your numerology will work, if not, not).
I think this works the same in my “dimensionless” utility tricks. If we are consistent about it, it lets us do (certain forms of) numerology without hazard.
Why are some unit systems “dimensionless”, and others not?
Some ratios are dimensionless because the numerator and denominator are in the same dimension, so they cancel. for example, a P/E (price to earnings) ratio of a stock. The numerator & denominator are both in $ (or other currency).
Radians are a ratio of lengths (specifically, arc length to radius) whereas degrees are the same ratio multiplied by an arbitrary constant (180/pi). We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians, but degrees and arc-minutes are right out.
Lengths offer one degree of freedom because they lack units but not an origin (all lengths are positive, and this pinpoints a length of 0). For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name. Thanks.
For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Ok good, that’s what I was intending to do, maybe it should be a bit clearer?
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
Shamelessly ripped from wikipedia; their typo. 10^3 does seem more reasonable. Thanks.
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name.
For the record, I was also making up a name when I said “halfradians”. And now that I think about it, it should probably be “twiceradians”, because two radians make one twiceradian. Oops.
Hmm. You are right, and I should fix that. When we did that trick in school, we always called it “dimensionless”, but you are right it’s distinct from the pi-theorem stuff (reynolds number, etc). I’ll rethink it.
Edit: Wait a minute, on closer inspection, your criticism seems to apply to radians (why radius?) and reynolds number (characteristic length and velocity are rather arbitrary in some problems).
Why are some unit systems “dimensionless”, and others not? More relevently, taboo “dimensionless”, why are radians better (as they clearly are) than degrees or grads or arc-minutes? Why is it useful to pick the obvious characteristic lengths and velocities for Re, as opposed to something else.
For radians, it seems to be something to do with euler’s identity and the mathematical foundations of sin and cos, but I don’t know how arbitrary those are, off the top of my head.
For Re, I’m pretty sure it’s exactly so that you can do numerology by comparing your reynolds number to reynolds numbers in other problems where you used the same charcteristic length (if you used D for your L in both cases, your numerology will work, if not, not).
I think this works the same in my “dimensionless” utility tricks. If we are consistent about it, it lets us do (certain forms of) numerology without hazard.
Some ratios are dimensionless because the numerator and denominator are in the same dimension, so they cancel. for example, a P/E (price to earnings) ratio of a stock. The numerator & denominator are both in $ (or other currency).
Radians are a ratio of lengths (specifically, arc length to radius) whereas degrees are the same ratio multiplied by an arbitrary constant (180/pi). We could imagine that halfradians (the ratio of arc length to diameter) might also be a natural unit, and then we’d have to go into calculus to make a case for radians, but degrees and arc-minutes are right out.
Lengths offer one degree of freedom because they lack units but not an origin (all lengths are positive, and this pinpoints a length of 0). For utilities, we have two degrees of freedom. One way to convert such a quantity to a dimensionless one is to take (U1 - U2)/(U1 - U3), a dimensionless function of three utilities.
This is more or less what you’re doing in your “dimensionless utility” section. But it’s important to remember that it’s a function of three arguments: 0.999 is the value obtained from considering Satan, paperclips, and whales simultaneously. It is only of interest when all three things are relevant to making a decision.
Incidentally, there’s a typo in your quote about Re: 103 should be 10^3.
I was actually thinking of diameter-radians when I wrote that, but I didn’t know what they were called, so somehow I didn’t make up a name. Thanks.
Ok good, that’s what I was intending to do, maybe it should be a bit clearer?
Shamelessly ripped from wikipedia; their typo. 10^3 does seem more reasonable. Thanks.
For the record, I was also making up a name when I said “halfradians”. And now that I think about it, it should probably be “twiceradians”, because two radians make one twiceradian. Oops.