3 ms·

Weird units are cool. I was pretty mindblown when I realized that radians are basically meters/meters and steradians are meters^2/meters^2

by fish45 6y ago

thaumasiotes 6y ago
> I was pretty mindblown when I realized that radians are basically meters/meters This is true in a dimensional analysis way, where e.g. arclength around a circle is equal to the subtended angle times the radius and, since the arclength and the radius have the same dimension, the angle must be dimensionless. But units are more than that. You can't do anything in chemistry without running into quantities like molarity or ppm that appear to be dimensionless but make no sense if you think about them as coefficients rather than as combinations of semantically-distinct units that happen to have the same name. Imagine some subtance that is 5% salt by mass and 8% salt by volume. ppm by mass is just kg / kg, right? And ppm by volume is m^3 / m^3. Those are the same unit -- in fact, no unit at all. Except that the implication of that non-fact is that ppm by volume and ppm by mass are the same thing, which they aren't. Similarly, radians are best conceived of as percentages of a rotation, not as dimensionless coefficients. Then we calculate an arclength around a circle by saying that the ratio between the arclength and the full perimeter is equal to the ratio between the subtended angle and the full rotation. Characterized so, we have both units present twice, canceling each other out on each side of the equation, and we don't need to clean up by saying that the angle has no unit at all. We divided part of a perimeter by all of a perimeter -- producing a dimensionless quantity -- and we divided part of a rotation by all of a rotation -- also producing a dimensionless quantity -- and then we correctly compared two dimensionless quantities to each other. (On this analysis, the equation s = rθ is actually incorrect; it should be written s = rθ/1 . You can extend this -- we all know the area under sin(x) from 0 to pi is 2. 2 whats? Well, the x axis is radians and the y axis is sine values. Sine values are computed by dividing the length of a line segment by the length of another line segment; they are legitimately dimensionless. So the area under sin(x) must be 2 radians.)
- QuesnayJr 6y ago
  This has always bothered me, that there were dimensionless units that a dimension-like quality, like radians.
  - thaumasiotes 6y ago
    Seriously, radians aren't dimensionless. If you calculate arclengths using angles measured in degrees, the whole problem disappears: a 60° arc around a circle of radius 3cm has length equal to (2·pi·3cm·60°)/(360°) = (2·pi·3cm)/6 = pi centimeters. Looking at it that way makes it very plain that the angles have units attached, and those units don't appear in the result of "pi centimeters" because they canceled out during the computation. This is overt when calculating in degrees because they are an inconvenient size. The size of a radian is chosen so that the constant terms (2·pi) in the numerator and (2·pi radians) in the denominator appear to cancel out, giving a (1 radian) in the denominator which is omitted from the formula because dividing by 1 is numerically pointless. But dividing by 1 radian is not dimensionally pointless; omitting the 1 from the formula leads to the illusion that θ has no unit attached.

Weird units are cool. I was pretty mindblown when I realized that radians are basically meters/meters and steradians are meters^2/meters^2

by fish45 6y ago

Weird units are cool. I was pretty mindblown when I realized that radians are basically meters/meters and steradians are meters^2/meters^2

thaumasiotes 6y ago

> I was pretty mindblown when I realized that radians are basically meters/meters This is true in a dimensional analysis way, where e.g. arclength around a circle is equal to the subtended angle times the radius and, since the arclength and the radius have the same dimension, the angle must be dimensionless. But units are more than that. You can't do anything in chemistry without running into quantities like molarity or ppm that appear to be dimensionless but make no sense if you think about them as coefficients rather than as combinations of semantically-distinct units that happen to have the same name. Imagine some subtance that is 5% salt by mass and 8% salt by volume. ppm by mass is just kg / kg, right? And ppm by volume is m^3 / m^3. Those are the same unit -- in fact, no unit at all. Except that the implication of that non-fact is that ppm by volume and ppm by mass are the same thing, which they aren't. Similarly, radians are best conceived of as percentages of a rotation, not as dimensionless coefficients. Then we calculate an arclength around a circle by saying that the ratio between the arclength and the full perimeter is equal to the ratio between the subtended angle and the full rotation. Characterized so, we have both units present twice, canceling each other out on each side of the equation, and we don't need to clean up by saying that the angle has no unit at all. We divided part of a perimeter by all of a perimeter -- producing a dimensionless quantity -- and we divided part of a rotation by all of a rotation -- also producing a dimensionless quantity -- and then we correctly compared two dimensionless quantities to each other. (On this analysis, the equation s = rθ is actually incorrect; it should be written s = rθ/1 . You can extend this -- we all know the area under sin(x) from 0 to pi is 2. 2 whats? Well, the x axis is radians and the y axis is sine values. Sine values are computed by dividing the length of a line segment by the length of another line segment; they are legitimately dimensionless. So the area under sin(x) must be 2 radians.)

QuesnayJr 6y ago
This has always bothered me, that there were dimensionless units that a dimension-like quality, like radians.
- thaumasiotes 6y ago
  Seriously, radians aren't dimensionless. If you calculate arclengths using angles measured in degrees, the whole problem disappears: a 60° arc around a circle of radius 3cm has length equal to (2·pi·3cm·60°)/(360°) = (2·pi·3cm)/6 = pi centimeters. Looking at it that way makes it very plain that the angles have units attached, and those units don't appear in the result of "pi centimeters" because they canceled out during the computation. This is overt when calculating in degrees because they are an inconvenient size. The size of a radian is chosen so that the constant terms (2·pi) in the numerator and (2·pi radians) in the denominator appear to cancel out, giving a (1 radian) in the denominator which is omitted from the formula because dividing by 1 is numerically pointless. But dividing by 1 radian is not dimensionally pointless; omitting the 1 from the formula leads to the illusion that θ has no unit attached.