But how about s in Laplace transform above? It is not magnitude. So why is it called frequency and have the unit?Complex numbers don't have units, but the magnitude of a complex number can have units.
Yes, s has units of 1/time. It is a complex number that has units of frequency, which is why it is referred to as "complex frequency".In Laplace transform s = σ + jω is complex number. e^(st) should be dimensionless. So s should have the unit of frequency (1/time).
Does it make sense to say that a complex number has unit?
You get the magnitude of a complex number (one way) by taking the Pythagorean sum of the real part and the imaginary part. If the complex number does not have units, then how do the units magically appear when you take the magnitude? They don't. The units of the magnitude arise directly from the units on the real part and on the imaginary part (which must be the same since these two quantities are added together to form the complex number).Complex numbers don't have units, but the magnitude of a complex number can have units.
That makes no sense -- and it isn't consistent with what you said before about only the magnitude having units.I don't strictly agree. I think it might be more correct to say that the real and imaginary parts have units but the complex number does not.
Is this only right for central angle?In the case of angular measure, there just happens to be a ratio that is very convenient, namely the ratio of the arclength subtended by an angle to the radius of the arc.
The definition is for the central angle of a circular arc. I don't see anyway to define something analogous for an arbitrary curve that would have any meaning. Just defining the "radius" would be very ambiguous.Is this only right for central angle?
With an arbitrary curve, is it possible to define angle this way?
I remember that there is a definition of radius for a curve and then according to the definition a straight line has a radius of infinity. So I thought that it may be right for an arbitrary curve.The definition is for the central angle of a circular arc. I don't see anyway to define something analogous for an arbitrary curve that would have any meaning. Just defining the "radius" would be very ambiguous.
Actually if I remember correctly there is a definition of radius for an arbitrary curve. According to the definition, a straight line has a radius of infinity. So I thought that there may be a similar definition for angle.The definition is for the central angle of a circular arc. I don't see anyway to define something analogous for an arbitrary curve that would have any meaning. Just defining the "radius" would be very ambiguous.
The radius of a curve is well-defined, but that is the radius at a point. That doesn't extend to the radius over an extended section of an arbitrary curve, which is what would be needed to define some kind of angle.I remember that there is a definition of radius for a curve and then according to the definition a straight line has a radius of infinity. So I thought that it may be right for an arbitrary curve.
I think these measures should be called relative measures. For example, relative length or relative angle.We could actually very easily define dimensionless measures for many things using quantum limits. For instance, we could define the length of an object to be the ratio of its length to the Planck length. Then, anytime you saw a length without units you would know that it is using this definition of its measure -- just like we do with an angle that doesn't have units given
Should we make a distinction between dimensionless and unitless?The first is a dimensionless angular measure, meaning that it is radians. The second has dimensions of degrees. Like other unit conversions, their is a scaling factor between them, namely 180°/Π radians.
I believe that degree is also dimensionless. It is also defined as the ratio of two quantities that have units of length.We simply have one particular measure of an angle that is normalized and, hence, that one particular measure of an angle is dimensionless.
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