Does it make sense to say that a complex number has unit?

Thread Starter

anhnha

Joined Apr 19, 2012
880
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?
 

WBahn

Joined Mar 31, 2012
25,923
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?
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".

Since s is the sum of a real, σ, part and an imaginary part, ω, both must likewise of units of 1/time.
 

WBahn

Joined Mar 31, 2012
25,923
Complex numbers don't have units, but the magnitude of a complex number can have units.
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).

As with any other quantity, the units are an integral part of the representation. You are not 72 tall, you are 72 inches tall. A circuit does not an impedance of 3+j4, it has an impedance of 3 Ω + j4 Ω.

When you take the magnitude of that impedance, you get

|Z| = √( (3 Ω)² + (4 Ω)² )
|Z| = √( 9 Ω² + 16 Ω² )
|Z| = √(25 Ω²)
|Z| = 5 Ω

Nothing magical about it.
 

Papabravo

Joined Feb 24, 2006
13,958
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.
 

WBahn

Joined Mar 31, 2012
25,923
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.
That makes no sense -- and it isn't consistent with what you said before about only the magnitude having units.

Would you agree that a complex quantity is the sum of a real part and an imaginary part?

Is so, then you are claiming that in

z = a + jb

that 'a' and 'b' have units but that 'z' doesn't.

That creates two big problems:

1) The units magically appear and disappear.
2) The left side of an equation is dimensionless but the right side has dimensions.

Furthermore, the angle also, in general, has dimensions (independent of whether the magnitude has dimensions).

Z = 10 ∠ 30

is VERY different from

Z = 10 ∠ 30°

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.
 

WBahn

Joined Mar 31, 2012
25,923
As an interesting side note, many people think that angular measure is somehow special because it can either have units or not have units. This really isn't the case. We can define measures many different ways and some of them are as ratios. When the two quantities in the ratio have the same units, then the result technically has units of (something)/(something) but since the two (somethings)'s "cancel" we often consider it dimensionless.

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.

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.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
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.
Is this only right for central angle?
With an arbitrary curve, is it possible to define angle this way?
 

WBahn

Joined Mar 31, 2012
25,923
Is this only right for central angle?
With an arbitrary curve, is it possible to define angle this way?
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.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
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.
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.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
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.
 

WBahn

Joined Mar 31, 2012
25,923
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 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.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
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
I think these measures should be called relative measures. For example, relative length or relative angle.
 

WBahn

Joined Mar 31, 2012
25,923
Measures like this are usually referred to as "normalized" measures -- and they are used quite frequently in many fields.

For instance, I have to take rat poison (though, hopefully, tonight's does was my last if the heart surgery on Wednesday goes as planned) and the measure of the effect it has had on my blood clotting is to measure the time it takes to reach a certain level of clotting divided by the time it takes a "normal" person's blood to clot. This is called the INR, or the International Normalized Ratio.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
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.
Should we make a distinction between dimensionless and unitless?
Angle is a dimensionless quantity but it may have units like degrees, radians, grad.
https://en.wikipedia.org/wiki/Dimensionless_quantity
 

WBahn

Joined Mar 31, 2012
25,923
I would disagree with the Wikipedia article that angle is a dimensionless quantity. The notion of measuring the magnitude of an angle is no less physical than is the notion of measuring the magnitude of a line segment. Most angular measures are NOT dimensionless. Just as 10 feet is not the same distance as 10 meters, so too is 10 degrees not the same angle as 10 grads. If these were dimensionless, then they would be equivalent. We simply have one particular measure of an angle that is normalized and, hence, that one particular measure of an angle is dimensionless.
 

Thread Starter

anhnha

Joined Apr 19, 2012
880
We simply have one particular measure of an angle that is normalized and, hence, that one particular measure of an angle is dimensionless.
I believe that degree is also dimensionless. It is also defined as the ratio of two quantities that have units of length.
Also degree and radian are related by a factor. So if radian is dimensionless then degree also is.
 
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