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

Discussion in 'Math' started by anhnha, Dec 28, 2015.

1. ### anhnha Thread Starter Active Member

Apr 19, 2012
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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?

2. ### Papabravo Expert

Feb 24, 2006
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Complex numbers don't have units, but the magnitude of a complex number can have units.

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3. ### anhnha Thread Starter Active Member

Apr 19, 2012
776
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Bu
But how about s in Laplace transform above? It is not magnitude. So why is it called frequency and have the unit?

4. ### nerdegutta Moderator

Dec 15, 2009
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Hi, I've moved this thread to the Math forum. It's more like a math discussion than General Electronics discussion.

5. ### WBahn Moderator

Mar 31, 2012
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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.

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6. ### WBahn Moderator

Mar 31, 2012
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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.

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7. ### Papabravo Expert

Feb 24, 2006
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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.

8. ### WBahn Moderator

Mar 31, 2012
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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.

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9. ### Glenn Holland Member

Dec 26, 2014
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The unit of a complex number is the same as the coefficient ("A") of the number such as "A times the Square Root of - N".

Reactive power is an example of an imaginary quantity that has units.

https://en.wikipedia.org/wiki/AC_power

10. ### WBahn Moderator

Mar 31, 2012
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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.

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11. ### anhnha Thread Starter Active Member

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

12. ### WBahn Moderator

Mar 31, 2012
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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.

13. ### anhnha Thread Starter Active Member

Apr 19, 2012
776
48
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.

14. ### anhnha Thread Starter Active Member

Apr 19, 2012
776
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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.

15. ### WBahn Moderator

Mar 31, 2012
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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.

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16. ### anhnha Thread Starter Active Member

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

17. ### WBahn Moderator

Mar 31, 2012
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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.

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18. ### anhnha Thread Starter Active Member

Apr 19, 2012
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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

19. ### WBahn Moderator

Mar 31, 2012
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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.

20. ### anhnha Thread Starter Active Member

Apr 19, 2012
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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.