Representing complex number as a vector.

[dE_logics]

In the above image the default format of a complex number is taken as a vector, since vector addition is only between 2 similar units (example 10 cos 30 cm + 10 sin 30 cm or 5 cms on y-axis and 2 cm on the x; which gives a resultant following vector addition), it can be said that that the real and imaginary part (taken as individual axes) are having the same units; what I mean here is that in a complex number, a + bj, a and bj are having the same units; i.e the magnitude is divided into real and imaginay parts and it gives a resultant following vector addition (a + bj).

Infact this is the reason why they get added, cause addition of 2 numbers is only possible if the units are common.

So if you get a complex number in physics, its (a+ib) units; it can't be a [unit 1] + ib [unit 2].


Am I right?

 

[HallsofIvy]

I'm not clear what you are saying. What units are you talking about? There are no "units" such as m, inches, etc. in number systems.

 

[dE_logics]

Yes

But, let's take the practical application, I mean number systems are used in physics right?...it its given a unit, when a complex number comes in a real life situation, it does have a unit, what I'm asking here is it necessary for both the real and imaginary parts to have the same unit?

If not that image might be wrong in a few place, that's not possible considering the graphical representation of every complex number.

 

[HallsofIvy]

Please give an actual situation in which a measured quantity, i.e. a quantity with units, is complex!

 

[csprof2000]

... Well, yes, if you have (a + bi), then a and bi both have the same units.

But a, b, and i are all dimensionless. Hence, they trivially have the same dimension.

 

[de_brook]

In the above image the default format of a complex number is taken as a vector, since vector addition is only between 2 similar units (example 10 cos 30 cm + 10 sin 30 cm or 5 cms on y-axis and 2 cm on the x; which gives a resultant following vector addition), it can be said that that the real and imaginary part (taken as individual axes) are having the same units; what I mean here is that in a complex number, a + bj, a and bj are having the same units; i.e the magnitude is divided into real and imaginay parts and it gives a resultant following vector addition (a + bj).

Infact this is the reason why they get added, cause addition of 2 numbers is only possible if the units are common.

So if you get a complex number in physics, its (a+ib) units; it can't be a [unit 1] + ib [unit 2].


Am I right?

Yes you are right. Don't forget that the real numbers R is a subset of the complex numbers C. The imaginary unit i is just a number which has a meaning which you know.
The fact that 'a is a real number' also mean that 'a is a complex number'. Two vectors can be added if they are of the same dimension, since they are from the same vector space. So if you take a real number 'a' which is in it sense a complex, and add it to complex number ib, the result will give a complex number a + ib.

 

[dE_logics]

Thank you people.

Problem solved.