Serving humanity with dimensional analysis

Yesterday, The Godsister asked about the aperture values on a camera. The result was “OHHHHHHHHHHHHHHHHHHHH [excess H's truncated]. That explains a lot.”

I would like to think I have imparted the concept of dimensional analysis upon a 16 year-old. I would like to think that I have made a difference.



Edited chat transcript below:


Hi!
Hello Cute One.
Hello Pink Man! Can I ask something? Going to ask anyways
What is the unit of the aperture number? Does it have a unit?
Let me get you a proper reference
Ok, wikipedia's aperture articles says the following:
area = (f / 2N)squared x pi
N is the aperture number, and f is the focal length of the lens
If you want units I will need to take you down the road of dimensional analysis. It’s not hard; bear with me...
The area of a circle is:
Area = pi x r squared
Area is measured in square metres, r is measured in metres. square the r and the units match.
The same principle must also apply for the equation for aperture area
area = pi x (f/2N)squared
The unit of (f/2N) should then be length
f is the focal length of the lens
I don’t get the f/2N part
f/2N is just how the aperture number is made
Ok.
Someone decided that using this correlation would give a convenient system
Ok, so area = pi x (f/2N) squared
Why is it 2N then? Why not just N?
Again, it falls to definition issues.
If the equation for aperture had be made using N instead of 2N, the aperture values would come out differently
Ok.
So, it is a manufactured value that people have decided is convenient. For example, the metre was initially defined as the length from the equator to the North Pole, divided by 10 million
If aperture is the area of the thingy... then why is the focal length involved?
Anyway, back to (f/2N)
I’ll finish the units of aperture first...
(f/2N) should have a length unit, then when the length is squared it gives the area
We know that f is the focal length, which is a length unit
This length is divided by 2N.
Since focal length divided by something still gives length, the aperture number has no unit
Jumping a few steps ahead, the aperture is actually the ratio between the focal length and the hole diameter
And ratios between things of the same units have no dimentions
OHHHHHHHHHHHHHHHHHHHHHHHHH [excess H's truncated]
That explains a lot.
I thought the aperture number was just the area of the opening