1. ## Dimensionless Numbers

Any dimensionless number multiplied by another dimensionless number is always another dimensionless number right?

Therefore any scale multiplied by a group of dimensionless numbers is also dimensionless.

A, B and C are arbitrary scales

x = (A/B)x' (x' is dimensionless variable),
y = Cy' (y' is dimensionless variable)

Then

xy = (AC/B)x'y' (term is also dimensionless right?)

2. Originally Posted by davefulton
Any dimensionless number multiplied by another dimensionless number is always another dimensionless number right?-
Yes, of course.

Therefore any scale multiplied by a group of dimensionless numbers is also dimensionless.
What do you mean by "scale"?

A, B and C are arbitrary scales

x = (A/B)x' (x' is dimensionless variable),
y = Cy' (y' is dimensionless variable)

Then

xy = (AC/B)x'y' (term is also dimensionless right?)
Assuming that A, B, and C are themselves dimensionless, yes.