Re: difference between saying that one quantity is proportional to another and =

"Proportional" means that there is some relation between the two sides of the "equation" (Although it's not strictly an equation without the "=" sign!). It's more obvious when the "=" is there, because we all know that that really means that the left side is the same as the right side. A proportion is a lot like an equation except that the left and right sides aren't necessarily equal, but they have some relation to each other, usually with a constant factor that is not included. I'll use an example to help illustrate the difference.

Let's say I know that the acceleration of gravity at the surface of the earth is directly proportional to the mass of the Earth and inversely proportional to the square of the radius of the Earth. That would be written
g &prop M / R²
"Directly proportional" means that the quantity is in the numerator, and "inversely proportional" means the quantity is in the denominator, of the right side. Okay, but it's not an "equation", yet, because there's some constant factor involved that maybe I haven't been able to determine yet. But when I do figure it out, it is usually multiplied times the right side. In this case I know it is the constant "G", which is the so-called "universal gravitational constant", and it gets into the act like this:
g = G M / R²

By adding the constant factor I know the relation between the left and right sides is exact. Prior to having the constant there was some relationship, but not yet necessarily exact, and we call that a "proportion".

John Link, MadSci Physicist