1) How and When do you multiply or divide or combine units? 2) Is there an intuitive understanding or experience for every permutation of derived units? example: distance is a natural phenomenon we experience and understand intuitively as space between two points or positions we've denoted the unit of meters to measure distance. ~~~ area is a natural phenomenon we experience and understand intuitively as space on a flat surface we've denoted the unit of meters x meters to measure area. ~~~ volume is a natural phenomenon we experience and understand intuitively as space in our 3D world. we've denoted the unit of meters x meters x meters to measure volume. ~~~ mass is a natural phenomenon we experience and understand intuitively as the amount of stuff that something has. we've denoted the unit of kilograms to measure mass. ~~~ density is a natural phenomenon we experience and understand intuitively as the amount of stuff that's inside* some volume of space. we've denoted the unit of kilograms / (meters x meters x meters) to measure density. *why inside? why not outside? what's governing how we even decide what density is? ~~~~~~~~~~~ what about... meters / kilograms? or... kilograms x kilograms? what would they describe? if anything? so am i correct to conclude that all base units and derived units are put together in such a way as to describe only these phenomenons we intuitively understand and experience because if we didn't understand there's no way to describe it ~~~~~~~~~~~ but then how are 'derived units' DERIVED in the first place? experiments? how? mathematically? how? ~~~~~~~~~~~ thanks for your time to read this and please help me understand! and please provide some examples... and please don't just send me to some link of long articles... thanks again.
Derived units are just that: they have been derived from the application of some mathematical or physical formula, like F = ma. In SI, the unit of force, the newton, is derived from this relation, so a newton is derived from base units of kg-m/s^2 http://en.wikipedia.org/wiki/SI_derived_unit
No, I don't think that's correct. Intuition will often mislead you. Units are defined based on the algebraic relationship used to define physical quantities. Density is defined as mass/volume, so its units are by definition kg/m^{3}. whether that jives with some pre-existing intuition is a separate question not relevant to the choice of units
When physicists state an equation, they want it to be correct no matter what conventions people are using in measuring the physical quantities. For example, if the equation is correct when mass is measured in kilograms, it should still be correct if an experimenter decides to measure mass in grams. This applies to equations where we make statements that are supposed to be "universally" true, like F = MA. It also applies to equations that are meant to apply only to very particular situations. For example, suppose you have some complicated piece of machinery and you determine by experiment that there as an empirical relation between the force F in newtons that you exert on a lever and the distance X in meters that the machine moves given by F = 3 X. This is not a universal physical law. And it isn't the correct equation if X is measured in centimeters. No fiddling with units can turn F = 3 X into a universal physical law. But we can at least state the equation so that it is valid for the particular machine regardless of what units of measure are used. This is done by assigning units to the constant 3. We state that the constant 3 has units of " newton/meter". That information tells a person who wants to measure distance in centimeters how to create a valid equation with a different constant (by using the usual rules for converting units). In this situation, you may or may not have an intuitive feel for what a newton/meter is. If you don't understand the complicated machine, you probably won't. Even equations that give so-called "universal" physical laws are not applicable to arbitrary situations. For example, F = MA doesn't claim that the force you use depress the lever on your toaster this morning is equal to the mass of physics textbook times the acceleration of your car on the highway next week. A lot of words are needed to specify the situation to which an equation applies. When people a state physical law merely as an equation, the audience is expected to have the cultural background to understand the situation without hearing a description. Whether you find the physical units in an equation intuitively understandable or not will vary with how well you understand the situation to which the equation applies. You might find the same units intuitively clear in one situation and not in another.