# With and between independent variables

This post continues the previous post here on independent and dependent variables.

The selection of a physical independent variable (or variables) applies to a context such as an experiment. Within that context all other variables are, at least potentially, dependent on the independent variable(s) selected. Functions with the physical independent variable as a functional independent variable are consistent and form a kind of closed system.

Nevertheless, an independent variable in one context may be the same as a dependent variable in another context. The functions in both of these contexts may be related. That is, there may be relations between the functions of one independent variable in one context and the functions of another independent variable in another context.

~For example, (time) speed is defined as the change in distance per unit of time with time as the independent variable. There is also the inverse pace or space speed, which is the change in distance per unit of time with distance as the independent variable.

~However, these two speeds are not the same. One difference is how the speeds are aggregated or averaged: speed requires arithmetic addition and arithmetic averaging but space speed requires reciprocal addition and harmonic averaging. The arithmetic average of speeds v1 and v2 is (v1 + v2)/2. The harmonic average of space speeds v1 and v2 is 2/(1/v1 + 1/v2).

A quasi-variable is a dependent variable in the denominator as if it were an independent variable. If it is a first-order variable, one can take the reciprocal to put it into its proper dependent position, the numerator. Second and higher order variables need to use their inter-functional relation.

One should always use mathematical variables and functions that mirror the physical variables and functions. A physically independent variable should always appear as the argument in a mathematical function. A physically dependent variable should always appear as a function, even if the functional relation is uncertain.

To continue the example above, the space speed should be stated mathematically as 1/(dt(x)/dx) so that distance is shown as an independent variable and time is shown as a dependent variable. Reciprocal addition (see reciprocal arithmetic) is for adding quotients whose denominator is a function of the numerator.