Rates and inverses

This post is the latest in a series on rates.

A rate is a variable quantity measured with respect to a quantity determined independently. A rate is expressed as a ratio of the quantity measured and the independent quantity. A rate of change is a difference of quantities measured with respect to a difference of quantities determined independently. The denominator is the independent quantity.

The converse rate is the rate formed by interchanging the names, units, and dependencies of the quantities. If x/y is a rate with independent quantity y, then y/x is the converse rate with independent quantity x. If Δxy is a rate of change, then Δyx is the converse rate of change. Note that a rate and its converse are isomorphic.

The inverse rate is the rate formed by taking the reciprocal of the rate. If x/y is a rate, then 1/(x/y) is the inverse rate. If Δxy is a rate of change, then 1/(Δxy) is the inverse rate of change. Note that the independent quantity is the same for a rate and its inverse.

The inverse converse rate is the reciprocal of the converse rate, which equals the converse of the inverse rate. If x/y is a rate, then 1/(y/x) is the inverse converse rate. If Δxy is a rate of change, then 1/(Δyx) is the inverse converse rate of change. Note that the inverse converse rate has the same dependencies as the converse rate.

Rate Converse Rate Inverse Rate Inverse Converse Rate
x/y y/x 1/(x/y) 1/(y/x)
Δxy Δyx 1/(Δxy) 1/(Δyx)

For example:

The speed (or time speed) of a body is the length traversed per unit of independent time without regard to direction, Δxt. The converse speed, pace (or slowness), of a body is the traversal time per unit of independent length without regard to direction, Δtx. The inverse speed of a body is the reciprocal speed, 1/(Δtx). The inverse pace (or space speed) of a body is the reciprocal pace, 1/(Δxt). The reverse speed of a body is the negation of its speed, −Δxt.