Ratios and quotients

The traditional ratio, x : y, represents both x / y and y / x. In order to represent a ratio as quotients, both forms are required. Let us define a ratio as an ordered pair of quotients:

x:y=\left [\frac{x}{y},\frac{y}{x} \right ]

For vectors this means

\mathbf{x}:\mathbf{y}= \left [\frac{\mathbf{x}}{y},\frac{\mathbf{y}}{x} \right ]

One can either exclude zero or include infinity as follows;

\begin{matrix} \frac{x}{o}=\infty & x\neq 0 \wedge \infty \\ & \\ \frac{x}{\infty}=0 & x\neq 0 \wedge \infty \end{matrix}

A rate of change is

\Delta x:\Delta t=\left [\frac{\Delta x}{\Delta t},\frac{\Delta t}{\Delta x} \right ]

A vector rate of change is

\Delta \mathbf{x}:\Delta \mathbf{y}= \left [\frac{\Delta \mathbf{x}}{\Delta y},\frac{\Delta \mathbf{y}}{\Delta x} \right ]

A point rate of change (speed, pace) is

dx:dt=\left [\frac{dx}{dt},\frac{dt}{dx} \right ]

A vector point rate of change (velocity, lenticity) is

d\mathbf{x}:d\mathbf{y}= \left [\frac{d\mathbf{x}}{dy},\frac{d\mathbf{y}}{dx} \right ]

A second order point rate of change (acceleration, relentation) is

d^2\mathbf{x}:d^2\mathbf{t}=\left [\frac{d^2\mathbf{x}}{dt^2},\frac{d^2\mathbf{t}}{dx^2} \right ]