# 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:

$x:y=\left&space;[\frac{x}{y},\frac{y}{x}&space;\right&space;]$

For vectors this means

$\mathbf{x}:\mathbf{y}=&space;\left&space;[\frac{\mathbf{x}}{y},\frac{\mathbf{y}}{x}&space;\right&space;]$

One can either exclude zero or include infinity as follows;

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

A rate of change is

$\Delta&space;x:\Delta&space;t=\left&space;[\frac{\Delta&space;x}{\Delta&space;t},\frac{\Delta&space;t}{\Delta&space;x}&space;\right&space;]$

A vector rate of change is

$\Delta&space;\mathbf{x}:\Delta&space;\mathbf{y}=&space;\left&space;[\frac{\Delta&space;\mathbf{x}}{\Delta&space;y},\frac{\Delta&space;\mathbf{y}}{\Delta&space;x}&space;\right&space;]$

A point rate of change (speed, pace) is

$dx:dt=\left&space;[\frac{dx}{dt},\frac{dt}{dx}&space;\right&space;]$

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

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

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

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