Rest means no motion, or at least no motion detected by an observer.
We know what rest for space means: staying in the same place. That is, rest means no change of position, no travel distance, no length of motion. So at rest the numerator of the speed is zero.
Yet clocks tick on. The denominator of speed is not zero. The speed of rest then equals zero, that is, a zero length of motion divided by a non-zero quantity of time. Speed v = Δx/Δt = 0/Δt = 0.
What is rest for time? Rest means no duration of motion, no travel time. So at rest the numerator of the pace is zero.
In this case, is the length of motion zero, too? No. For pace the length is the independent quantity. It doesn’t depend on the motion. It depends on the given length or unit of length. So the pace of rest is zero, that is, a time of zero divided by a non-zero length. Pace u = Δt/Δx = 0/Δx = 0.
Yet a zero pace seems to say one gets a change of place with no lapse of time. What gives?
Length of motion in the pace ratio is the independent variable. Whether length is conceived to be continually increasing, as if it were a clock, or just a quantity of length for comparison, it is independent of the motion measured. The numerator, the time, is what is measured and compared with a quantity of length to determine the pace.
It is similar with speed. Whether or not there is a clock ticking away, the denominator is a quantity of time compared with a quantity of length. All the clocks in the world could be broken, yet the denominator of speed, the change in time, would still be non-zero.
Consider a vehicle with an odometer and a stopwatch that is running whenever the vehicle is in motion. Both the odometer and the stopwatch would record no additional time for a vehicle at rest. This could not be represented as a ratio since 0/0 is not a valid ratio. Such a state has an indeterminate rate of motion.