Physical Numbers

A physical number is one whose displacement on the numberline closes to zero in a finite number of iterations. Each digit placed reduces the uncertainty in its position; eventually the uncertainty is gone and the number is perfectly described. It is exactly where it is.

Measuring Displacement

Consider the number $7423$. We iterate digit by digit and track the remaining displacement — the width of the interval on the numberline within which the true value still lies:

Iteration	Value known	Displacement
1	7___	within $[7000, 8000)$, width $10^3$
2	74__	within $[7400, 7500)$, width $10^2$
3	742_	within $[7420, 7430)$, width $10^1$
4	7423	exactly $7423$, width $0$

$\frac{d}{dN}$ — the precision per digit — is constant and nonzero throughout, then drops to zero at step 4. The number is physical because that drop happens at finite $N$.

The Infinite Zoom Analogue

The same idea has a spatial reading. Zoom into the numberline by a factor of 10 at each step, centering on the value. Each zoom reduces our displacement by one order of magnitude:

Zoom level	View window	Resolution
$1/1$	$[0, 10)$	ones
$1/10$	$[7, 8)$	tenths
$1/100$	$[7.4, 7.5)$	hundredths
$1/1000$	$[7.42, 7.43)$	thousandths
$1/10000$	$[7.423, 7.424)$	ten-thousandths

At each step the window closes. For a physical number the window reaches a point — a single location — in finite steps. The zoom terminates. The number has an address.