We go beyond merely asserting the exchangeability of units, we demonstrate it through their concrete realization. Concrete units operationalize additivity. Obtaining total length by concatenating inches demonstrates that additivity holds. Since combining inches by physical concatenation gives the result predicted by additivity, rulers implement what Physicist Norman Campbell called fundamental measurement. We count by means of the ordered numerals to obtain the total.
Rulers implement a correspondence between units and numbers.
The left-most edge or marker of the ruler is the beginning and indicates the absence of any units, i.e., none. The natural numbers mark the unit-inches as successive amounts concatenated to the right. The ruler is a sophisticated device. It contains a history of ideas and concepts that are fundamental to measurement. The crucial concept is that the abstract mathematical properties of a ruler transcend the concrete instrument.