Properly conceived, the real numbers (i.e., rational and irrational numbers) are tools of measurement, means of expressing and specifying quantitative relationships among magnitudes. Yet mathematicians in the late 19th century, offering a new conception of mathematical rigor, transformed the number system into a purely formal construct, satisfying the required axioms, but bearing no particular relationship to objects in the world.

Drawing an alternative from his book in progress, Mathematics: The Measure of All Things, Dr. Knapp begins by offering a geometric perspective on magnitudes, showing that relationships between real numbers reflect, and are determined by, relationships between the magnitudes that they measure. On this foundation, he presents the real numbers as a system of measurements, explaining, in particular, the role, meaning, and validity of irrational numbers in mathematics, especially in light of the fact that no concrete measurement ever distinguishes a rational number from an irrational number.