Mathematical Mensuration

the art of measuring

[under construction, and arithmetic needs re-checking]

NOTIONAL-development is a natural mathematics learning tool, used by better students for introducing themselves to computational studies.

MEASUREMENT - a meter-stick, a slide-rule, a photo-detector - graduated guide, logarithmic convenience, systematic device. Each is precision limited to what can be visibly distinguished in the region of marked interest. The meter-stick suffers parallax: the ability of the user to hold the stick near enough to the object to obtain an accurate reading. A vernier on a caliper can add a digit of precision if the object itself has parallel faces (or facets) or honed vertices, and registers readily. The slide-rule suffers focus: the adjacent, finely drawn marks are already free by design of most parallax, and (if the rule is a top quality purchase) all that remains to challenge the user is to visually interpolate between knife-edge stripes almost on top of each other - they are so close. The photo-detector suffers thresholding: a certain quanta of photons and electric charge (or current) is required to ascertain sufficiency. More or less, at the point of the "yes-no" decision cross-over, can be exceedingly small differences, and yet in-between is a "metastable" position, wherein the electronic circuit has not sufficient energy to report either choice. Very similar constraints on the ability to determine a measurement - but we will consider only the validity of the choice, in this article, as such is the usual case: metastability is usually resolved by setting additional constraints, time limits, repetitions, and random dither (or circuitry noise).

MATHEMATICALLY: we begin with our notion of measurement within a frame of resolution. (We may prefer the slide-rule example for its nearness to plain arithmetic). Presuming an accurately drawn scale at maximum resolution, and graduations identified as a, a+1, a+2, ... , and a measurable value x: a x a+1 , or x e [a,a+1], and m(x), our (attempted) measurement of x, we reason our notion (and prove it to ourselves). If a m(x) a+0.5, if x measures distinguishably nearer a than to a+1, then we could have resolved further the interval [a,a+1] into [a,a+0.5], .... Therefore only, a m(x) or m(x) a+0.5 but not both, either, x is indistinguishably around a, or else, indistinguishably between a's. Then again, we could have resolved even further yet by assigning a+0 to [a-0.5,a+0.5] and a+1 to [a,a+1], that is, gain a bit of precision (one bit = a factor of two) in resolution by attempting to estimate the balancing of x around-vrs.-between a's, and calling out our choice. Therefore

[under construction]

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