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Consider an original abstract data type whose values are rational numbers. Store an ordered pair $(n,d)$ of integers, interpreted as a numerator and a denominator. Use exact, unbounded integer arithmetic in this model. The storage space, representation invariant and abstraction function areThe representation invariant does not require a positive denominator or a reduced fraction. Consequently $(1,2)$, $(2,4)$ and $(-1,-2)$ are three different concrete states with the same abstract rational number $1/2$. Similarly, $(0,1)$ and $(0,-3)$ both represent zero. The abstraction function is a surjection: every rational number has an integer numerator and a nonzero integer denominator, so every abstract value has a representation.

Both $(1,0)$ and $(0,0)$ violate the representation invariant; neither has a value under this abstraction function. By contrast, $(2,4)$ is completely valid. Simplifying it to the reduced fraction $1/2$ may be a useful implementation choice, but is not a requirement of this representation. Strengthening the representation invariant to require a positive denominator and a greatest common divisor of one would choose a unique stored representative for each rational number; every constructor and operation would then need to establish that stronger condition.

Equality of abstract values is therefore different from equality of stored ordered pairs. For valid $(a,b)$ and $(c,d)$,This defines an equivalence relation on $R_{\mathrm{valid}}$: each equivalence class contains all representations of one rational number. An API equality operation can compare these cross-products using exact integer arithmetic. Comparing the two fields individually would incorrectly declare $(1,2)$ and $(2,4)$ unequal as rational numbers. The general reason is that an abstraction function may map several concrete values to one abstract value.^[ref]

The peak-to-peak amplitude is the difference between a signal's largest and smallest values over a stated interval. For $u(t)=u_c+A\cos \omega t$, it equals $2A$ and is independent of the offset $u_c$. It is thus distinct from the maximum value measured relative to zero.

An oscilloscope trace that runs from $-2\mathrm{V}$ to $+4\mathrm{V}$ has peak-to-peak amplitude $6\mathrm{V}$. If the trace is sinusoidal, its offset is $1\mathrm{V}$, its peak amplitude about that offset is $3\mathrm{V}$, and its alternating component has root mean square $3/\sqrt{2}\mathrm{V}$. For a nonsinusoidal or noisy signal these conversion factors need not hold. Reporting the observation interval matters because a rare noise spike can change the extrema without much change in the root mean square.

For a synthetic illustration, choose $m=0.3000\mathrm{g}$, $V=125.0\mathrm{mL}$, $p=101325\mathrm{Pa}$ and $T=373.15\mathrm{K}$. Using molar gas constant $R\approx 8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ gives $n\approx 0.00408234\mathrm{mol}$ and $M_{\mathrm{inferred}}\approx 73.4873\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$ under the ideal gas law. These are chosen model inputs, not observations of a particular chemical substance.

An error's direction follows the model. Extra exterior water raises the apparent $m$ and hence $M_{\mathrm{inferred}}$. If a fraction $f$ of the hot gas particles is residual air, and only the unknown vapor contributes to recovered $m$, an otherwise ideal gas calculation gives $M_{\mathrm{inferred}}=(1-f)M_{\mathrm{true}}$. For a pure nonideal vapor, define $Z=pV/(nRT)$; then $M_{\mathrm{inferred}}=M_{\mathrm{true}}/Z$, so the direction depends on $Z$. The dye endpoint alone cannot establish gas purity, thermal equilibrium, or ideality. These conditional checks explain why the same apparatus cannot provide a universal molar mass measurement for arbitrary mixtures or substances that decompose on heating.

Online text: openstax.org/books/university-physics-volume-3/pages/2-3-images-formed-by-refraction

Cited passages: OpenStax: Images formed by refraction.

Online text: openstax.org/books/university-physics-volume-3/pages/2-8-microscopes-and-telescopes

Section of University Physics Volume 3 by Samuel J. Ling and others. The existing book record supplies the whole-book authors and 2016 publication year; no separately credited section author or date is inferred.

Cited passages: Microscopes, Figure 2.38; equations 2.35–2.37 and near-point versus relaxed viewing.

Online text: openstax.org/books/university-physics-volume-2/pages/8-4-capacitor-with-a-dielectric

Cited passages: §8.4, Figure 8.17 and equation (8.11): isolated capacitor, unchanged free plate charge and increased capacitance.

Online text: openstax.org/books/university-physics-volume-2/pages/8-5-molecular-model-of-a-dielectric

Cited passages: §8.5, Figures 8.19–8.20 and the thermal-fluctuation qualification following Figure 8.20; §8.5, dielectric breakdown paragraph immediately before the definition of dielectric strength; §8.5, paragraph following Table 8.1: high dielectric constant does not ensure a good insulator; mobile ions permit conduction; §8.5: dielectric strength definition and voltage limit for a specified plate separation.

Online text: openstax.org/books/university-physics-volume-2/pages/2-1-molecular-model-of-an-ideal-gas

A section of University Physics Volume 2, published 6 October 2016, by Samuel J. Ling, William Moebs and Jeff Sanny. These are verified whole-book authors; no distinct chapter author is inferred.

Cited passages: OpenStax University Physics Volume 2, section 2.1: departures from ideal-gas behavior; OpenStax University Physics Volume 2, section 2.1: molecular model and ideal gas law; OpenStax University Physics Volume 2, section 2.1: particle-count and molar forms; OpenStax, University Physics Volume 2, section 2.1: the ideal gas law; §2.1, The Gas Laws, Figure 2.4, and equations 2.1–2.2: fixed temperature and particle number, absolute pressure, and the dilute-gas limit.

Online text: openstax.org/books/university-physics-volume-2/pages/10-1-electromotive-force

Online text: openstax.org/books/university-physics-volume-1/pages/2-2-coordinate-systems-and-components-of-a-vector

Section of the authored University Physics Volume 1. The year 2016 comes from the existing whole-book record; no section revision date is asserted.

Cited passages: §2.2, Cartesian coordinates: equations (2.10)–(2.13) and Figure 2.16, unit basis vectors and scalar components.

Online text: openstax.org/books/university-physics-volume-1/pages/2-1-scalars-and-vectors

Section of the authored University Physics Volume 1. The year 2016 comes from the existing whole-book record; no section revision date is asserted.

Cited passages: §2.1, Algebra of Vectors in Two Dimensions: Figure 2.12 and the head-to-tail construction; §2.1, Figures 2.2–2.5: vector arrows, equal vectors and displacement.

Online text: openstax.org/books/university-physics-volume-1/pages/6-2-friction

Cited passages: Ling, Sanny and Moebs, OpenStax University Physics 1, §6.2: responsive static friction.

Cited passages: §6.2, Static and Kinetic Friction, equations 6.1–6.2; empirical-law limitations before Example 6.10; Friction and the Inclined Plane, Example 6.11 strategy and Figure 6.14 description: normal-force balance.

Online text: www.math.toronto.edu/mgualt/courses/MAT1300F-2012/docs/1300-2012-notes-2.pdf

Official course notes for the 2012 offering. Marco Gualtieri is the course instructor, rather than an inferred author of these notes.

Cited passages: Topology I: smooth manifolds, Week 2, Section 1.2, equation (11) and Definitions 1.15–1.16, pages 5–6; Topology I: smooth manifolds, Week 2, Section 1.3, equation (21), Definition 1.21 and Remark 1.24, page 7.

Online text: www.math.toronto.edu/mgualt/courses/MAT1300F-2012/docs/1300-2012-notes-1.pdf

Official course notes for the 2012 offering. Marco Gualtieri is identified as the course instructor; no author byline for these notes is established. University of Toronto hosts the course.

Cited passages: Topology I: smooth manifolds, Week 1, Example 1.6, equations (4)–(5), page 2; specialize the sphere charts to dimension one; Topology I: smooth manifolds, Week 1, Section 1.1, Definition 1.1, page 1.

Online text: farside.ph.utexas.edu/teaching/sm1/Thermalhtml/node68.html

The HTML footer is dated 25 January 2016. This is the inspected page version, not an independently established print publication date.

Cited passages: Fitzpatrick, Joule-Thompson Throttling, equations 6.167–6.174: inlet/outlet pressure work and enthalpy conservation; Fitzpatrick, Joule-Thompson Throttling, equations 6.175–6.182: ideal-gas limit and coefficient; Fitzpatrick, Joule-Thompson Throttling, equations 6.188–6.190 and following paragraph: inversion locus, maximum temperature and limits of the van der Waals prediction; Fitzpatrick, Joule-Thompson Throttling: steady-flow enthalpy balance.