ISO 80000-2: Mathematical signs and symbols for quantities and units

ISO 80000-2 assembles the mathematical grammar needed to interpret measurements, equations, and technical reports without ambiguity. The scope goes beyond a list of symbols: it describes how to write values, indices, and operators so quantities from the other chapters carry the same meaning in any language or scientific software.

Pair this guide with the introduction to ISO 80000-1 to understand how notation relates to base quantities. For a panoramic view of every discipline covered by the series, explore the compendium of all 13 parts of ISO 80000. When you need to revisit official unit definitions, consult our summary of the International System of Units (SI).

New to metrological terminology? Start with what a unit of measurement is before adopting the formal notation that Part 2 requires.

What ISO 80000-2 covers

Part 2 is the semantic backbone of the entire standard. Before speaking about length, mass, or electric charge, we need consensus on how equations express quantities, how vectors differ from scalars, and what a logical symbol means in a unit table. These highlights summarize the scope.

  • Unified mathematical language

    Part 2 specifies how to write operators, functions, vectors, and sets so an equation means the same thing in lab notes, CAD documentation, or scientific software.

  • Bridge to physical quantities

    It defines dimensionless quantities—such as angles, ratios, and logarithmic levels—that make it possible to compare measurements across the remaining ISO 80000 parts.

  • Consistency with SI and ISQ

    Every mathematical symbol aligns with the typographic rules of the International System of Units and the International System of Quantities to remove ambiguity.

Essential typographic conventions

ISO 80000-2 details the correct way to present quantities to avoid interpretation errors in drawings, reports, and digital applications. Adopt these guidelines before publishing documentation or feeding analytical models.

Variables and constants

Use italic letters for algebraic quantities (such as v or p) while keeping fundamental constants like e and π upright to separate measured values from universal numbers.

Vectors, matrices, and tensors

Vectors appear in bold or with an arrow overhead (\mathbf{F} or →F), matrices use bold uppercase letters (\mathbf{A}), and tensors require clearly spaced subscripts and superscripts.

Functions and operators

Write trigonometric and logarithmic functions in upright type (sin, log) and separate them with thin spaces to avoid confusing them with multiplication or multiple arguments.

Relations and sets

Symbols such as ∈, ⊂, ∀, or ∑ follow spacing rules that preserve readability in long derivations and unit tables.

Families of symbols and operators

These groups of symbols appear in every discipline covered by ISO 80000. Keeping their syntax consistent ensures calculations in mechanical, chemical, or biomedical engineering stay aligned.

  • Arithmetic and algebra

    Addition, subtraction, dot products (⋅), and cross products (×) follow an agreed order of precedence reflected in calculators and spreadsheets.

  • Differential and integral calculus

    Partial derivatives (∂), the differential symbol (d), and integrals maintain consistent spacing around limits to document rates of change and accumulated quantities.

  • Complex mathematics

    ISO 80000-2 prescribes the upright i for the imaginary unit, a centered modulus |z|, and arguments arg(z) measured in radians—essential for electromagnetism and signal analysis.

  • Statistics and logic

    Quantifiers (∀, ∃), logical operators (¬, ∧, ∨), and probability symbols (P, E, Var) align with conventions that reduce mistakes when reporting uncertainty or hypotheses.

Dimensionless quantities and core units

While ISO 80000-2 does not introduce new base quantities, it standardizes essential dimensionless units for describing physical phenomena. Use these definitions as a reference when documenting measurements and experimental results.

Plane angle (rad)

Defined as the ratio of an arc length to the radius of a circle. ISO 80000-2 treats the radian as a fundamental dimensionless quantity for rotations and phase shifts.

Applications: Kinematics, gear design, wave analysis, and angular frequency conversions (ω).

Convert radians to degrees

Solid angle (sr)

Ratio between the projected area on a sphere and the square of its radius. It quantifies light or radiation beams without extra base units.

Applications: Photometry (lumens, lux), radiometry, and optical sensor design.

Lux to lumens calculator

Pure number (1)

Any ratio of homogeneous quantities (efficiency, friction factor, or Reynolds number) that requires no explicit unit but still needs clear indices and subscripts.

Applications: Transfer coefficients, refractive indices, and adiabatic exponents in thermodynamics.

ISO 80000 parts overview

Neper (Np)

Logarithmic unit based on the natural logarithm for amplitudes and signal levels. ISO 80000-2 clarifies its relationship with exponentials and complex numbers.

Applications: Telecommunication attenuation, vibration analysis, and electronic filters.

Logarithm base conversion

Bel and decibel (B / dB)

Logarithmic units based on log10 that describe power or intensity ratios. The decibel is one tenth of a bel, making it practical for sound, radio, and electronics reports.

Applications: Acoustic measurements, losses in electrical networks, and signal levels in audio systems.

Decibel to power percentage

Reference table of principal quantities and units

The table below captures the dimensionless quantities most frequently cited in ISO 80000-2. Each entry links to tools or articles that help you apply the notation in engineering and scientific workflows.

Quantity Symbol Coherent SI unit How ISO 80000-2 defines it Example use
Plane angle θ radian (rad) Dimensionless ratio of arc length to radius; ISO 80000-2 formalizes that the symbol rad should accompany numerical values even though the unit is technically 1.

Documenting angular position, wave phase, and rotational speed in conjunction with ISO 80000-3 on space and time.

Radians to Degrees Calculator

Solid angle Ω steradian (sr) Dimensionless measure of how large an object appears from a point, defined as area divided by radius squared. The steradian underpins photometric quantities like lumen and lux.

Calculating luminous flux, radiated power, and antenna patterns alongside ISO 80000-7 on light and related electromagnetic radiation.

Lux to Lumens Calculator

Logarithmic amplitude level L neper (Np) ISO 80000-2 defines levels via natural logarithms of amplitude ratios to maintain coherence between mechanical vibrations and control-system signals.

Assessing structural damping, signal attenuation, and modal analysis where exponential decay appears.

Logarithm Base Conversion

Power level L_{10} decibel (dB) Power levels rely on common logarithms. ISO 80000-2 sets the syntax for reference quantities, subscripts, and annotation when comparing energy flux or sound pressure.

Reporting noise exposure, RF link budgets, and amplifier gain alongside ISO 80000-7 photometric definitions.

Decibel to Power Percentage

Relative refractive index n 1 Expressed as the ratio of phase velocities in different media. ISO 80000-2 formalizes the use of dimensionless ratios with subscripts to track reference materials.

Designing optical systems, calibrating sensors, and connecting to ISO 80000-5 on light and radiation.

ISO 80000 parts overview

How to integrate ISO 80000-2 into your processes

Applying the standard demands alignment between engineering, metrology, and documentation teams. Use these steps as a checklist so notation never becomes a bottleneck.

  1. Normalize templates and styles

    Update spreadsheets, reports, and CAD libraries to use the italics, bold type, and spacing defined by ISO 80000-2. Add concrete examples to your technical brand guide.

  2. Document dimensionless quantities

    When publishing efficiencies or phase numbers, reference radians, nepers, or decibels explicitly and follow the syntax prescribed by the standard.

  3. Validate with digital tools

    Pair reports with interactive calculators that respect typographic conventions to avoid misinterpretation during peer review or auditing.

When preparing calibration reports or performance dashboards, verify that indices, exponents, and dimensionless quantities follow the recommendations of Part 2. Cross-reference our article on ISO 80000: quantities and units to connect notation with industrial applications.

Recommended calculators

Practice ISO 80000-2 notation with tools that already apply its rules explicitly. Each calculator links to related documentation so you can reinforce your understanding of quantities and units.

  • Arc length of a circle

    Reinforces the role of radians by linking central angles to arc distances while applying ISO 80000-2 notation.

  • Circle sector area

    Applies standardized trigonometric symbols to compare surfaces bounded by coherent plane angles.

  • Radians to degrees conversion

    Shows how plane angles remain dimensionless yet meaningful when converted to sexagesimal units.

  • Logarithm base conversion

    Checks logarithmic factors that must stay consistent with bel and neper units described by the standard.

Connect notation with ISO’s dimensionless guides

Reinforce how Part 2 treats angles and logarithmic ratios by pairing this reference with our radian overview, the steradian explainer, and the decibel deep dive so geometry, photometry, acoustics, and signal processing all share the same symbol rules.

Use the radians-to-degrees calculator and the decibel-to-power converter alongside these articles to see how ISO 80000-2 notation stays intact from raw numbers to published specifications.

Keep exploring the ISO 80000 series

Strengthen your reference library by linking this guide to chapters that explain specific physical quantities and to SI overviews.