ISO 80000-1: General Principles for Quantities and Units

Introduction

ISO 80000-1 (“Quantities and units — Part 1: General”) is the keystone of the ISO 80000 series. It sets out the foundational rules for how scientific and technical quantities are named, symbolized, and combined with SI units in a coherent, unambiguous way. If you design instruments, publish research, prepare technical documentation, or build software that handles measurements, Part 1 is the style guide and rulebook that keeps your formulas and data interoperable across disciplines and borders.

This article provides a rigorous, practitioner-oriented overview of ISO 80000-1: what it covers, why it exists, how it relates to the International System of Units (SI) and the International System of Quantities (ISQ), and the conventions you should apply in daily work. It emphasizes clarity of notation, dimensional consistency, and good measurement practice, with concrete examples and pitfalls to avoid.

Scope and Purpose of ISO 80000-1

ISO 80000 comprises a multi-part standard that harmonizes the International System of Quantities (ISQ) with the International System of Units (SI) across major scientific domains (mechanics, electromagnetism, acoustics, photometry, thermodynamics, etc.). Part 1 provides the cross-cutting general rules that all other parts adopt:

• Definitions and relationships among quantity, dimension, unit, and value.
• Coherence of equations: how quantities combine so that units never require hidden conversion factors.
• Symbols and typography for quantities (italic), units (upright/roman), constants, functions, and operators.
• Formatting rules for numbers, decimal markers, grouping of digits, multiplication/division signs, powers, and the use of parentheses.
• SI prefixes, their symbols, and constraints on their usage.
• Treatment of dimensionless quantities (including angles and counting quantities).
• Guidance on rounding, significant digits, and reporting conventions.

By unifying these basics, ISO 80000-1 makes it possible for a materials scientist, a metrologist, and a control engineer to read each other’s equations without ambiguity.

Historical Context and Position within Standards

The 80000 series superseded earlier ISO 31 standards to better align with modern SI practice and to reflect the consolidation of metrology concepts across disciplines. ISO 80000-1 sits at the hub of this series, ensuring that every domain-specific part (ISO 80000-2 mathematics, -3 space and time, -4 mechanics, -5 thermodynamics, -6 electromagnetism, and so on) adheres to common rules for symbols, units, and notation. It is also consistent with broader metrology resources (e.g., the international vocabulary of metrology and contemporary SI definitions based on fixed constants).

The 2019 SI revision—which fixed the numerical values of several defining constants—did not change Part 1’s fundamental guidance; rather, it reinforced the advantages of coherent SI usage and exact relationships that Part 1 already promotes.

Core Concepts Standardized in Part 1

Quantities, Units, and Values

A quantity is a property of a phenomenon, body, or substance that can be measured or calculated. Every quantity has a dimension (e.g., length L, mass M, time T) and a unit (e.g., metre, kilogram, second). A value of a quantity is expressed as the product of a numerical value and a unit:

Q = {Q}[Q]

Example: The length L of a rod may be written L = 1.234 m, where {L} = 1.234 and [L] = m.

Key implications:

• Numerical values depend on the unit; the quantity itself does not.
• Always carry the unit explicitly; avoid “naked numbers” for physical quantities.

Dimensions and Dimensional Equations

Each quantity has a dimension—a symbolic representation in terms of the base dimensions L, M, T, I, Θ, N, J (length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity). Part 1 uses dimensional equations to formalize physical relationships. For example, force has dimension M L T⁻², corresponding to the SI derived unit newton (N):

[N] = kg · m · s⁻²

Dimensional consistency is non-negotiable: both sides of any physical equation must have the same dimension.

Coherence and Derived Units

A set of units is coherent if equations between numerical values have the same form as equations between the corresponding quantities (i.e., no extraneous factors appear). The SI is built to be coherent: when you express all quantities in SI units, the algebra mirrors the physics. Examples:

• Work W = ∫ F · ds yields joule J = N · m.
• Pressure p = F/A yields pascal Pa = N · m⁻².
• Frequency in hertz is s⁻¹ coherently; no 2π appears unless you define angular frequency.

Special Names and Dimensionless Quantities

Some dimensionless derived quantities have special names for clarity and tradition, e.g.:

• radian (rad) for plane angle, steradian (sr) for solid angle.
• neper (Np) and decibel (dB) for logarithmic ratio quantities (accepted with SI).

Part 1 emphasizes that such units are dimensionless but not meaningless: retaining unit symbols like “rad” or “sr” is encouraged where it prevents ambiguity (e.g., distinguishing s⁻¹ from rad · s⁻¹).

Units Accepted for Use with the SI

Part 1 recognizes that certain non-SI units are accepted for use with the SI for practical reasons (e.g., minute, hour, degree of arc, litre, tonne). Their symbols and conversions are standardized, but the principle remains: use SI units wherever practical, and state conversions clearly when non-SI units are necessary.

Symbols, Typography, and Mathematical Conventions

Quantities vs Units

• Quantity symbols (e.g., m for mass, V for volume, T for temperature) are set in italic.
• Unit symbols (e.g., kg, m, s, K) are set in upright (roman) and are not pluralized. It is 5 kg, never “5 kgs”.

Spacing and the Number–Unit Relationship

• Use a space between a number and its unit: 25 °C, 1.00 m, 3.6 MJ. The degree sign and the unit “°C” count as a unit; write 20 °C (not 20° C or 20°C without a space).
• No space for plane and solid angles written with symbols degree (°), arcminute (′), arcsecond (″): 30°, 20′, 10″ (these are not SI units but accepted).

Multiplication, Division, and Powers

• For multiplication of units, use a half-high dot (·) or a space: N·m or N m. Avoid Nm (it can be mistaken for a new unit).
• For division, use a solidus (/) or negative exponents, but avoid multiple solidi in the same expression. Prefer kg·m·s⁻² or kg m/s², not kg/m/s².
• Group exponents clearly: m·s⁻² or m/s². Parentheses resolve ambiguity: J/(mol·K).

Decimal Marker and Grouping

• Either decimal point (.) or decimal comma (,) is permitted by international convention; use one consistently.
• Group digits in threes with a thin space, not commas: 12 345 678.9. Do not group the digits of four-digit numbers to the left of the decimal (both 1234 and 1 234 are acceptable; choose a consistent style).

SI Prefixes

• Prefixes represent integer powers of ten: k (10³), M (10⁶), m (10⁻³), µ (10⁻⁶), etc.
• Do not double prefix: write nm (10⁻⁹ m), not “mµm”.
• Do not attach prefixes to non-prefixed base names that already contain a prefix (e.g., kilogram is the unique base unit with a prefix in its name; you write mg as 10⁻³ g, not as “µkg”).
• Prefixes apply to unit symbols, not quantity symbols: 5 mm, not “5 m m”.

Functions, Constants, and Operators

• Mathematical constants e, π, operators sin, ln, exp, and function names are upright/roman.
• Use parentheses appropriately to avoid ambiguity: sin(ωt); for products and quotients combine with · and clear exponents.

Quantity Calculus and Dimensional Consistency

The Value Equation Q = {Q}[Q]

This identity separates physics from arithmetic. When you change units, {Q} changes accordingly; [Q] keeps track of the unit. Always transform units explicitly to keep equations and code unit-safe.

Dimensional Analysis

• Validates equations (both sides must share dimensions).
• Guides scaling and similarity (e.g., constructing Reynolds, Mach, and Nusselt numbers).
• Helps diagnose mistakes: if you add m to s, something is wrong.

Angles, Counts, and “Dimensionless” Quantities

• Angles (radian, steradian) are dimensionless but should retain their unit symbols in contexts where confusion could arise.
• Counts (e.g., “5 molecules”) are not “units”; use wording such as “count” or specify entities: N = 5 molecules. When normalized (per unit volume, per unit time), use the appropriate unit of denominator (m⁻³, s⁻¹).

Ratios, Percent, ppm

Percent, parts per million, etc., are ratios (pure numbers) that implicitly reference unity. Use with care, define the reference quantity if ambiguity is possible, and avoid chaining ratios that mix contexts.

Reporting Values: Rounding, Significant Digits, and Uncertainty

Rounding and Significant Digits

• Round only at the end of a calculation, not at intermediate steps.
• Match the significant digits of reported results to the measurement uncertainty: avoid spurious precision (“12.345000 mm” when the instrument resolves to 0.01 mm).
• When reporting a product or quotient of measured quantities, propagate uncertainties and round coherently.

Uncertainty Notation

While full uncertainty evaluation is covered elsewhere, Part 1 is compatible with standard metrology notation:

• Standard uncertainty u and expanded uncertainty U = k u with a coverage factor k.
• Concise parenthetic notation for last digits is acceptable where context is clear: 1.2345(12) m means 1.2345 ± 0.0012 m.

Always state the unit, the type of uncertainty, and the coverage factor or confidence level if it affects interpretation.

Practical Guidance and Common Pitfalls

Temperature and Temperature Difference

Absolute thermodynamic temperature uses kelvin (K): T = 293.15 K. Celsius temperature uses degree Celsius: t = 20 °C. A difference of 1 K equals a difference of 1 °C; write ΔT = 10 K or Δt = 10 °C consistently. When you need conversions on the fly, our Celsius to Kelvin converter keeps the relationship straight.

Frequency, Angular Frequency, and Cycles

Frequency f in hertz (Hz = s⁻¹). Angular frequency ω in radians per second (rad·s⁻¹). Do not conflate f and ω; the relationship is ω = 2πf.

Work, Energy, Power

Energy in joule (J = N·m), power in watt (W = J·s⁻¹). Avoid “kW/h” when you mean kilowatt-hour kW·h, a unit of energy, not of power rate of change. For more real-world practice, run a scenario through the horsepower to watts converter to translate legacy units into coherent SI terms.

Pressure and Stress

Pascal (Pa = N·m⁻²) is the SI unit. The bar (1 bar = 100 kPa) is accepted for use, but prefer Pa in equations. For sound and logarithmic levels, declare the reference explicitly (e.g., dB re 20 µPa).

Angles and Trigonometric Functions

Use radians in calculus for coherent derivatives and series without hidden factors (e.g., d(sin x)/dx = cos x only if x is in radians). Retain “rad” in units for clarity where mixing with s⁻¹ could confuse. When you want more context, our ISO 80000-3 space and time guide expands on how angular measures interact with spatial quantities.

Worked Examples (Do/Don’t)

Applications and Impact Across Disciplines

Research Communication and Peer Review

A manuscript that follows ISO 80000-1 instantly communicates professional quality: symbols are unambiguous, units are coherent, and numbers are formatted for international readability. Reviewers can check dimensional consistency at a glance; readers can reproduce results without guessing about notation.

Instrumentation, Control, and Data Systems

• Firmware and drivers should expose quantities with units and clearly documented prefixes.
• Data pipelines benefit from unit metadata and conversion layers that respect the value identity Q = {Q}[Q].
• Human–machine interfaces that follow Part 1 avoid misreads (e.g., “kW·h” vs “kW/h”) that can cause costly errors.

Standards, Regulation, and Compliance

In sectors from pharmaceuticals to aviation, specifications rely on clear units and traceable quantities. Adherence to ISO 80000-1 reduces compliance risk by eliminating notation ambiguities in design dossiers, test reports, and labels.

Education and Training

Part 1’s conventions form the grammar of measurement. Teaching them early prevents entrenched bad habits (pluralized unit symbols, mixed solidi, missing spaces) and equips students to read the literature fluently across fields. Link these lessons with topical explainers such as our unit of measurement primer to reinforce good habits.

Relationship to Other ISO 80000 Parts

• Part 2 (Mathematics) complements Part 1 by standardizing mathematical signs and symbols used in formulas that carry units.
• Parts 3–12 define domain-specific quantities and units (e.g., space and time, mechanics, thermodynamics, electromagnetism, acoustics, photometry, physical chemistry, atomic/nuclear physics, characteristic numbers, solid-state). Each inherits Part 1’s general rules.
• The series together ensures that a quantity defined in one part fits seamlessly into equations drawn from another, with symbols and units that obey the same formatting and coherence rules.

Dive into neighboring chapters like ISO 80000-4 mechanics or ISO 80000-7 light and radiation to see how the principles manifest in specific disciplines.

Implementation Checklist

Use the following as a quick audit against ISO 80000-1 principles:

• Quantity symbols italic; unit symbols upright, correctly cased, no plural.
• Space between number and unit; correct placement for °C.
• Coherent SI units in equations; special names (e.g., N, Pa, J) when helpful.
• Single solidus in unit expressions; otherwise use exponents or parentheses.
• Thin-space grouping for digits; consistent decimal marker.
• SI prefixes applied correctly; no double prefixes; mindful of “kg” special case.
• Angles and other dimensionless quantities labeled when needed (rad, sr).
• Uncertainty clearly stated; significant digits matched to uncertainty.
• Dimensional checks performed for every new or imported equation.

Conclusion

ISO 80000-1 is not merely a typographic nicety; it is the infrastructure of measurement literacy. By enforcing coherent SI, precise symbols, disciplined formatting, and dimensional rigor, Part 1 enables scientists, engineers, educators, and regulators to speak the same quantitative language. When you adopt its rules:

• Your equations become unit-safe and reproducible.
• Your documents are internationally readable and review-ready.
• Your software and data systems gain clarity and interoperability.

From drafting a calibration certificate to coding a simulation, the principles in ISO 80000-1 are the quiet, essential standards that make modern measurement culture possible.