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Metrology

Understanding Safety Levels in Physical Units Libraries

Physical quantities and units libraries exist primarily to prevent errors at compile time. However, not all libraries provide the same level of safety. Some focus only on dimensional analysis and unit conversions, while others go further to prevent representation errors, semantic misuse of same-dimension quantities, and even errors in the mathematical structure of equations.

This article explores six distinct safety levels that a comprehensive quantities and units library can provide. We'll examine each level in detail with practical examples, then compare how leading C++ libraries and units libraries from other languages perform across these safety dimensions. Finally, we'll analyze the performance and memory costs associated with different approaches, helping you understand the trade-offs between safety guarantees and runtime efficiency.

We'll pay particular attention to the upper safety levels—especially quantity kind safety (distinguishing dimensionally equivalent concepts such as work vs. torque, or Hz vs. Bq) and quantity safety (enforcing correct quantity hierarchies and scalar/vector/tensor mathematical rules)—which are well-established concepts in metrology and physics, yet remain widely overlooked in the C++ ecosystem. Most units library authors and users simply do not realize these guarantees are achievable, or how much they matter in practice. These levels go well beyond dimensional analysis, preventing subtle semantic errors that unit conversions alone cannot catch, and are essential for realizing truly strongly-typed numerics in C++.

Introducing Absolute Quantities

Until now, mp-units forced users to choose between points (no arithmetic) and deltas (no physical semantics) — missing the most common case: a non-negative absolute amount.

An absolute quantity represents an absolute amount of a physical property — measured from a true, physically meaningful zero. Examples include mass in kilograms, temperature in Kelvin, or length in meters (as a size, not a position). Such quantities live on a ratio scale and are anchored at a physically meaningful zero; negative values are typically meaningless.

Absolute quantities stand in contrast to:

  • Affine points (e.g., \(20\ \mathrm{°C}\), \(100\ \mathrm{m}\ \mathrm{AMSL}\)) — values measured relative to an arbitrary or conventional origin.
  • Deltas (e.g., \(10\ \mathrm{K}\), \(–5\ \mathrm{kg}\)) — differences between two values.

Arithmetic on absolute quantities behaves like ordinary algebra: addition, subtraction, and scaling are well-defined and map naturally to physical reasoning. This article proposes making absolute quantities the default abstraction in mp-units V3, reflecting how scientists express equations in practice.

Note: Revised on March 23, 2026 for clarity, accuracy, and completeness.

Bringing Quantity-Safety To The Next Level

All quantities and units libraries need to be unit-safe. Most of the libraries on the market do this correctly. Some of them are also dimension-safe, which adds another level of protection for their users.

mp-units is probably the only library on the market that additionally is quantity-safe. This gives a new quality and possibilities. I've described the major idea behind it, implementation details, and benefits to the users in the series of posts about the International System of Quantities.

However, this is only the beginning. We've always planned more and worked on the extensions in our free time. In this post, I will describe:

  • What a quantity character is?
  • The importance of using proper representation types for the quantities.
  • The power of providing character-specific operations for the quantities.
  • Discuss implementation challenges and possible solutions.

International System of Quantities (ISQ): Part 5 - Benefits

In the previous articles, we introduced the International System of Quantities, described how we can model and implement it in a programming language, and presented the issues of software that does not use such abstraction to implement a units library.

Some of the issues raised in Part 2 of our series were addressed in Part 3 already. This article will present how our ISQ model elegantly addresses the remaining problems.

International System of Quantities (ISQ): Part 3 - Modeling ISQ

The physical units libraries on the market typically only focus on modeling one or more systems of units. However, as we have learned, this is not the only system kind to model. Another, and maybe even more important, is a system of quantities. The most important example here is the International System of Quantities (ISQ) defined by ISO/IEC 80000.

This article continues our series about the International System of Quantities. This time, we will learn about the main ideas behind the ISQ and describe how it can be modelled in a programming language.