Units of Information¶

Choose a logarithmic reference base and unit convention for information quantities so the same amount can be expressed coherently in bits or shannons, nats, hartleys, and their standardized multiples without confusing information with storage capacity.

Version: v3 · 2026-09-06 · History
Domain-specific #: 3036
Origin domain: information science
Subdomain: information theory
Aliases: Units of information and data, Information units

Core Idea¶

Units of information are reference units used to express logarithmic information quantities and closely related data quantities. If an information measure is written with logarithm base \(b\), changing the base rescales every value by a constant: \(\log_c x=\log_b x/\log_b c\). Shannon made the principle explicit: the choice of logarithmic base corresponds to the choice of a unit.^[1] Base two yields the bit or, under standards that distinguish the information-theoretic unit name, the shannon; base \(e\) yields the nat; and base ten yields the hartley.

For equiprobable alternatives, the Hartley quantity is \(I=\log_b N\); for an event of probability \(p\), self-information is \(-\log_b p\); and entropy is an expectation of that quantity. A unit therefore does not assert that every physical device stores or communicates exactly that much information. The same bit word is used in computing for a binary digit or storage unit, and a byte is standardized as eight bits in modern practice. Storage capacity counts available symbol positions, while Shannon information depends on probabilities and coding assumptions. One eight-bit byte can carry less than eight bits of entropy when its possible values are not equiprobable.

IEC 80000-13 standardizes names, symbols, definitions, conversions, and binary prefixes for information science and technology.^[2] Decimal prefixes such as kbit denote powers of ten, while binary prefixes such as Kibit denote powers of \(2^{10}\); NIST documents the distinction and the eight-bit byte convention.^[3] The abstraction is the coherent unit system—quantity, logarithmic base, conversion factor, prefix rule, and semantic qualifier—not a list of fashionable file-size labels.

Structural Signature¶

The information quantity. Entropy, self-information, capacity, data amount, or storage quantity is named before a unit is attached.
The logarithmic base. For logarithmic information measures, the base fixes the scale factor.
The unit name and symbol. Bit/shannon, nat, hartley, byte, and prefixed forms follow a declared convention.
The reference event count. One base-\(b\) unit corresponds to resolving among \(b\) equiprobable alternatives in the elementary case.
The conversion law. Values in different logarithmic units are related by constant base-change factors.
The quantity-kind qualifier. Information content, entropy, data storage, and transmission capacity are not silently treated as identical physical claims.
The prefix convention. Decimal SI prefixes and binary IEC prefixes are kept distinct.
The byte convention. A byte is eight bits under the modern standard convention, not an arbitrary historical machine word.
The uncertainty or distribution context. Information-theoretic values require the relevant probability model or coding ensemble.
The traceability record. A calculation names the standard edition, symbol, base, and conversions used.

What It Is Not¶

Not information itself. A unit is a scale convention used to express a quantity.
Not always a physical storage cell. A bit of entropy and a binary memory position answer different questions.
Not a guarantee that one byte has eight bits of entropy. Distribution and coding determine realized information.
Not permission to mix kB and KiB. Decimal and binary prefixes encode different multipliers.
Not dimensionless in the sense of meaningless. Logarithmic values have quantity-kind and unit conventions even when ratios underlie them.
Not one universally collision-free vocabulary. Standards and subfields sometimes distinguish 'bit' and 'shannon' differently, so the convention must be named.

Scope of Application¶

The unit family is literal in information theory, coding, data storage, communication engineering, and software measurement when quantitative claims must be compared or converted.

Source coding. Reporting entropy and average code length in base-two units.
Channel analysis. Expressing capacities per symbol or per second with a named logarithmic base.
Data storage. Distinguishing bit and byte capacities from statistical information content.
File and memory reporting. Applying decimal or binary prefixes without ambiguity.
Cryptography. Expressing key-space or uncertainty measures while preserving probability assumptions.
Software metrology. Relating data quantities to units and declared measurement procedures.

Clarity¶

Name the quantity first, then give its mathematical definition, logarithm base, unit name, symbol, and any rate denominator. State whether 'bit' means binary digit, storage capacity, or base-two information unit, and use 'shannon' when the governing standard or audience requires that distinction. Write conversions explicitly: one nat equals \(\log_2 e\) base-two units and one hartley equals \(\log_2 10\) base-two units. For bytes and prefixes, state that \(1\,\mathrm B=8\,\mathrm{bit}\), \(1\,\mathrm{kbit}=10^3\,\mathrm{bit}\), and \(1\,\mathrm{Kibit}=2^{10}\,\mathrm{bit}\). Do not infer entropy from storage size without a distribution or code model. Cite the standard edition when terminology has changed.

Manages Complexity¶

The unit system allows results derived with different logarithmic bases or engineering conventions to be compared by multiplication rather than re-derivation. It separates the invariant information relation from a conventional numeric scale: changing from bits to nats changes the number but not the underlying ordering or coding conclusion. Standard names and prefixes also prevent a binary-versus-decimal factor from growing into large capacity discrepancies. The compression is powerful because a symbol such as bit can carry several adjacent meanings. That is also its main risk. Treating a storage count as entropy ignores redundancy; treating a byte as a probability-dependent unit confuses architecture with information theory; and writing MB when MiB is intended creates a measurable mismatch. The plural title is coherent only as a conversion-governed family. It would collapse into a glossary if it merely enumerated labels without the common logarithmic scale, quantity distinctions, and standardization rules. A reference-grade use keeps those roles visible and records where vocabulary is conventional rather than mathematical.

Abstract Reasoning¶

Identify the information or data quantity being expressed.
Write the defining formula and its logarithm base, if applicable.
Select the unit name and symbol under a declared standard convention.
Convert between bases using the constant logarithm base-change factor.
Separate entropy or self-information from storage positions and encoded length.
Apply decimal or binary prefixes exactly as declared.
Attach probability, coding, time, bandwidth, or symbol denominators required by the quantity.
Check that every numerical comparison uses compatible quantity kinds and units.

Knowledge Transfer¶

The strict parent is Measurement: information units provide the reference scale and unit convention by which a defined target attribute is assigned a quantitative value. The domain-specific residual is the logarithmic-base family, its conversion law, and the information-versus-storage distinctions. Measurement also requires procedure and uncertainty when an empirical target is observed; a purely mathematical entropy calculation supplies its value from a stated distribution.

Examples¶

Canonical¶

A choice among eight equiprobable alternatives has information \(\log_2 8=3\) base-two units, \(\ln 8\approx2.079\) nats, or \(\log_{10}8\approx0.903\) hartleys. These are the same information amount on three scales because \(1\) nat is \(\log_2 e\approx1.443\) bits and \(1\) hartley is \(\log_2 10\approx3.322\) bits.^[1]

Mapped back: same eight-way distinction → selected logarithm base → different numerical unit value → exact base-change equivalence.

Applied / In Practice¶

A storage specification lists \(64\,\mathrm{GiB}\). Converting under IEC binary prefixes gives \(64\times2^{30}\) bytes and eight times as many storage bits. That capacity does not establish that a stored dataset contains the same number of Shannon-information bits; compression may reveal strong redundancy. A network figure in \(\mathrm{Gbit/s}\) uses a decimal prefix and a time denominator, so it is not compared to the storage number until units and quantities are aligned.^[3]

Mapped back: standardized prefix and byte rule → physical capacity count → explicit boundary from statistical entropy → compatible rate conversion.

Structural Tensions¶

Invariant amount vs. conventional scale. Changing log base changes the number but not the encoded information relation. Diagnostic: Is the base and unit stated?
Information vs. storage. Probability-weighted uncertainty and available binary positions can diverge sharply. Diagnostic: Is the claim about entropy, code length, or capacity?
Decimal vs. binary prefixes. Familiar labels invite factors of \(1000\) and \(1024\) to be mixed. Diagnostic: Are SI and IEC symbols used literally?
Stable mathematics vs. evolving terminology. Base-change laws persist while standards refine unit names. Diagnostic: Which edition and vocabulary convention govern the claim?
Autonomous unit family vs. generic Measurement. Measurement travels; logarithmic information bases and storage boundaries define this residual. Diagnostic: Does the account preserve quantity kind, base, conversion, and prefix roles?

Structural–Framed Character¶

Units of Information are mixed. Logarithm base-change and information formulas are structural; unit names, symbols, byte conventions, and prefix governance are standardized frames maintained by technical communities. The entry remains domain-specific because it concerns information and data quantities, not every measurement scale or logarithmic ratio.

Structural Core vs. Domain Accent¶

The skeleton is target quantity + scale + unit + procedure/conversion → interpretable value. The accent is entropy, self-information, base-two/e/ten logarithms, bits or shannons, nats, hartleys, bytes, and IEC prefixes. Removing those features yields Measurement or Scale.

Measurement is the strict parent because the unit family maps a defined information attribute onto a conventional quantitative scale that enables comparison and conversion. Its logarithmic basis and information/storage semantics are the autonomous residual.

The prospective workspace queue contains one strict upward edge to prime:measurement. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Units of Information Domain-specific

Parents (1) — more general patterns this builds on

Units of Information is a kind of Measurement Prime

Measurement is the strict parent because the unit family maps a defined information attribute onto a conventional quantitative scale that enables comparison and conversion.

Hierarchy path (1) — routes to 1 parentless root

Units of Information → Measurement

Neighborhood in Abstraction Space¶

Units of Information sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Logarithmic Information & Scale (9 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Bit as binary digit. A two-valued symbol position rather than necessarily one bit of Shannon information.
Byte. An eight-bit storage grouping under the modern convention.
Binary prefix. Kibi, mebi, and related multipliers applied to a unit, not independent information units.
Bit rate. A quantity per unit time, not an amount of information or storage by itself.
Entropy. A probability-derived quantity expressed in one of these units.
Channel capacity. A maximum reliable information rate under a channel model and constraints.

References¶

[1] Claude E. Shannon, ‘A Mathematical Theory of Communication,’ Bell System Technical Journal 27 (1948): 379–423 and 623–656, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x and https://doi.org/10.1002/j.1538-7305.1948.tb00917.x. registry ↩a ↩b

[2] IEC 80000-13:2025, Quantities and Units—Part 13: Information Science and Technology, 2^nd ed., https://www.iso.org/standard/87648.html. registry ↩

[3] National Institute of Standards and Technology, ‘Prefixes for Binary Multiples,’ https://physics.nist.gov/cuu/Units/binary.html. registry ↩a ↩b