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Prime (symbol)
Prime (symbol)
Prime (symbol)
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′
Prime
″ ‴ ⁗
Double prime Triple prime Quadruple prime

The prime symbol ′, double prime symbol ″, triple prime symbol ‴, and quadruple prime symbol ⁗ are used to designate units and for other purposes in mathematics, science, linguistics and music.

Although the characters differ little in appearance from those of the apostrophe and single and double quotation marks, the uses of the prime symbol are quite different.[1] While an apostrophe is now often used in place of the prime, and a double quote in place of the double prime (due to the lack of prime symbols on everyday writing keyboards), such substitutions are not considered appropriate in formal materials or in typesetting.

Designation of units

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See also: Positional notation § Sexagesimal system

The prime symbol ′ is commonly used to represent feet (ft), and the double prime ″ is used to represent inches (in).[2] The triple prime ‴, as used in watchmaking, represents a ligne (1⁄12 of a "French" inch, or pouce, about 2.26 millimetres or 0.089 inches).[3]

Primes are also used for angles. The prime symbol ′ is used for arcminutes (1⁄60 of a degree), and the double prime ″ for arcseconds (1⁄60 of an arcminute).[4] As an angular measurement, 3° 5′ 30″ means 3 degrees, 5 arcminutes and 30 arcseconds. In historical astronomical works, the triple prime was used to denote "thirds" (1⁄60 of an arcsecond)[5][6] and a quadruple prime ⁗ "fourths" (1⁄60 of a third of arc),[a] but modern usage has replaced this with decimal fractions of an arcsecond.

Primes are sometimes used to indicate minutes, and double primes to indicate seconds of time, as in the John Cage composition 4′33″ (spoken as "four thirty-three"), a composition that lasts exactly 4 minutes 33 seconds. This notation only applies to duration, and is seldom used for durations longer than 60 minutes.[8][better source needed]

Use in mathematics, statistics, and science

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In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with x′ generally meaning something related to (or derived from) x. For example, if a point is represented by the Cartesian coordinates (x, y), then that point rotated, translated or reflected might be represented as (x′, y′).

Usually, the meaning of x′ is defined when it is first used, but sometimes, its meaning is assumed to be understood:

  • A derivative or differentiated function: in Lagrange's notation, f′(x) and f″(x) are the first and second derivatives of the function f(x) with respect to x. The pattern may be continued, such as in f ‴(x) and f ⁗(x), with each additional prime denoting the next higher derivative. Similarly, if y = f(x), then y′ and y″ denote the first and second derivatives of y with respect to x. Other notations for derivatives also exist (see Notation for differentiation).
  • Set complement: A′ is the complement of the set A (other notations also exist).[9]
  • The negation of an event in probability theory: Pr(A′) = 1 − Pr(A) (other notations also exist).
  • The result of a transformation: Tx = x′
  • The transpose of a matrix (other notations also exist)
  • The dual of a vector space

The prime is said to "decorate" the letter to which it applies. The same convention is adopted in functional programming, particularly in Haskell.

In geometry, geography and astronomy, prime and double prime are used as abbreviations for minute and second of arc (and thus latitude, longitude, elevation and right ascension).

In physics, the prime is used to denote variables after an event. For example, vA′ may indicate the velocity of object A after an event. It is also commonly used in relativity: the event at (x, y, z, t) in frame S, has coordinates (x′, y′, z′, t′) in frame S′.

In chemistry, it is used to distinguish between different functional groups connected to an atom in a molecule, such as R and R′, representing different alkyl groups in an organic compound. The carbonyl carbon in proteins is denoted as C′, which distinguishes it from the other backbone carbon, the alpha carbon, which is denoted as Cα. In physical chemistry, it is used to distinguish between the lower state and the upper state of a quantum number during a transition. For example, J′ denotes the upper state of the quantum number J while J″ denotes the lower state of the quantum number J.[10]

In molecular biology, the prime is used to denote the positions of carbon on a ring of deoxyribose or ribose. The prime distinguishes places on these two chemicals, rather than places on other parts of DNA or RNA, like phosphate groups or nucleic acids. Thus, when indicating the direction of movement of an enzyme along a string of DNA, biologists will say that it moves from the 5′ end to the 3′ end, because these carbons are on the ends of the DNA molecule. The chemistry of this reaction demands that the 3′ OH be extended by DNA synthesis. Prime can also be used to indicate which position a molecule has attached to, such as 5′-monophosphate.

Use in linguistics

[edit]

The prime can be used in the transliteration of some languages, such as Slavic languages, to denote palatalization. Prime and double prime are used to transliterate Cyrillic yeri (the soft sign, ь) and yer (the hard sign, ъ).[11] However, in ISO 9, the corresponding modifier letters are used instead.

Originally, X-bar theory used a bar over syntactic units to indicate bar-levels in syntactic structure, generally rendered as an overbar. While easy to write, the bar notation proved difficult to typeset, leading to the adoption of the prime symbol to indicate a bar. (Despite the lack of bar, the unit would still be read as "X bar", as opposed to "X prime".) With contemporary development of typesetting software such as LaTeX, typesetting bars is considerably simpler; nevertheless, both prime and bar markups are accepted usages.

Some X-bar notations use a double prime (standing in for a double-bar) to indicate a phrasal level, indicated in most notations by "XP".

Use in music

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You may need rendering support to display the uncommon Unicode characters in this section correctly.
Prime, double prime and triple prime

The prime symbol is used in combination with lower case letters in the Helmholtz pitch notation system to distinguish notes in different octaves from middle C upwards. Thus c represents the ⟨C⟩ below middle C, c′ represents middle C, c″ represents the ⟨C⟩ in the octave above middle C, and c‴ the ⟨C⟩ in the octave two octaves above middle C. A combination of upper case letters and sub-prime symbols is used to represent notes in lower octaves. Thus C represents the ⟨C⟩ below the bass stave, while C ͵ represents the ⟨C⟩ in the octave below that.

In some musical scores, the double prime ″ is used to indicate a length of time in seconds. It is used over a fermata 𝄐 denoting a long note or rest.[b]

Computer encodings

[edit]

Unicode and HTML representations of the prime and related symbols are as follows.

  • U+2032 ′ PRIME (&prime;) (lower case p)
  • U+2033 ″ DOUBLE PRIME (&Prime;) (upper case P)
  • U+2034 ‴ TRIPLE PRIME (&tprime;)
  • U+2035 ‵ REVERSED PRIME (&backprime;, &bprime;)
  • U+2036 ‶ REVERSED DOUBLE PRIME
  • U+2037 ‷ REVERSED TRIPLE PRIME
  • U+2057 ⁗ QUADRUPLE PRIME (&qprime;)
  • U+02B9 ʹ MODIFIER LETTER PRIME
  • U+02BA ʺ MODIFIER LETTER DOUBLE PRIME

The "modifier letter prime" and "modifier letter double prime" characters are intended for linguistic purposes, such as the indication of stress or the transliteration of certain Cyrillic characters.[citation needed]

In a context when the character set used does not include the prime or double prime character (e.g., in an online discussion context where only ASCII or ISO 8859-1 [ISO Latin 1] is expected), they are often respectively approximated by ASCII apostrophe (U+0027) or quotation mark (U+0022).

LaTeX provides an oversized prime symbol, \prime (), which, when used in super- or sub-scripts, renders appropriately; e.g., f_\prime^\prime appears as . When in math mode, an apostrophe, ', is a shortcut for a superscript prime; e.g., f' appears as .

See also

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Notes

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References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Prime (symbol)

View on Grokipedia
from Grokipedia
The prime symbol (′), also known as the single prime or minute mark (Unicode U+2032), is a diacritical mark used across , sciences, , and to denote distinctions between related quantities, such as of functions, transformed coordinates, or complements of sets. In , it commonly appears as a superscript following a variable, like f′(x) to indicate the first of function f with respect to x. Beyond , the symbol represents units of and : a single prime denotes feet in imperial measurements (e.g., 6′ for six feet) or minutes of arc in angular notation (e.g., 30′ for thirty minutes). It is also used in to denote minutes and seconds of latitude and longitude. The double prime (″, Unicode U+2033) extends this for inches or seconds (e.g., 6′2″ for six feet two inches), while triple (‴, U+2034) and quadruple (⁗, U+2057) primes denote further subdivisions or higher-order distinctions in specialized contexts. Distinct from the apostrophe (', Unicode U+0027), which curves and serves grammatical roles like contractions or possessives in writing, the prime is typically straight or slightly slanted and positioned as a true superscript in professional typography to avoid confusion in technical fields. This typographical precision ensures clarity, as substitutions with apostrophes or straight quotes often occur in informal digital text due to keyboard limitations but are discouraged in formal publishing. In , the prime can mark glottal stops, ejective , or tone variations (e.g., in phonetic transcriptions), while in music it indicates octaves above a note or prime form in . Historically rooted in astronomical and navigational notations for arcminutes, the symbol's versatility has made it indispensable in modern STEM disciplines, though official standards like those from NIST prefer abbreviations such as "ft" and "in" over primes for measurements to promote metric consistency.

History and Origin

Etymology

The prime symbol (′) derives its name from the Latin primus, meaning "first" or "foremost," which evolved through prime (first in rank or time) and prime, ultimately tracing to Proto-Italic *prī́semos, linked to concepts of primacy and initial position in sequences or divisions. This linguistic root underscores the symbol's function as the marker for the primary or first subdivision, particularly in angular measurements where it denotes a minute of arc—the first 1/60th part of a degree, originating from the Medieval Latin phrase pars minuta prima ("first small part"). Historically, the term "prime" became a for the symbol itself due to its longstanding association with these "prime minutes" in astronomical notation, a usage documented as early as the late . By the early , this shorthand extended broadly, with notations like x′ read aloud as "x prime" to indicate the first in a series of related quantities.

Development in Astronomy

The prime symbol (′) was introduced in the by astronomers such as to denote minutes of arc, the first sexagesimal subdivision of a degree into 60 equal parts. Brahe adopted this notation in his 1573 work De nova stella, using a single accent to mark minutes as the initial division, building on earlier Greek and Byzantine conventions where accents indicated fractional parts. This system expanded to include the double prime (″) for seconds of arc (1/60 of a minute) and the triple prime (‴) for thirds of arc (1/60 of a second), as documented in astronomical texts from the through the . These multiple primes allowed precise expression of small angular measurements in observations, reflecting the need for finer in positional astronomy. The framework, including these notations, traces its conceptual roots to Ptolemy's (2nd century AD), which employed similar divisions for angular calculations, though Ptolemy himself used abbreviated letters rather than accents. In the 20th century, the formalized the use of the prime (′) for arcminutes and double prime (″) for arcseconds in standard astronomical nomenclature. A key application persists in celestial coordinates, where the symbols denote subdivisions in ; for instance, the position of Sirius is expressed as α = 06h 45m 08.9s, δ = −16° 42′ 58″, with the primes indicating arcminutes and arcseconds.

Notation and Typography

Symbol Variants and Multiples

The single prime symbol (′) is typographically rendered as a straight vertical mark, often with a slight rightward slant for visual distinction from curved apostrophes, ensuring clarity in both mathematical and textual contexts. This form contrasts with more curved variants that may appear in non-standard fonts, emphasizing its role as a precise rather than . Multiples of the prime extend this notation: the double prime (″) combines two primes for seconds or inches, the triple prime (‴) and quadruple prime (⁗), used in some historical contexts for finer subdivisions in angular measurements, such as thirds and fourths of an arcsecond in sexagesimal systems. These multiples maintain proportional spacing and alignment, with each additional prime stacked vertically to avoid overcrowding. In mathematical typesetting, typographic guidelines recommend an upright orientation for the prime symbol to differentiate it from italicized variables, achievable in LaTeX via the \prime command, which renders it consistently across fonts like STIX Two Math or Libertinus Math. Font variations can affect positioning—such as slight shifts in baseline alignment—but upright primes ensure legibility in expressions, preventing confusion with sloped accents. Rare variants include the reversed prime (‵), a typographic symbol resembling a raised grave accent with limited use. Such variants highlight the prime's adaptability but underscore the preference for conventional upright symbols in modern typography. Note that the prime's straight form can occasionally lead to mix-ups with apostrophes in casual typesetting.

Distinctions from Apostrophe and Quotes

The prime symbol (′) differs from the apostrophe (' or ’) in both form and function. The prime is a straight, tapered mark with a slight rightward slant, designed for technical notations such as denoting feet or minutes of arc, whereas the apostrophe is typically curved, resembling a right-facing comma, and used for contractions (e.g., don't) or possessives (e.g., child's). This distinction ensures clarity in typography, as the apostrophe's curve integrates better with surrounding text in prose. Similarly, the prime contrasts with the single ('), which is directional and often paired with a mirrored opening mark for or emphasis. Primes remain neutral and non-directional, avoiding the curvature of that hooks toward enclosed text, thereby preventing confusion in mathematical or scientific contexts where quotes might imply textual . Historically, limitations of typewriters, which lacked dedicated prime glyphs, led to frequent substitutions with apostrophes or straight quotes, a practice that persisted into early digital typography until the widespread adoption of in the enabled proper rendering. This substitution has contributed to ongoing errors in modern digital text, where ambiguous straight marks are misinterpreted. Style manuals like recommend reserving primes exclusively for technical uses, such as angular measurements or derivatives, while directing apostrophes and quotes to roles to maintain typographic precision.

Uses in Measurement

Angular Units

The prime symbol (′) is employed to denote the arcminute, a unit of angular measurement equivalent to 1/60 of a degree, facilitating precise expression of angles in fields such as , , and astronomy. For instance, 30′ represents half a degree, illustrating its utility in subdividing coarser degree measurements. The double prime (″) signifies the arcsecond, defined as 1/60 of an arcminute or approximately 4.85 × 10^{-6} radians, essential for detailing small angular scales in astronomical observations. A representative application appears in stellar positioning, where the angular separation between Alpha Centauri A and B varies from 2″ to 22″ across their 79.9-year . This notation enables accurate cataloging of celestial coordinates and relative positions. Historically, the triple prime (‴) has denoted a third of an arcsecond (1/60 of an arcsecond) in precision astronomy, though its use has become rare in contemporary contexts favoring decimal or alternative subdivisions. Standard conventions for these symbols, as established in ISO 80000-2 since , require placement immediately after the numerical value without intervening space, such as in 45° 30′ 15″ for a compound angle. This practice ensures clarity and consistency across international scientific communication, with exceptions limited to specific disciplinary traditions like astronomy.

Linear Units

In the imperial system of measurement, the single prime symbol (′) denotes feet, as exemplified by 6′ representing six feet. This convention stems from customary units where the foot serves as a fundamental linear measure. The double prime symbol (″) indicates inches, equivalent to one-twelfth of a foot, commonly written as 6′ 2″ to express a of six feet two inches. In and applications, these symbols follow the numerical value without spaces, such as 10′ 6″ for ten feet six inches, adhering to practices that prohibit substituting apostrophes to maintain clarity and precision. Standards from institutions like emphasize inserting the exact prime symbols via specialized tools rather than typographic approximations. This notation predominates in the United States and within imperial frameworks, while metric-dominant systems internationally rely on centimeters and meters, eschewing primes entirely.

Uses in Mathematics and Science

Derivatives and Calculus

In , the prime symbol (′) serves as a key notation for denoting derivatives, particularly in Lagrange's notation, which was introduced by in his 1797 work Théorie des fonctions analytiques as an alternative to Gottfried Wilhelm Leibniz's differential notation dydx\frac{dy}{dx}. This approach treats the derivative as a functional operation, where if ff is a function, its first derivative is written as f′(x)f'(x), equivalent to dfdx\frac{df}{dx}. The notation emphasizes the derivative as a new function derived from the original, facilitating clarity in expressions involving rates of change. For higher-order derivatives, successive primes indicate repeated differentiation: the second derivative is f′′(x)f''(x), the third is f′′′(x)f'''(x), and so on, with the nn-th derivative often denoted as f(n)(x)f^{(n)}(x) for readability when nn is large. These are read aloud as "f double prime of x" for the second derivative and "f triple prime of x" for the third, reflecting the accumulation of the prime symbols. This convention allows compact representation of concepts like concavity in second derivatives or jerk in third derivatives of position functions. A representative example illustrates the application: consider f(x)=x2f(x) = x^2. To find f′(x)f'(x), apply the power rule for differentiation, which states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = n x^{n-1}. Here, n=2n = 2, so f′(x)=2x2−1=2xf'(x) = 2x^{2-1} = 2x. The second follows similarly: differentiate 2x2x using the power rule with n=1n = 1, yielding f′′(x)=2f''(x) = 2, a constant indicating linear change in the first derivative. In physics, the prime notation is employed to denote rates of change, such as as the of , written as a=v′(t)a = v'(t) or equivalently dvdt\frac{dv}{dt}. The prime notation is used for differentiation with respect to variables other than time, while Newton's dot notation is typically reserved for time derivatives.

Distinct Elements and Ordering

In and , the prime symbol serves to distinguish variables that are related to but distinct from their unprimed counterparts, often indicating a transformation, adjustment, or sequential ordering without implying a operation. This notation facilitates clear labeling in contexts where multiple similar quantities appear, such as in coordinate systems or iterative approximations. In linear algebra and physics, primed notation commonly denotes coordinates or vectors in a transformed reference frame. For instance, x′x' represents the position coordinate after a linear transformation, such as a or boost, where the primed system is expressed relative to the original unprimed basis using direction cosines or transformation matrices. This convention is particularly prevalent in discussions of vector spaces and , ensuring that components in the new frame are systematically offset from the original. The prime symbol also appears in sequencing and estimation contexts to indicate successive or modified elements. In , the notation can distinguish related parameters, such as moments of distributions. Scientific applications extend this distinction to specialized potentials and functions. In chemistry, E′E' signifies the formal , which accounts for non-ideal solution conditions like activity coefficients, differing from the standard potential E∘E^\circ by environmental factors such as ion strength. In physics, particularly , the prime notation is used to denote approximate or perturbed states, such as in where corrections to the wave function are indicated. Conventions for multiple distinctions employ successive primes, such as double or triple primes, to label further iterations. In , for multi-lens systems, L′′L'' represents the final image distance after two transformations, building on the intermediate L′L' to track through sequential elements. This multi-prime approach maintains clarity in chained calculations, with each additional prime indicating a subsequent stage in the process.

Uses in Linguistics and Music

Phonetic and Linguistic Notation

The modifier letter prime (ʹ, U+02B9) is a spacing modifier in phonetic notations, primarily associated with stress or emphasis, and is canonically equivalent to the Greek numeral sign in some contexts. It is important to distinguish this from the modifier letter apostrophe (ʼ, U+02BC), which is the standard IPA diacritic for ejective consonants—produced with glottalic egression—such as the bilabial ejective [pʼ]. While the prime may appear in non-IPA or orthographic representations for similar glottal features, the apostrophe is reserved for ejectives in formal IPA usage, and a separate glottal stop symbol (ʔ, U+0294) or apostrophe variant denotes glottal stops to avoid confusion.

Pitch and Octave Designation

In the system, the prime symbol (′) is affixed to lowercase letters to denote pitches in successively higher s, providing a compact method for specifying musical registers. For instance, the note middle C, which serves as a central reference point, is represented as c′, while the C one above is c″ (using the double prime ″), and the next higher is c′″ (triple prime). This ascending convention allows for clear differentiation of pitch heights without reliance on numerical or staff-based visuals. Conversely, lower s employ commas placed below the baseline, such as c, for the below middle C and c,, for the one below that, ensuring a balanced symbolic framework for the full range of the . The system was formalized within 19th-century German to standardize pitch description across theoretical and scientific discussions. This notation emerged from Hermann von Helmholtz's foundational treatise Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (1863), in which he developed the primes and commas to precisely articulate tone sensations and their physiological underpinnings in acoustics and music theory. Helmholtz's approach emphasized the octave as the fundamental unit of pitch organization, with each cycle beginning on C, reflecting the perceptual hierarchy of musical tones. An English translation by Alexander J. Ellis, On the Sensations of Tone as a Physiological Basis for the Theory of Music (1885), further disseminated the system internationally, adapting it for broader scholarly use. In practical musical contexts, such as orchestral scores, the prime notation specifies exact registers to avoid in parts that span wide ranges, particularly where ledger lines might clutter the staff; for example, a double prime ″ elevates a bass line by two s in analytical annotations or transposed excerpts. This utility extends to theoretical analyses of works, where precise octave designation aids in studying intervallic structures and timbral balances without visual staff representation. The Helmholtz system has profoundly influenced subsequent conventions, including , which adapts its octave-centric logic for numerical labeling while retaining conceptual parallels to prime-based elevation.

Encoding and Representation

Unicode Standards

The prime symbol and its variants are formally encoded in the Standard within the General Punctuation block (U+2000–U+206F). The single prime is assigned to U+2032 ′ PRIME, categorized as Po (Other Punctuation), with bidirectional class ET (European Number Terminator), indicating it terminates numeric sequences in , and mirroring property N (not mirrored). This character was introduced in Unicode 1.1 in June 1993 and is represented in as ′. According to line breaking rules in Unicode Standard Annex #14, U+2032 prohibits a line break immediately before it when following a number, ensuring it remains attached in measurements like 5′ (five feet). Variants extend this notation for multiple instances: U+2033 ″ DOUBLE PRIME (for seconds or inches, also Po and ET, introduced in Unicode 1.1), U+2034 ‴ TRIPLE PRIME (for lines, equivalent to three single primes, Po and ET, Unicode 1.1), and U+2057 ⁗ QUADRUPLE PRIME (Po and ET, introduced in Unicode 3.2 in March 2002). For phonetic and linguistic applications, the modifier variant U+02B9 ʹ MODIFIER LETTER PRIME is used, classified as Lm (Letter, Modifier) with bidirectional class ON (Other Neutrals) and mirroring N, also from Unicode 1.1; it denotes palatalization or stress in transliterations. These primes must be distinguished from similar characters to avoid substitution errors: U+0027 ' APOSTROPHE (straight vertical punctuation, Ps category) and U+2019 ’ RIGHT SINGLE QUOTATION MARK (curved typographic quote, Ps category), which differ in glyph shape, bidirectional behavior, and intended use. The Unicode Consortium recommends using the dedicated prime codes for mathematical, measurement, and scientific contexts to preserve semantic accuracy.

Input and Display Methods

The prime symbol (′), officially named PRIME in Unicode (U+2032), is a distinct character from the apostrophe (') or acute accent (`), and its input and display require specific methods to ensure accurate rendering across systems. On Windows systems, the prime symbol can be inputted using Alt codes with the numeric keypad enabled: hold Alt and type 8242 on the numpad for the single prime (′), or 8243 for the double prime (″, U+2033). Alternatively, in applications supporting Unicode hex input, such as , users can type 2032 followed by Alt+X to insert the prime. The Character Map utility (charmap.exe) allows visual selection and copying of the symbol from the General Punctuation block. For macOS, switch the keyboard input source to "Unicode Hex Input" via System Preferences, then hold Option and type 2032 for the single prime or 2033 for the double prime. In text editors like , the Emoji & Symbols viewer (accessed via Control+Command+Space) provides a searchable interface to insert the symbol by name or category. Linux distributions vary, but common methods include using the (if configured) with sequences like Compose + ' + ' for the prime, or the character map tools in (gucharmap) or (kcharselect) for selection. via Ctrl+Shift+U followed by 2032 also works in many environments like . In and , the prime is inserted using entities such as ′ or ′ () or ′ (), ensuring compatibility across browsers without relying on font-specific rendering. For CSS, the symbol can be referenced via content: "\2032". In programming languages like Python or , it is directly embeddable as a Unicode literal, e.g., '\u2032'. Mathematical typesetting systems like handle the prime differently for precision. In standard LaTeX with unicode-math package (for XeLaTeX or LuaLaTeX), the command \prime produces the symbol, or the raw Unicode character U+2032 can be used directly in math mode, e.g., f′(x)f'(x). The package maps input to appropriate font glyphs, supporting ranges from single to quadruple primes. Display positioning in math contexts, such as superscripts for derivatives, depends on the math font (e.g., STIX Two Math or Latin Modern Math), where primes are often raised and spaced to avoid confusion with apostrophes. Display of the prime symbol relies on font support for U+2032 in the General Punctuation block, with most modern fonts (e.g., , Noto Sans, or DejaVu Serif) rendering it as a raised, slanted mark distinct from straight quotes. In mathematical rendering engines like MathJax or , it inherits font metrics for proper superscript alignment, but legacy fonts may fallback to apostrophes, causing visual inconsistencies. For units like minutes (′) or feet, typographic best practices recommend avoiding curly smart quotes from word processors to maintain straight, upright rendering.

References

  1. https://mathworld.wolfram.com/Prime.html
  2. https://carsonparkdesign.com/quotation-marks-apostrophes-versus-primes/
  3. https://practicaltypography.com/foot-and-inch-marks.html
  4. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors
  5. https://www.etymonline.com/word/prime
  6. https://www.etymonline.com/word/minute
  7. https://languagehat.com/prime/
  8. https://hsm.stackexchange.com/questions/8104/origin-of-arcminutes-arcseconds-arcthirds-arcfourths-etc
  9. https://astro4edu.org/resources/glossary/term/79/
  10. https://www.unicode.org/charts/PDF/U2000.pdf
  11. https://polytype.dev/prime-symbol/
  12. https://www.math.toronto.edu/mathit/symbols-letter.pdf
  13. https://creativepro.com/the-definitive-guide-to-quotes-apostrophes-and-primes/
  14. https://www.typewolf.com/cheatsheet
  15. https://www.chicagomanualofstyle.org/qanda/data/faq/topics/Punctuation/faq0098.html
  16. https://www.bipm.org/documents/20126/2071204/SI_Brochure_9_EN.pdf
  17. https://www2.phys.canterbury.ac.nz/~physacp/alphacen.html
  18. https://www.iso.org/standard/31887.html
  19. https://www.inchcalculator.com/inch-background-and-definition/
  20. https://chec.engineering.cornell.edu/units-of-measurement/
  21. https://www.math.hawaii.edu/~tom/history/calculus.html
  22. https://web.ma.utexas.edu/users/m408n/CurrentWeb/LM2-8-8.php
  23. https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/a/derivative-notation-review
  24. https://www.cliffsnotes.com/study-guides/calculus/calculus/applications-of-the-derivative/distance-velocity-and-acceleration
  25. https://www.sciencedirect.com/topics/engineering/primed-system
  26. https://openacttexts.github.io/Loss-Data-Analytics/ChapModelSelection.html
  27. https://www.palmsens.com/knowledgebase-article/origins-of-electrochemical-potentials/
  28. https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture31.pdf
  29. https://www.unicode.org/charts/PDF/U02B0.pdf
  30. https://teaching.ncl.ac.uk/ipa/consonants-non-pulmonic.html
  31. https://www.dolmetsch.com/musictheory1.htm
  32. https://archive.org/details/b21717114
  33. https://archive.org/details/onsensationsofto00helmrich
  34. https://www.fileformat.info/info/unicode/char/2032/index.htm
  35. https://www.unicode.org/reports/tr14/tr14-16.html
  36. https://www.fileformat.info/info/unicode/char/2033/index.htm
  37. https://www.fileformat.info/info/unicode/char/2034/index.htm
  38. https://www.fileformat.info/info/unicode/char/2057/index.htm
  39. https://www.fileformat.info/info/unicode/char/02b9/index.htm
  40. https://www.unicode.org/charts/nameslist/n_2000.html
  41. https://www.webnots.com/how-to-type-prime-symbols-with-keyboard/
  42. https://graphemica.com/%25E2%2580%25B2
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