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One half

One half, denoted as 1/2, is a fundamental rational number in mathematics that represents a single equal part when a whole is divided into two parts.^[1] It serves as the multiplicative inverse of the integer 2, meaning that $1/2 \times 2 = 1. This fraction is irreducible, with a numerator of 1 and a denominator of 2, and it is commonly used to express halves in contexts such as sharing, measurements, and proportions. In decimal notation, one half is exactly equivalent to 0.5, obtained by dividing 1 by 2.^[2]^[3] As a percentage, it corresponds to 50%, reflecting half of 100 parts.^[4] In binary representation, which is the base-2 numeral system used in computing, one half is precisely 0.1₂, making it one of the few fractions with an exact finite binary expansion without rounding errors.^[5] One half plays a central role in arithmetic operations, geometry, and probability; for instance, it denotes the area of a semicircle relative to a full circle or the likelihood of an even-odds event.^[1] Its equivalents facilitate conversions across numerical systems, underscoring its versatility in education, engineering, and everyday applications like dividing resources equally.^[6]

Linguistic Aspects

Etymology and Definitions

The term "one half" in English derives from the combination of the numeral "one" and "half," with the latter tracing its roots to Old English healf, meaning "side" or "part," often implying a division without initial emphasis on equality. This Old English form evolved from Proto-Germanic *halba-, denoting "something divided," and ultimately connects to the Proto-Indo-European root *(s)kel- or *kel-, associated with cutting or splitting. Over time, the meaning shifted to specifically signify one of two equal parts, reflecting linguistic adaptations in early Germanic languages where spatial or partial concepts of division predominated.^[7]^[8] Major dictionaries define "half" consistently with this evolved sense of equality in partition. The Oxford English Dictionary describes it as "either of two equal parts into which something is or can be divided," underscoring the balanced division central to the term. Similarly, Merriam-Webster defines "half" as "either of two equal parts that compose something," with additional allowance for approximations, emphasizing the precise halving in quantitative contexts. These definitions highlight the term's transition from a broader notion of partiality to a standardized measure of equivalence.^[9] In other languages, equivalents of "one half" reveal shared Indo-European heritage while showing regional variations. In French, it is "un demi," from Latin dimidius (divided in half), a term influenced by Roman numeral systems and later Old French adaptations. German uses "eine Hälfte," directly from the same Proto-Germanic *halba- as English, preserving the original sense of a divided part. Comparative linguistics traces these to common Indo-European roots like *sem- (one) in forms such as Latin sēmi- (half), illustrating how duality concepts spread across branches of the language family.^[10] The linguistic evolution of "one half" parallels ancient concepts of duality in Mesopotamian and Egyptian numerals, where division into two parts emerged in practical computations. In ancient Egypt around 1650 BCE, the Rhind Papyrus demonstrates halving through unit fractions, with 1/2 represented symbolically as part of the Eye of Horus system for equitable shares in problems like dividing loaves among groups. Mesopotamian Babylonians, using a sexagesimal system by the Old Babylonian period (c. 1800–1600 BCE), expressed half as 0;30 (30/60), integrating it into reciprocal tables and geometric calculations for balanced divisions. These early numeral practices laid foundational ideas of duality that influenced later linguistic terms for "one half."^[11]^[12]

Idiomatic and Cultural Usage

In English-language idioms, "better half" commonly refers to one's spouse, often used affectionately to denote a partner as the more valuable or complementary portion of a marital union. This expression emerged in the late 16th century, initially applied to close friends before shifting to romantic partners, as seen in Sir Philip Sidney's Arcadia (1580), where it implies a superior or cherished counterpart.^[13] Similarly, "half a mind" conveys hesitation or a tentative inclination toward an action, typically arising from annoyance or indecision, as in the phrase "I have half a mind to leave," indicating partial resolve without full commitment. Another informal idiom, "go halfsies," describes sharing something equally between two people, such as costs or items, emphasizing equitable division in casual agreements. Proverbs involving "one half" often highlight themes of compromise and sufficiency. The saying "Half a loaf is better than no bread" advises accepting partial success or provision over none at all, originating in John Heywood's 1546 collection A Dialogue conteinyng the nomber in effect of all the Prouerbes in the Englishe tongue.^[14] In literature, "one half" appears in cultural references symbolizing incompleteness or duality. William Shakespeare's King John (c. 1596) employs "half part" to describe fragmented perfection, as in the line "He is the half part of a blessed man, / Left to be finished by such as she," portraying lovers as complementary halves of an ideal whole.^[15] Folklore similarly uses halves to depict divided realms or inheritances, as in the Brothers Grimm's "The Two Brothers" (1812), where a kingdom is split between siblings, underscoring themes of rivalry and reconciliation in traditional European tales. Icelandic folklore echoes this in "Half a Kingdom," a tale where a prince's reward involves claiming a divided realm after heroic deeds.^[16] Societally, phrases like "weaker half" have historically reinforced gendered roles, portraying women as the subordinate portion of humanity in 19th-century discourse. This usage, critiqued in early feminist writings such as Matilda Joslyn Gage's "Voluntary Motherhood" (1870), which laments woman as "the gentler and weaker half" burdened by unwilling reproduction, highlights patriarchal implications now challenged in modern feminism for perpetuating inequality.^[17]

Mathematical and Scientific Contexts

Representation as a Fraction

One half, denoted as \frac{1}{2}, is a rational number expressed as the ratio of the integers 1 and 2, where the denominator is nonzero.^[18] It serves as the multiplicative inverse, or reciprocal, of the integer 2, meaning $2 \times \frac{1}{2} = 1. In decimal notation, \frac{1}{2} is equivalent to 0.5, a terminating decimal that arises from dividing 1 by 2.^[2] As the simplest unit fraction—a fraction with numerator 1—\frac{1}{2} holds a foundational position in the theory of rational numbers.^[19] Its binary representation in base-2 is exactly 0.1_2, which equals [1](/page/1) \times 2^{-[1](/page/1)}, allowing precise computation without approximation in binary systems due to the denominator being a power of 2.^[5] The use of \frac{1}{2} dates back to ancient Egyptian mathematics, where it appeared as a core unit fraction in the Rhind Mathematical Papyrus, composed around 1650 BCE, to solve practical problems of division and portioning.^[20] Later, in ancient Greek mathematics, Euclid formalized the properties of such ratios in Book V of the Elements (c. 300 BCE), establishing a rigorous theory that treats \frac{1}{2} and similar rationals through proportional magnitudes.^[21] Basic arithmetic operations with \frac{1}{2} illustrate its role in foundational computations. For addition, \frac{1}{2} + \frac{1}{2} = 1, showing how it combines to form unity.^[22] In multiplication, \frac{1}{2} \times n = \frac{n}{2} for any integer n, as in halving 6 to yield 3, which supports even division in integer arithmetic.^[22]

Applications in Measurements and Geometry

In everyday measurements, the concept of one half frequently appears in standardized units. A half-liter, equivalent to 500 milliliters, is a common volume measure in the metric system, often used for beverages and small containers. Similarly, a half-mile represents 0.5 miles, approximately 804.67 meters, serving as a distance unit in transportation and athletics within customary systems.^[23] A half-hour denotes 30 minutes, a standard subdivision of time in both daily scheduling and scientific timing protocols.^[24] In geometry, one half manifests in spatial divisions and properties of shapes. A semicircle is half of a full circle, with its area calculated as \frac{1}{2} \pi r^2, where r is the radius, making it essential for analyzing curved surfaces in architecture and engineering.^[25] The diameter of a circle, defined as twice the radius, implies that the radius is inherently one half of the diameter, a fundamental relation underpinning circle theorems and calculations in Euclidean geometry. Scientific applications extend this to dynamic processes. In physics, the half-life t_{1/2} of a radioactive substance is the time required for half of the nuclei to decay, given by the formula t_{1/2} = \frac{\ln 2}{\lambda}, where \lambda is the decay constant, a key parameter in nuclear stability assessments.^[26] In biology, mitosis divides a parent cell into two genetically identical daughter cells by partitioning replicated DNA and cellular components equally, supporting growth and repair.^[27] Standards organizations incorporate halves into unit definitions for practicality. The International Organization for Standardization (ISO) recognizes subdivisions like the half-tonne, 500 kilograms in metric contexts, used in logistics and manufacturing for load specifications.^[28] Customary units similarly employ half-ton measures, around 1,000 pounds or 453.6 kilograms, in North American trade and vehicle ratings.^[29]

Computing and Typography

Unicode and Character Encoding

In digital character encoding, the symbol for one half, known as the vulgar fraction one half (½), is assigned the Unicode code point U+00BD.^[30] This character was introduced in Unicode version 1.0, released in October 1991, and belongs to the Latin-1 Supplement block (U+0080 to U+00FF).^[31] It is classified as an Other Number (No) with a numeric value of 1/2, functioning primarily as a mathematical symbol for typographic representation of the fraction.^[32] The 7-bit ASCII standard, limited to 128 characters (codes 0–127), does not include U+00BD or any precomposed fraction symbols like ½, necessitating approximations such as the plain text sequence "1/2".^[33] This limitation persisted in early computing environments, where extended ASCII variants like ISO-8859-1 (code 189) later incorporated ½ to support Latin-1 typography.^[34] Historically, IBM's EBCDIC encoding, developed in the 1960s for mainframe systems, included the one half symbol at hexadecimal B8 (decimal 184) in code pages like EBCDIC 1047, enabling its use in early business and scientific computing.^[35] As web technologies emerged, this evolved into HTML named entities such as ½, which renders U+00BD in browsers supporting ISO-8859-1 or UTF-8, facilitating consistent display across documents.^[36] Font rendering of ½ varies by typeface family: serif fonts often depict it with a more traditional, slightly curved or stacked slash for enhanced legibility in print, while sans-serif fonts typically use a straighter, linear slash aligned with the numeral for modern, screen-optimized simplicity.^[37] These differences arise from OpenType features like font-variant-numeric, which can influence fraction glyph substitution, though the precomposed U+00BD remains a fixed symbol in most implementations.^[38]

Usage in Digital Formatting

In word processors such as Microsoft Word, the one half symbol (½) can be inserted using the keyboard shortcut Alt+0189 on Windows systems, which enters the character's decimal code while holding the Alt key and typing on the numeric keypad.^[39] Additionally, Word features auto-conversion for common fractions; typing "1/2" followed by a spacebar automatically formats it as the stacked symbol ½ through its AutoFormat As You Type settings.^[40] In spreadsheet applications like Microsoft Excel, the formula =1/2 computes the decimal value 0.5, which can be displayed as a fraction by applying the Fraction number format to the cell via the Home tab's format options.^[41] For data processing, the value 0.5 is exactly representable in binary floating-point arithmetic under the IEEE 754 standard, as its binary form is 0.1 with no fractional bits required beyond the implicit leading 1 in the significand, avoiding rounding errors common in other decimal fractions.^[42] In web development and markup languages, fractions like one half are rendered using the HTML entity ½ or ½, which displays as ½ in browsers, while CSS can style custom stacked fractions through properties like display: inline-block on container elements with positioned numerator and denominator spans for vertical alignment.^[43] In LaTeX document preparation, the command \frac{1}{2} produces a properly formatted fraction, adjustable in size with variants like \tfrac{1}{2} for inline text or \dfrac{1}{2} for display mode.^[44] For accessibility in digital documents, screen readers such as JAWS or NVDA pronounce the ½ symbol as "one half" when encountered, leveraging Unicode semantic properties to convey the fraction's meaning rather than reading it as individual characters.^[45] The symbol corresponds to the Unicode code point U+00BD in the Latin-1 Supplement block.^[30]

Other Specialized Uses

In Music Notation

In music notation, the half note, known as the minim in British English, is represented by an open (unfilled) oval note head with a straight vertical stem extending from its right side.^[46] This symbol denotes a duration that is twice as long as a quarter note and half as long as a whole note, equating to two beats in common 4/4 time, where the basic fraction 1/2 expresses the proportional relationship in rhythmic values.^[46] The half note's design allows it to be filled when dotted, extending its value by an additional half, resulting in three beats.^[47] In certain time signatures, such as 2/2 (cut time or alla breve), the half note serves as the primary beat unit, with each measure accommodating exactly two half notes to fill the bar.^[48] This notation simplifies faster tempos by effectively halving the perceived pulse compared to 4/4, emphasizing half-note groupings in orchestral and choral works.^[48] The half note's historical roots lie in medieval mensural notation developed in the late 13th century, where the minim emerged as the shortest note value, dividing longer notes like the semibreve into halves to indicate proportional durations.^[49] This system built upon the foundational staff notation innovations of Guido d'Arezzo in the 11th century, which enabled precise pitch and rhythmic representation in polyphonic music. Regarding intervals, the perfect fifth spans seven half steps and corresponds to a frequency ratio of 3:2 (or 1.5 times the fundamental pitch), forming a consonant foundation in harmonic theory.^[50] In contrast, a half step—also called a semitone—represents the smallest interval in the Western chromatic scale, dividing the octave into twelve equal parts for microtonal adjustments in melodies and chords.^[51]

In Sports and Timekeeping

In sports, the concept of "one half" frequently refers to the division of match time into equal periods separated by a halftime break, a practice originating in 19th-century English public schools where intervals allowed for rule changes and player rest.^[52] In association football (soccer), matches are structured as two 45-minute halves totaling 90 minutes of regulation play, with a 15-minute halftime interval as specified by the International Football Association Board (IFAB) Laws of the Game.^[53] Similarly, in basketball, the National Basketball Association (NBA) divides games into two halves of 24 minutes each—comprising four 12-minute quarters—with a 15-minute halftime between the second and third quarters.^[54] Timekeeping conventions also employ "one half" to denote midway points in hourly or daily cycles. The expression "half past" indicates 30 minutes after a given hour, such as 3:30 being half past 3, a usage rooted in analog clock mechanics where the minute hand completes half its cycle.^[55] A half-day conventionally represents 12 hours, aligning with the 12-hour clock format that splits the 24-hour day into two equal periods marked by a.m. and p.m. This division facilitates scheduling in work, education, and daily routines. Variations in sports highlight adaptations of halved structures to game dynamics. In American football, National Football League (NFL) games consist of two 30-minute halves (four 15-minute quarters), though actual ball-in-play time averages about 11 minutes due to stoppages, with a 12- to 15-minute halftime.^[56]^[57] Cricket's Test format, governed by the International Cricket Council (ICC), divides each day's approximately six hours of play into three 2-hour sessions—morning, afternoon, and evening—with a 40-minute lunch interval after the morning session and a 20-minute tea interval between the afternoon and evening sessions.^[58] Cultural events like the half-marathon embody "one half" as a halved distance of the full marathon's 42.195 kilometers, set at precisely 21.0975 kilometers by World Athletics standards to promote accessible long-distance running.^[59] This event, often held as a standalone race, underscores halved measurements in athletic challenges while maintaining rigorous certification for accuracy.^[60]

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ASCII Code 189. In the ISO-8859-1 character set, ASCII code 189 is represented by the character ½ also known as the fraction one half.
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... Fraction One Quarter B8 ½ Vulgar Fraction One Half B9 ¾ Vulgar Fraction Three Quarters BA [ Left Square Bracket BB ] Right Square Bracket BC ‾ Overline BD ...
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