長さ

Length — Grokipedia Fact-checked by Grok 3 months ago Length Ara Eve Leo Sal 1x Length is a fundamental physical quantity that measures the distance between two points or the extent of an object along a single dimension. [1] In the International System of Units (SI), length is one of the seven base quantities, with the metre (symbol: m) defined as the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. [2] This definition, adopted in 1983, ties length to the universal constant of the speed of light , ensuring precision and invariance across reference frames. [3] Historically, length measurements originated from human body parts and natural objects, such as the cubit (approximately the length of a forearm ) used in ancient Egypt around 3000 BCE or the foot derived from an average human foot in early Saxon times. [4] Standardization efforts culminated in the metric system during the French Revolution in 1791, establishing the metre based on one ten-millionth of the distance from the equator to the North Pole , later refined through prototypes and now the light-based definition. [4] In everyday applications, length underpins engineering , architecture , and navigation , while in physics, it is essential for describing motion, forces, and spacetime in relativity, where length contraction occurs at high velocities relative to the observer. [5] In mathematics , length extends beyond physical distance to include the magnitude of vectors—calculated as the square root of the dot product with itself—and the arc length of curves, approximated by integrals for precise computation. [6] [7] These concepts form the basis for geometry , where Euclid defined a line as "breadthless length" in Elements (c. 300 BCE), emphasizing its one-dimensional nature without width. [8] Across disciplines, length scales range from the Planck length (about 1.6 × 10⁻³⁵ m), the smallest meaningful distance in quantum gravity , to cosmic distances like the observable universe's diameter (approximately 8.8 × 10²⁶ m). [5] Fundamentals Definition and Concept Length is a fundamental quantity that measures the one-dimensional extent between two points or along a continuous path in space. Intuitively, it represents the separation between objects, such as the distance across a room or the height of a building, providing a basic sense of spatial scale in everyday observations. [9] In physical contexts, length is defined as a basic property independent of other quantities, capturing the spatial interval without inherent direction. [10] Formally, in mathematical terms, length corresponds to the separation between elements in a metric space , where a distance function d ( x , y ) d(x, y) d ( x , y ) quantifies the "length" between points x x x and y y y , adhering to axioms such as non-negativity ( d ( x , y ) ≥ 0 d(x, y) \geq 0 d ( x , y ) ≥ 0 ), identity of indiscernibles ( d ( x , y ) = 0 d(x, y) = 0 d ( x , y ) = 0 if and only if x = y x = y x = y ), symmetry , and the triangle inequality . [11] This concept extends to higher dimensions while remaining one-dimensional in nature, as it evaluates extent along a single axis or direction within the space. Length thus operates as a scalar value, yielding a non-negative real number that describes magnitude without vectorial components. [12] While related, length differs from distance in that the latter typically denotes the shortest straight-line separation ( geodesic distance) between points, whereas length can refer to the total extent along any specified path, such as a curve . It also contrasts with broader notions of size , which encompass multi-dimensional measures like area or volume , focusing instead on linear extension . In vector spaces, length manifests as the norm of a vector, a scalar that quantifies its magnitude, reinforcing its role in abstract geometric structures. [13] Basic Properties In Euclidean geometry , the length of a line segment is characterized by several fundamental properties that ensure its consistency and utility as a measure of extent. These properties include additivity for collinear segments, invariance under rigid transformations, positivity, homogeneity under scaling, and the triangle inequality . They form the basis for defining length as a metric function on the space , applicable prior to more advanced geometric constructions. Additivity states that for any three collinear points A, B, and C, with B between A and C, the length of segment AC equals the sum of the lengths of AB and BC, expressed as $ \length(AC) = \length(AB) + \length(BC) $. This property arises from the order axioms in foundational systems, allowing the continuous extension of segments along a line. It ensures that length behaves like a one-dimensional measure along straight paths. Length is invariant under rigid transformations, such as translations and rotations, in Euclidean space . Specifically, if T is a rigid motion ( isometry ), then for any points A and B, $ \length(T(A)T(B)) = \length(AB) $. This preservation follows from the congruence axioms, which equate segments that can be superimposed by such motions without distortion . Positivity requires that the length of any segment is non-negative, with $ \length(AB) > 0 $ for distinct points A and B, and $ \length(AB) = 0 $ only if A coincides with B. This axiom establishes length as a strict measure of separation, excluding negative or zero values for non-degenerate segments. Homogeneity implies that scaling a figure by a positive factor k proportionally affects its lengths, so $ \length(k \cdot AB) = k \cdot \length(AB) $. Derived from the correspondence between geometric segments and real numbers in axiomatic frameworks, this property supports similarity transformations and dimensional analysis . The triangle inequality provides that for any points A, B, and C, $ \length(AC) \leq \length(AB) + \length(BC) $, with equality holding if and only if B lies on the segment AC. As a derived theorem from the metric properties and order axioms, it bounds the direct extent between points by indirect paths, foundational for path minimization in geometry . Historical Development Ancient and Pre-Modern Measures In ancient Mesopotamia, length measurements were primarily empirical and derived from body parts, with the cubit (known as kush ) serving as a fundamental unit equivalent to approximately 0.5 meters, based on the length of the forearm from elbow to fingertip. [14] This unit was subdivided into smaller components, such as the shu-si (finger, about 1.67 cm) and she (barleycorn, roughly 0.28 cm), and scaled up to larger measures like the nindan (rod, about 6 meters), facilitating applications in construction, agriculture, and land surveying. [14] Similarly, in ancient Egypt, the cubit was a core unit, with the royal cubit standardized at around 52.3 cm—slightly longer than the common cubit to account for the pharaoh's forearm plus a hand span—and used extensively for building monuments and surveying Nile floodplains. [3] Variations in the royal cubit appeared in practical artifacts, such as New Kingdom jars with circumferences ranging from 48.6 to 59.8 cm, reflecting regional adaptations while maintaining a base of 28 fingers (each about 1.87 cm). [15] Greek metrology built on these traditions, introducing units tied to athletics and architecture, notably the stadion—a track length of approximately 185 meters, or 600 Greek feet (each foot around 0.308 meters)—which defined the shortest sprint race at events like the Olympics and Pythian Games. [16] This measure varied slightly by locale, with the Olympic stadion measured at 192.27 meters and the Delphic at about 177.65 meters, underscoring early inconsistencies in foot lengths across city-states. Hero of Alexandria advanced metrology in the first century CE through his work Metrica and Dioptra , describing anthropometric systems that proportioned units like the finger (daktylos, ~1.9 cm), palm (4 fingers), and foot (16 fingers, ~30.8 cm) to the human body, while devising instruments for precise surveying of lengths in engineering and astronomy. [17] These contributions emphasized proportional relationships, such as the cubit as 18 inches or 24 fingers, aiding in the measurement of distances for trade routes and public works. [18] The Romans adapted Greek and earlier systems into a more militaristic framework, with the pes (foot) standardized at about 29.6 cm—derived from an average adult foot—and serving as the base for engineering feats like roads and aqueducts. [19] Larger distances were reckoned in paces ( passus , two steps or 1.48 meters) and miles ( mille passus , 1,000 paces or roughly 1,480 meters), enabling efficient legionary marches and territorial mapping. [3] Into the medieval period, these body-based units persisted across Europe but with growing inconsistencies; the foot varied from 25 to 35 cm by region (e.g., shorter in England at ~30.5 cm post-Norman Conquest, longer in parts of Germany), the hand (palm width, 7-10 cm) differed by occupation, and the pace (step length, 70-80 cm) fluctuated with terrain and individual gait, complicating trade and leading to disputes among merchants. [20] Such variability arose from local customs and the absence of durable standards, as measures were often calibrated against rulers' bodies or common tools rather than fixed artifacts. [21] Efforts toward standardization emerged in ancient trade contexts, particularly in Greece , where the Attic foot (approximately 29.5 cm) was promoted for consistency in commerce and architecture , as evidenced by its use in Athenian markets and temples to align measurements with imported goods from across the Mediterranean. [22] This unit, described by Plutarch as derived from proportional divisions of the human form, helped mitigate discrepancies in exchanges but remained one of many local variants until later reforms. [23] Modern Standardization The modern standardizatio