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58-451: The magnitude of an angle is called an angular measure or simply "angle". Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius , and may be a negative number . In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation , the arc is centered at the center of the rotation and delimited by any other point and its image by

116-462: A logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels ), the brightness of a star , and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. In the natural sciences , a logarithmic magnitude is typically referred to as a level . Orders of magnitude denote differences in numeric quantities, usually measurements, by

174-412: A real number r is defined by: Absolute value may also be thought of as the number's distance from zero on the real number line . For example, the absolute value of both 70 and −70 is 70. A complex number z may be viewed as the position of a point P in a 2-dimensional space , called the complex plane . The absolute value (or modulus ) of z may be thought of as the distance of P from

232-434: A ,  b ,  c , . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C)

290-461: A clever didactical presentation of his teacher's ideas. Later authors who wrote commentaries on Aristotle often made good use of Eudemus's preliminary work. It is for this reason that, though Eudemus's writings themselves are not extant, we know many citations and testimonia regarding his work, and are thus able to build up a picture of him and his work. Aristotle, shortly before his death in 322 BC, designated Theophrastus to be his successor as head of

348-1346: A constant η equal to 1 inverse radian (1 rad) in a fashion similar to the introduction of the constant ε 0 . With this change the formula for the angle subtended at the center of a circle, s = rθ , is modified to become s = ηrθ , and the Taylor series for the sine of an angle θ becomes: Sin ⁡ θ = sin ⁡   x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = η θ − ( η θ ) 3 3 ! + ( η θ ) 5 5 ! − ( η θ ) 7 7 ! + ⋯ , {\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,} where x = η θ = θ / rad {\displaystyle x=\eta \theta =\theta /{\text{rad}}}

406-403: A factor of 10—that is, a difference of one digit in the location of the decimal point. In mathematics , the concept of a measure is a generalization and formalization of geometrical measures ( length , area , volume ) and other common notions, such as magnitude, mass , and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in

464-450: A full turn are not equivalent. To measure an angle θ , a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses . The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: θ = s r r a d . {\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} Conventionally, in mathematics and

522-504: A magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm , such as the Euclidean space, is called a normed vector space . The norm of a vector v in a normed vector space can be considered to be the magnitude of v . In a pseudo-Euclidean space , the magnitude of a vector is the value of the quadratic form for that vector. When comparing magnitudes,

580-435: A north-west orientation corresponds to a bearing of 315°. For an angular unit, it is definitional that the angle addition postulate holds. Some quantities related to angles where the angle addition postulate does not hold include: Magnitude (mathematics) In mathematics , the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of

638-538: A pair of (often parallel) lines and is associated with exterior angles , interior angles , alternate exterior angles , alternate interior angles , corresponding angles , and consecutive interior angles . The angle addition postulate states that if B is in the interior of angle AOC, then m ∠ A O C = m ∠ A O B + m ∠ B O C {\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } I.e.,

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696-419: A point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of

754-500: A reliable form, by recording it in a long series of publications. These were based on Aristotle's writings, their own lecture notes, personal recollections, et cetera. Thus one of Aristotle's writings is still called the Eudemian Ethics , probably because it was Eudemus who edited (though very lightly) this text. More important, Eudemus wrote a number of influential books that clarified Aristotle's works: A comparison between

812-466: A single mathematical context. Measures are foundational in probability theory , integration theory , and can be generalized to assume negative values , as with electrical charge . Far-reaching generalizations (such as spectral measures and projection-valued measures ) of measure are widely used in quantum physics and physics in general. Eudemus of Rhodes Eudemus of Rhodes ( ‹See Tfd› Greek : Εὔδημος ; c. 370 BC - c. 300 BC )

870-502: A triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle. The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle. The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure . In some contexts, such as identifying

928-421: A two-dimensional Cartesian coordinate system , an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis , while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward

986-424: Is ⁠ 1 / 256 ⁠ of a turn. Plane angle may be defined as θ = s / r , where θ is the magnitude in radians of the subtended angle, s is circular arc length, and r is radius. One radian corresponds to the angle for which s = r , hence 1 radian = 1 m/m = 1. However, rad is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using

1044-491: Is "pedagogically unsatisfying". In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s), and torsional stiffness (N⋅m/rad), and not in

1102-468: Is called the vertical angle theorem . Eudemus of Rhodes attributed the proof to Thales of Miletus . The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure

1160-493: Is clear that the complete form is meant. Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1 . This radian convention allows the omission of η in mathematical formulas. It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference. In

1218-426: Is denoted ∠BAC or B A C ^ {\displaystyle {\widehat {\rm {BAC}}}} . Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). In other ways, an angle denoted as, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A,

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1276-509: Is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s / r is unaltered. Throughout history, angles have been measured in various units . These are known as angular units , with the most contemporary units being the degree ( ° ), the radian (rad), and the gradian (grad), though many others have been used throughout history . Most units of angular measurement are defined such that one turn (i.e.,

1334-401: Is most commonly defined as its Euclidean norm (or Euclidean length): For instance, in a 3-dimensional space, the magnitude of [3, 4, 12] is 13 because 3 2 + 4 2 + 12 2 = 169 = 13. {\displaystyle {\sqrt {3^{2}+4^{2}+12^{2}}}={\sqrt {169}}=13.} This is equivalent to the square root of

1392-554: Is not certain. First, he is said to have written a History of Theology , that discussed the Babylonian, Egyptian, and Greek ideas regarding the origins of the universe. Secondly, he is said to have been the author of a History of Lindos (Lindos is a town on the Greek island of Rhodes) To Eudemus is also ascribed a book with miraculous stories about animals and their human-like properties (exemplary braveness, ethical sensitivity, and

1450-403: Is supplementary to both angles C and D , either of these angle measures may be used to determine the measure of Angle B . Using the measure of either angle C or angle D , we find the measure of angle B to be 180° − (180° − x ) = 180° − 180° + x = x . Therefore, both angle A and angle B have measures equal to x and are equal in measure. A transversal is a line that intersects

1508-460: Is the angle in radians. The capitalized function Sin is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while sin is the traditional function on pure numbers which assumes its argument is a dimensionless number in radians. The capitalised symbol Sin {\displaystyle \operatorname {Sin} } can be denoted sin {\displaystyle \sin } if it

1566-404: Is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie. In navigation , bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so

1624-460: The Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics , magnitude can be defined as quantity or distance. An order of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale. Ancient Greeks distinguished between several types of magnitude, including: They proved that

1682-681: The SI , the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted. The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form ⁠ k / 2 π ⁠ , where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians ): θ = k 2 π ⋅ s r . {\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} The value of θ thus defined

1740-481: The area of a circle , π r . The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. In particular, Quincey identifies Torrens' proposal to introduce

1798-654: The complement of the angle. If angles A and B are complementary, the following relationships hold: sin 2 ⁡ A + sin 2 ⁡ B = 1 cos 2 ⁡ A + cos 2 ⁡ B = 1 tan ⁡ A = cot ⁡ B sec ⁡ A = csc ⁡ B {\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\[3pt]&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} (The tangent of an angle equals

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1856-445: The cotangent of its complement, and its secant equals the cosecant of its complement.) The prefix " co- " in the names of some trigonometric ratios refers to the word "complementary". If the two supplementary angles are adjacent (i.e., have a common vertex and share just one side), their non-shared sides form a straight line . Such angles are called a linear pair of angles . However, supplementary angles do not have to be on

1914-475: The dot product of the vector with itself: The Euclidean norm of a vector is just a special case of Euclidean distance : the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x : A disadvantage of the second notation is that it can also be used to denote the absolute value of scalars and the determinants of matrices , which introduces an element of ambiguity. By definition, all Euclidean vectors have

1972-546: The Peripatetic School. Eudemus then returned to Rhodes, where he founded his own philosophical school, continued his own philosophical research, and went on editing Aristotle's work. At the insistence of Aristotle, Eudemus wrote histories of Greek mathematics and astronomy. Though only fragments of these have survived, included in the works of later authors, their value is immense. It is only because later authors used Eudemus's writings that we still are informed about

2030-566: The angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n . Two exceptions are the radian (and its decimal submultiples) and the diameter part. In the International System of Quantities , an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in dimensional analysis . The following table lists some units used to represent angles. It

2088-513: The anticlockwise (positive) angle from B to C about A and ∠CAB the anticlockwise (positive) angle from C to B about A. There is some common terminology for angles, whose measure is always non-negative (see § Signed angles ): The names, intervals, and measuring units are shown in the table below: When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other. The equality of vertically opposite angles

2146-490: The area of a circular sector θ = 2 A / r gives 1 radian as 1 m/m = 1. The key fact is that the radian is a dimensionless unit equal to 1 . In SI 2019, the SI radian is defined accordingly as 1 rad = 1 . It is a long-established practice in mathematics and across all areas of science to make use of rad = 1 . Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of

2204-486: The clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see § Signed angles ). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, ∠BAC always refers to

2262-532: The early history and development of Greek science. In his historical writings, Eudemus showed how the purely practically oriented knowledge and skills that earlier peoples such as the Egyptians and the Babylonians had known, were by the Greeks given a theoretical basis, and built into a coherent and comprehensive philosophical building. Two other historical works are attributed to Eudemus, but here his authorship

2320-450: The final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation , which

2378-507: The first two could not be the same, or even isomorphic systems of magnitude. They did not consider negative magnitudes to be meaningful, and magnitude is still primarily used in contexts in which zero is either the smallest size or less than all possible sizes. The magnitude of any number x {\displaystyle x} is usually called its absolute value or modulus , denoted by | x | {\displaystyle |x|} . The absolute value of

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2436-513: The inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus , an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes , who regarded an angle as a deviation from a straight line ; the second, angle as quantity, by Carpus of Antioch , who regarded it as

2494-432: The interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. In mathematical expressions , it is common to use Greek letters ( α , β , γ , θ , φ , . . . ) as variables denoting the size of some angle (the symbol π is typically not used for this purpose to avoid confusion with the constant denoted by that symbol ). Lower case Roman letters (

2552-479: The like). However, as the character of this work does not at all fit in with the serious scientific approach that is apparent from Eudemus's other works, it is generally held that Eudemus of Rhodes cannot have been the author of this book (it may have been another Eudemus — his was a fairly common name in ancient Greece). Eudemus, Theophrastus, and other pupils of Aristotle took care that the intellectual heritage of their master after his death would remain accessible in

2610-591: The magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate , z ¯ {\displaystyle {\bar {z}}} , where for any complex number z = a + b i {\displaystyle z=a+bi} , its complex conjugate is z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi} . (where i 2 = − 1 {\displaystyle i^{2}=-1} ). A Euclidean vector represents

2668-458: The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. Three special angle pairs involve the summation of angles: The adjective complementary is from the Latin complementum , associated with the verb complere , "to fill up". An acute angle is "filled up" by its complement to form a right angle. The difference between an angle and a right angle is termed

2726-467: The negative y -axis. When Cartesian coordinates are represented by standard position , defined by the x -axis rightward and the y -axis upward, positive rotations are anticlockwise , and negative cycles are clockwise . In many contexts, an angle of − θ is effectively equivalent to an angle of "one full turn minus θ ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although

2784-494: The origin of that space. The formula for the absolute value of z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space : where the real numbers a and b are the real part and the imaginary part of z , respectively. For instance, the modulus of −3 + 4 i is ( − 3 ) 2 + 4 2 = 5 {\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5} . Alternatively,

2842-546: The position of a point P in a Euclidean space . Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n -dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P ): x = [ x 1 , x 2 , ..., x n ]. Its magnitude or length , denoted by ‖ x ‖ {\displaystyle \|x\|} ,

2900-432: The quantities of torque (N⋅m) and angular momentum (kg⋅m/s). At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a base unit of measurement for a base quantity (and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for

2958-403: The radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by y = rθ centimetres, where r is the magnitude of the radius of the pulley in centimetres and θ is the magnitude of the angle through which the pulley turns in radians. When multiplying r by θ , the unit radian does not appear in the product, nor does

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3016-635: The rotation. The word angle comes from the Latin word angulus , meaning "corner". Cognate words include the Greek ἀγκύλος ([ankylοs] Error: {{Lang}}: Non-latn text/Latn script subtag mismatch ( help ) ) meaning "crooked, curved" and the English word " ankle ". Both are connected with the Proto-Indo-European root *ank- , meaning "to bend" or "bow". Euclid defines a plane angle as

3074-414: The same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero. In vector spaces,

3132-582: The same line and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary. The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs. In Euclidean geometry, any sum of two angles in

3190-487: The unit centimetre—because both factors are magnitudes (numbers). Similarly in the formula for the angular velocity of a rolling wheel, ω = v / r , radians appear in the units of ω but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge

3248-518: The vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x , the measure of angle C would be 180° − x . Similarly, the measure of angle D would be 180° − x . Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B

3306-521: Was an ancient Greek philosopher, considered the first historian of science. He was one of Aristotle 's most important pupils, editing his teacher's work and making it more easily accessible. Eudemus' nephew, Pasicles, was also credited with editing Aristotle's works. Eudemus was born on the isle of Rhodes , but spent a large part of his life in Athens , where he studied philosophy at Aristotle's Peripatetic School . Eudemus's collaboration with Aristotle

3364-467: Was long-lasting and close, and he was generally considered to be one of Aristotle's most brilliant pupils: he and Theophrastus of Lesbos were regularly called not Aristotle's "disciples", but his "companions" (ἑταῖροι). It seems that Theophrastus was the greater genius of the two, continuing Aristotle's studies in a wide range of areas. Although Eudemus too conducted original research, his forte lay in systematizing Aristotle's philosophical legacy, and in

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