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R - radian to ruled surface - Mathematics Dictionary

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radian: A measure of angle which is the ratio of the arc length to the radius formed by a circular arc with the angle.

radical: The symbol representing an nth root,.

radicand: The operand of a radical.

radius: 1. Any line segment where one endpoint is the centre and the other on a circle/sphere.

2. The equivalent line segment in an ellipse/ellipsoid, seen as an elongated circle/sphere.

radius of convergence: The radius of the circular region on an argand diagram where all points within that region (in the interior of the disc), as argument, provides a convergent power series of a specified function.

random variable: A quantity having a numerical value for each member of a group, esp. one whose values occur according to a frequency distribution.

range: 1. The codomain of a function - the set of values (outputs) that a function may take.

2. The image of the domain of a function - the set of values (outputs) that a function takes.

3. A statistical measure of dispersion - the difference between the greatest and the least elements within a data set of real values.

rank: Have a specified 'rank' or place/position within a grading system.

ratio: A relationship between 2 numbers, expressed either as 2 numbers (usually integers) or a single number:

1. a : b or "a to b" - the first quantity multiplied by the scalar a is the same as the quantity multiplied by the scalar value b

2. r - the scalar which the second quantity has to multiply by so that it equals to the first quantity.

rational function: A function which is the ratio of 2 polynomials. The analogue of a rational number by considering polynomials as analogues to integers.

rational number: A number which can be expressed as a ratio of 2 integers.

Note that it refers to the property of a number, a number is rational as long as it can be represented as a ratio of 2 integers, the fact that it is not expressed as such does not prevent a number from being rational. (As opposed to a fraction which is a type of representation of a number.)

rational zero theorem: Also known as the rational root theorem - it ia a theorem which states that any rational roots of a polynomial must be a rational number which can be expressed as a ratio of 2 integers where the numerator is a factor (including negatives) of the constant term, and the denominator is a factor (also including negatives) of the coefficient of the highest order term.

If the constant term is 0, then either the numerator is 0 - a rational root of 0, or the numerator divides the coefficient of the lowest order non-zero term..

ray: A half-line - a line segment where exactly one end is extended indefinitely. The term half-line is mostly used in polar/spherical coordinates or their derivatives, while the term ray has less of such conotations.

real numbers: A complex number for which the imaginary part is 0. It is the set of all numbers which can be formed by application of the elementary operations (addition, subtraction, multiplication, division) as well as taking limits of sequences of numbers which can be formed using the natural numbers.

real part: The component of a complex number the square of which is a positive number, such that the square of the only other component of the complex number is a negative number.

reciprocal: The multiplicative inverse of a number, i.e. the number such that the product of this number and the specified number is 1.

rectangle: An equiangular quadrilateral. Some (a minority) considers a rectangle to be an equiangular quadrilateral that is not also equilateral (or regular), i.e. a square is not a special type of rectangle.

rectangular co-ordinates: A descriptive class="d-title" name of the Cartesian coordinates , also called rectilinear coordinates.

recursive formula: An equation for the purpose of evaluating a term of a sequence using the previous term(s) of the sequence.

reducible equation: An equation which can be written in a simpler way. e.g. dividing a non-zero expression on both sides; one of both sides can be written as a product of simpler expressions etc. Specifically, an equation of a reducible polynomial.

reducible fraction: A fraction where the numerator and the denominator are not coprime, i.e. they have a common factor greater than 1, so that the fraction can be written equivalently with 2 smaller positive integers.

reducible radical: A quantity expressed as the radical of an integer, which can be written as the multiple of a radical of a smaller integer.

reductio ad absurdum: Reducing to the absurd - commonly a form of argument that shows that the complement is necessarily contradictory and therby proving an assertion, also known as proof by contradiction.

reduction formulae: A set of formulae which relates the integral of some general expression to a simpler integral of the same type. e.g. A reduction formula which turns an integral of the nthn power of a trigonometric function into another integral of the same trigonometric function of a lower power (and some other expressions not involving integrals).

re-entrant angle: An interior reflex angle, i.e. an angle of a geometric figure which is greater than 180°.

reference angles: An acute angle whose absolute value of the trigonometric value (for a particular trigonometric function ) is the same as that of the angle specified. The relationship between angles as reference angles remains the same regardless of the choice of the trigonometric function .

reference axis: An axis in a geometric figure or coordinate system that is used as a reference.

reflection: A transformation where a line/plane of invariant points (the line/plane of reflection) is symmetric to any point and their image..

reflection matrix: A matrix which represents a reflection where the line/plane of reflection contains the origin, it is an orthonormal matrix with a determinant of -1.

reflection property: Alternative methods of defining the conic sections through loci of circles with a specified relation to a circle or a line (which can be considered as a great circle in the projective plane).

reflex angle: An angle greater than half a revolution (180°) but less than one revolution (360°).

region: A connected subset of points in a topological space.

region of convergence: The region of points, as arugments, provide a convergent series.

regressand: Another class="d-title" name for a dependent variable

regression: A family of techniques designed to uncover the relationship between a dependent variable with one or more independent variables.

regression equation: A equation that links the dependent variable with independent variables by paramaters. e.g. A linear regression is a regression model where the regression equation is a linear equation.

regular graph: In graph theory, a graph where the valency (degree) is the same for all vertices. i.e. a graph where the number of edges adjacent to a vertex is the same for all vertices.

regular polygon: An equiangular and equilateral polygon, i.e. a polygon where all interior angles are the same and all edges are of the same length.

regular polyhedron: A polyhedron where a tranformation (combining rotation, reflection and translation only) of a vertex to another vertex and an adjacent edge (to the original vertex) is transformed to an adjacent edge (to the image vertex) means that the polyhedra coincide.

regular prism: A prism whose base is a regular polygon.

regular pyramid: A pyramid where the base is a regular polygon.

related angles: Another class="d-title" name for reference angles.

relation: A relation between two or more specified sets/classes (which may be the same, in the case of a relation on a set/class) is a set/class of ordered sets of elements of those specified sets/classes, always in the same order.

Some important properties of binary relations over a set X are:

  • reflexive: x ~ x for all x in X.

  • symmetric: x ~ y implies y ~ x for all x, y in X.

  • transitive: x ~ y and y ~ z implies x ~ z for all x, y, z in X..

An equivalence relation is a binary relation which is reflexive, symmetric and transitive.

relative acceleration: The acceleration of an object in the frame of reference of another object.

relative complement: Also known as set theoretic difference. The relative complement of B in A is the intersection of B and the (absolute) complement of A.

relative density: The ratio of the density of a system/object/material to the density of a reference system/object/material. The density of water or air is often used as a reference.

relative error: The error of a value compared to a reference value, as a proportion (when it exists) of the reference value. i.e. the absolute value of the ratio of the error to the reference value.

relative frequency: The ratio of the frequency of an event to the total frequency (sum) of all possible (partioioning - mutually exclusive and exhaustive) events.

relatively prime: Also known as coprime, the relation between 2 numbers which do not share a common factor greater than 1,

relative maximum: Also known as a local maximum.

relative velocity: The velocity of an object in the frame of reference of another object. GIven vA, the velocity of an object A and vB, the velocity of an object B, the velocity of A relative to B is vA r B = vA - vB.

relativistic mass: The measure of mass of an object as observed from another frame of reference.

relativistic mechanics: 1. A system of non-classical mechanics which is compatible with the Theory of Relativity.

2. The study of mechanics in such a system.

relativity: The idea that physical laws should be isotropic, homogeneous and time-independent (in all directions, at all points, at all times). Note that the idea of whether this is also true to only inertial frame of reference is essentially the difference between Special and General Relativity.

It (or statements to that effect) was assumed in formulation and application of classical mechanics and explicitly stated as a postulate in Albert's Einstein's theories. It's influence is particularly important that Albert Einstein's theories of Special Relativity and General Relativity were class="d-title" named after this idea (the Principle of Relativity).

remainder: 1. The minimum non-negative amount to take away from a dividend so that it is an exact multiple of the divisor.

2. The subseries ak where k ≥ N for some positive integer N, in the series an where n is an integer

remainder theorem: A theorem which states that f(k) is the number to take away from the polynomial f(x) so that the quotient of dividing the resulting polynomial by (x - k) is another polynomial.

removable singularity: A "hole" (undefined point) in a function where it is possible to fill in the hole (define the function as a piecewise function to have a certain value just for that point) so that the resulting function is differnetiable.

repeated decimal: Also known as a recurring decimal, a number whose decimal representation consist of regularly repeating digits beyond a certain decimal place. All recurring decimal is a rational numbers.

repeated root: A multiple root, a root whose multiplicity is greater than 1.

representative sample: A sample for which it is assumed that there is no reason the expectations of estimations from the sample differ from the parameters of the population.

repunit: Repeated unit in recreational mathematics. A number whose recreational representation contains only the digit 1. The term also applies to numbers not represented in base 10.

residual: See residuals in statisitcs..

residuals in statistics: The differences a sample and the results inferred by a estimation/model.

resonance: A physical phenomenon where an oscillating system oscillates at greater amplitude at particular frequencies.

response variable: Another class="d-title" name for a dependent variable.

rest mass: The mass of an object when it is considered to be stationary from a frame of reference.

restricted domain: A subset of the largest possible domain of a function, given its other definitions. A domain may be restricted such that an inverse functon exists.

resultant: 1. The sum of more than one vector.

2. An operation which takes 2 polynomials and returns a number.

revolution: The rotation of a geometric figure by 360o.

rhombohedron: A parallelepiped whose edges have the same length. Informally, it is a slumped cube (consider a cube made of jelly on a surface at an angle where the edges are not parallel to horizontal).

rhomboid: A parallelogram which is not a rectangle or rhombus.

rhombus: An equilateral quadrilateral. Sometimes considered to not include the square as a special case.

Riemann sum: 1. The area of rectangles, where its width is identified with a specific sub-interval between the upper and lower limits of the definite integral of an integrable function, whose height equals to the value of the function at a point within the sub-interval. e.g. lower Roemann sum, upper Riemann sum, middle Riemann sum etc.

2. The numerical method of approximating the area underneath a curve (value of a definite integral) by areas of rectangles whose widths are not necessarily all equal (known as Riemann sums in the sense of 1).

Riemannian geometry: The study of Riemann manifolds.

right angle: An angle of 90°, as in a corner of a square or at the intersection of two perpendicular straight lines.

right-angled triangle: A triangle which contains a right angle (90°).

right prism: A prism where all lines contained in the lateral edges are perpendicular to the base.

right pyramid: A pyramid where the line segment joining the apex with the centre of the base of the pyramid is perpendicular to the base itself.

rigid body: A system of particles, considered as a whole (body). where distances between any 2 points in the body is constant, regardless of velocity or external forces acting upon it (or part of it).

rise: The difference in the coordinates of the independent variable between 2 points.

robustness: The measure of the ability of a statistic to remain minimally affected by small deviations from assumptions within the model.

rod: An idealized material object with only length, having no width or height, and zero curvature.

rolling friction: The kinetic (or dynamic) friction on an object rolling (relatively) on a surface.

root: 1. A number whose substitution into an expression/function returns a value of zero. (This definition extends to an equation consisting of a function only.)

2. The number which, when raised to a specified power, results in a specified number (operand). e.g. The square root is a number which equals to a specified number when raised to the power of 2.

root mean square deviation: The square root of the mean of the squares of distances from a measure of central tendency, when the measure is the arithmetic mean, the root mean square deviation is called the standard deviation.

rose: A curve with equation r = cos(Rθ+k) (therefore includes r = sin(Rθ)) in the polar coordinate system, where R is a rational number and k a real number.

rotation: A transformation in which the coordinate axes are rotated by a fixed angle about the origin.

rotation matrix: A matrix representing the linear transformation of a rotation where the origin is an invariant point. Such a matrix is orthonormal and have a determinant of 1.

rotation of axes: A change from one coordinate system to another where the axes remain at the same angle to each other, in the same orientation.

rough: A surface modelled to produce a resistance to the relative motion between the surface and another rought object.

roulette: The locus of a point fixed on (relative to) a curve, as it rolls along another curve without slipping. It generalises the ideas behind epicycloids, epitrochoids, hypocycloids, hypotrochoids etc. to curves of any shape.

round angle: The angle of exactly one revolution (360°), also known as a perigon.

rounding: A numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation. For example $25.5486 by $25.55.

row: A set of entries all sharing the same first index in a matrix. In the usual representation of a matrix, the set of entries in a horizontal line within the rectangular array.

row echelon: A matrix which can be derived from applying row elementary operations such that the number of leading zero entries in each row must be either greater than the row before, or the same number as the row before in the case where both the row itself and the one before contain all zero entries.

row rank: The dimension of the row space of a given matrix.

row space: The set of vectors spanned by all linear combinations of the rows (considered as row vectors) of a matrix.

row vector: A matrix which has only one row and more than one column of elements.

RPM: Revolutions per minute - a measure of the speed of rotation, and indrectly as a measure of the frequency of rotation.

ruled surface: A surface where there is at least one (straight) line through any given point on the surface.

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