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product – Riemann integral
product
The result of applying a multiplication operation to two or more numbers or quantities.
proper factor
See factor.
p-series
An infinite series of the form
with p a positive real number. See the related article for details.
Related article: Series
Pythagorean theorem
In Euclidean geometry, the sum of the areas of the squares on the legs of any right triangle is equal to the area of the square on the hypotenuse. This is arguably the most important theorem of classical mathematics, and perhaps of all time.
Pythagorean triple
An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples.
quadratic formula
Given a quadratic function, i.e., a polynomial function of second degree y = ax 2 + bx + c, the zeros of the function are given by
The expression under the radical is called the determinant. If the determinant is positive, both solutions are real; if negative, both solutions are complex; and if zero, there is a single solution of multiplicity two.
quadrilateral
A closed, plane figure with four straight sides.
Cf. polygon.
quotient
The number that results from dividing one number by another.
Cf. division algorithm.
radian
A dimensionless unit of measure of angles. An angle of one radian is given by the central angle of a circle subtending an arc of length equal to the radius of the circle. Consequently, 360 degrees is the same as 2p radians. See the related article for a more extensive exposition.
Related article: Trig Functions and Identities
range
The set of elements to which a function maps the elements of its domain set.
rational exponent
An exponent of the form p/q, with p and q integers and q not zero. Evaluated as the qth root of the base, raised to the pth power, or equivalently, as the qth root of the pth power of the base. For a negative base, this operation is not defined except when q is odd. Irrational roots may be considered as limits of sequences of rational roots.
Cf. laws of exponents.
rational number
An element of the set Q consisting of ordered pairs (p, q) of integers, with q not 0, and with the order relation (p, q) < (r, s) if and only if ps < rq as integers. (The ordered pairs are usually written p/q, i.e., as a fraction (ratio) with integer numerator and denominator.) The rational numbers are countably infinite.
Cf. natural number, real number.
Related MiniText: Number -- What Is How Many?
ratio test
ARTICLE
A test for the convergence of a series. See the related article for a complete description.
real number
An element of the set R consisting of all of the rational numbers together with all of the irrational numbers. Sometimes called the continuum. Usually defined formally by a Dedekind cut of rational numbers. The real numbers form (uniquely) a complete ordered field, but are not algebraically complete.
It is a famous theorem of Georg Cantor that the real numbers are not countable.
Cf. complex number.
Related MiniText: Number -- What Is How Many?
real number line
A geometrical line graphically representing the set of real numbers, in which every real number corresponds to a unique point on the line, and every point on the line corresponds to a unique real number.
reflexive relation
A relation “ ~ ” on a set X is reflexive if for every element x in X we have x ~ x. The relation “ ~ ” is called irreflexive if for every x we have that x ~ x is false. Note that a relation may be neither reflexive nor irreflexive.
Cf. symmetric relation, transitive relation.
regular polygon
A polygon all of whose sides are equal in length and all of whose interior angles are equal.
regular solid
A polyhedron having congruent faces, which are themselves regular polygons. Also called Platonic solid.
Related article: Platonic Solids
relation
An n-place relation is defined on a Cartesian product of n sets, and is represented by a set of ordered n-tuples. For example, the less-than (“<”) relation is a binary relation on numbers, and the membership relation (“e”) is a binary relation on sets. The property of forming a Pythagorean triple is a ternary relation on natural numbers, of which for example (3,4,5) is a member since 32 + 42 = 52.
In a binary (two-place) relation, the set from which the abscissae are taken is called the domain, and the set providing the ordinates is called the range. Binary relations are classified according to whether they are reflexive, transitive, and/or symmetric.
Cf. function, partial order, lattice.
relatively prime
Two natural numbers a and b are relatively prime if their greatest common divisor is 1.
Riemann integral
See integral.
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