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range – sequence
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.
Riemann sum
Let f be a real-valued function defined on the closed interval [a, b], and let D be a partition of [a, b], i.e., a = x0 < x1 < ... < xn = b, and where Dxi is the width of the i th subinterval. If c i is any point in the i th subinterval, then the sum
is called the Riemann sum of f for the partition D.
right angle
An angle of 90 degrees (p/2 radians). Equivalently, it can be said that two right angles are supplemental angles, i.e., they add up to a straight line (180 degrees or p radians).
Cf. complementary angles, acute, obtuse.
root
An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.
Of a polynomial p: A number x such that p(x) = 0.
root test
A test for the convergence of a series. See the related article for a complete description.
Related article: Series
scalar
A quantity having only magnitude, not direction (typically an element of a field, such as the real numbers or complex numbers).
Cf. vector.
scalar product
The scalar product, also called dot product, of two vectors is the sum of the products of the corresponding components of the two vectors. I.e., given two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn), their scalar product is the scalar x1y1 + x2y2 + ... + xnyn.
Cf. vector product.
scalene
A triangle is called scalene if all of its sides are unequal (equivalently, if all of its angles are unequal).
scientific notation
A number is written in scientific notation when it is written as the product of a real number between 1 and 10 and a power of 10. E.g., 320 is written in scientific notation as 3.2 × 102. On some calculators and in some textbooks, this may be written as 3.2E2. Scientific notation is a convenient way to represent very large and very small numbers.
sequence
A sequence is a set (of numbers, or sets, or functions, etc.) indexed by the natural numbers. Sequences may be infinite, and may be regarded as a function with domain the set of natural numbers and range the set of objects in the sequence.
An infinite sequence of numbers is said to converge to a number L provided that, given any positive e, we may find a natural number N such that for all terms of the sequence after the N th one, their difference from L is less than e. Naively, the terms of the sequence eventually become “arbitrarily close” to L. Such a sequence is called convergent, and the number L is called the limit of the sequence, or the limit point, or sometimes the accumulation point of the sequence.
 Alternatively, a cluster point or accumulation point P of a sequence may be defined as a point with the property that infinitely many terms of the sequence lie in any neighborhood of P. A sequence may have more than one such cluster point (even infinitely many).
A sequence is called Cauchy if, for every e greater than zero, we may find a natural number N so that the difference between any two terms following the N th term is smaller than e. Every convergent sequence is Cauchy; the converse is true in complete spaces.
Cf. series.
Related article: Limits
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