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regular polygon – space
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
series
ARTICLE
A series is an infinite sum, where the nth summand is the nth term of a sequence. A series is usually denoted using “sigma notation,” i.e.,
The index n may begin with 0, 1, or k for any natural number k, as a matter of convenience. The nth partial sum Sn of a series is the (finite) sum of the first n terms of the series. A series is said to converge if and only if its sequence of partial sums {S 1, S 2, . . . , Sn, . . . } is a convergent sequence. There are several important types of series and several tests for the convergence of a series. Additionally, most useful functions have Taylor series representations, which makes them very important in the study of differential equations. See the article for a complete description.
set
Naively, any well-defined collection considered as a single, abstract object. By “well-defined” is meant that it is always possible to determine for a given set when something is an element of the set and when not. In formal set theory, the term “set” is not defined, but is a primitive term whose meaning is informed purely by the axioms in which it appears.
Cf. ZF, ZFC.
set difference
The set difference of two sets A and B, denoted A - B or A \ B, is the set of elements that is contained in A but not in B. This is equivalent to the intersection between A and the complement of B.
set theory
Naive set theory: The study of sets (i.e., well-defined collections of objects) which have a binary extensional relation (set membership) defined on them.
Abstract set theory: As naive set theory, but with all sets built using only elements which are themselves sets (beginning with the empty set, which has no members).
Formal set theory: Any of several axiom systems of abstract set theory in the language of first-order logic, such as Zermelo Fraenkel set theory, Gödel-Bernays set theory, Quine’s New Foundations, etc.
similar
Graph Theory: Two vertices or edges of a graph are called similar if there is an automorphism of the graph that takes one to the other.
slope
A line in the Cartesian plane which passes through two points (x 1, y 1) and (x 2, y 2) has a slope m given by
The slope may easily be remembered as “rise over run.” It is evident that the slope of a horizontal line is 0, and the slope of a vertical line is undefined.
Cf. linear function.
space
Any abstract set with a structure defined on it, such as an order relation, metric, etc.
Cf. Euclidean space, Hilbert space, metric space, topological space.
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