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product – real number line
product
The result of applying a multiplication operation to two or more numbers or quantities.
proper class
A collection of elements that is not a set. For example, the collection of all sets must be thought of as a proper class in order to avoid the Russell paradox.
proper factor
See factor.
propositional calculus
The formal system of symbolic logic in which sentences are treated as objects related by the logical connectives:
The last two symbols are called the conditional and the biconditional respectively, but they are not essential; indeed all the connectives are fully expressible by the use of the two connectives for “or” and “not.” For example, “p implies q” may be equivalently expressed as “not p or q.” In the notation of propositional calculus, this equivalence may be written,
Cf. predicate calculus.
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.
quantifier
See predicate calculus.
quasi-disjoint family
Set Theory: See D-system.
Quine atom
In set theory, a set whose only member is itself, i.e., x = { x }. More generally, some phrases may be “Quined” to form meaningful sentences, e.g., “is a five-word phrase” is a five-word phrase.
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
Ramsey cardinal
A cardinal k is called a Ramsey cardinal if
Cf. Ramsey’s Theorem.
Ramsey number
See Ramsey’s Theorem.
Ramsey’s Theorem
Graph theory: given any two positive integers m and n, there exists a number, called the Ramsey number of m and n, and denoted R(m, n), such that any simple graph with R(m, n) vertices either contains a clique with m vertices (all vertices adjacent) or an independent set with n vertices (no vertices adjacent).
The question of how to determine the Ramsey number for arbitrary m and n is unsolved and believed to be very difficult, but various lower and upper bounds on it have been proven over the years. In particular, it is known that R(m, m)  2m/2, and that R(m, n)  (m + n - 2)C(m - 1).
Also, certain specific Ramsey numbers are known, e.g., R(3, 3) = 6. Thus, in any group of six people, either three are mutual acquaintances or three are mutual strangers.
Set theory: Let n be a positive integer, k and l be (finite or transfinite) cardinals, and denote by k ln the assertion that the complete graph of size k in which each edge has been “colored” by one of n colors contains a complete subgraph of size l all of whose edges are the same color. More generally still, denote by
the assertion that if the subsets of k of size b have each been colored with one of a-many colors, then there is a subset of X of size l all of whose b-sized subsets are the same color (homogeneous). Ramsey’s Theorem (infinite case) may now be stated as: For all positive integers k and l, we have
i.e., for every coloring of the k-sized subsets of the natural numbers (w) into l colors, there is an infinite homogeneous set (an infinite collection of k-sized subsets all the same color).
This more general notion gives rise to a field of set theory known as Ramsey Theory, in which far more is known about infinite sets than finite ones.
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?
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.
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