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parabola – rational exponent
parabola
The locus of points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are equal.
Like the ellipse and hyperbola, the parabola is a conic section. See the related article for a full exposition.
Related article: Conics
parallelogram
A quadrilateral with opposite sides (and opposite angles) equal.
parallel postulate
The 5th postulate of Euclidean geometry.
partition
General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.
Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.
Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.
Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4.
perpendicular
Two lines are perpendicular if they intersect in a right angle. In particular, two lines in the Cartesian plane are perpendicular if their respective slopes m1 and m2 satisfy m 1 × m2 = –1.
Two curves intersect perpendicularly if their tangent lines at the point of intersection are perpendicular.
Perpendicularity of planes and surfaces is defined analogously. Cf. normal.
perpendicular distance
Geometry: The shortest distance between a point and a line, or between a point and a plane. Sometimes called directed distance.
Pi (p)
The real number that is equal to the ratio of the circumference of any circle to its diameter, approximately equal to 3.1415926. It is known to be a transcendental number.
plane figure
A figure, i.e., a graphical (visual) representation of points, lines, curves, or regions in the geometric plane. A plane figure, such as a polygon, that bounds a finite region is called closed.
Platonic solid
ARTICLE
A solid having congruent, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube.
polygon
A closed plane figure having straight sides. A polygon that has equal sides and equal angles is called a regular polygon. Examples: square, hexagon, nonagon, etc.
polyhedron
A closed, solid figure consisting of four or more planar faces, pairs of which meet at edges. The faces form polygons, and the points where three or more edges meet are called vertices. The number of faces, edges, and vertices of a polyhedron are related by the Euler polyhedron formula.
A polyhedron with four faces is called a tetrahedron, a pentahedron has five faces, a hexadron six, a heptahedron seven, an octahedron eight, a dodecahedron twelve, and an icosahedron twenty. If all the faces of a polyhedron are congruent, it is called a regular solid or Platonic solid.
polynomial
General: A function of the form
p(x) = a0 + a1x + a2x2 + ... + anxn
where the ai (called coefficients) are real numbers or complex numbers, and the exponents are all natural numbers. The highest exponent is called the degree of the polynomial, and the coefficient an on the highest degree term is called the leading coefficient. Polynomials of degree two are called quadratic polynomials, of degree 3 cubic, of degree 4 quartic, and those of degree 5 are called quintic.
More generally, a polynomial may be in several variables x1, ... ,xk, and may be thought of as a sum of the form
where all but finitely many of the ai are 0. In this case, the degree of the polynomial is the highest sum of exponents appearing in any term. For example, 2x4 and 3xy2z are both fourth-degree terms.
Abstract Algebra (formal polynomial): A formal sum of the form
The ai are called coefficients, and are elements of some commutative ring R. p is said to be a polynomial over R, or with coefficients in R. The x is just a formal symbol. Only finitely many of the ai can be non-zero, and if an is the last non-zero coefficient, n is called the degree of p.
Polynomials can be added and multiplied in the natural way, by using the cummutative and distributive laws of addition and multiplication in R. Explicitly, if p and q are polynomials over the same ring, and the ai and bi are the coefficients of p and q, respectively, then
and
These operations allow us to define R[x], the polynomial ring over R.
The Substitution Principle allows us evaluate a polynomial p on any element of R, and we can use this to define a function corresponding to p, thereby capturing the informal notion of a polynomial.
Cf. Diophantine equation, quadratic formula.
power series
An infinite series of the form
See the related article for a complete description.
Related article: Series
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.
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.
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