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.
Geometry: The shortest distance between a point and a line, or between a point and a plane. Sometimes called directed distance.
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.
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.
A solid having congruent, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube.
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.
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.
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
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.
A statement in a mathematical theory that is assumed without proof. Essentially synonymous with “axiom.”
An infinite series of the form
See the related article for a complete description.
Given a set X, the power set of X, denoted P(X), is the collection of all subsets of X. If X is a finite set with n elements, then P(X) is a finite set with 2n elements.
Cf. power set axiom.
In a structure with an order relation defined upon it, the predecessor of an element b is the greatest element less than b.
Any natural number greater than 1 that is evenly divisible only by itself and 1. There are infinitely many prime numbers. The number of primes less than a given number n is denoted p(n), and approaches the value n/lnn for sufficiently large n.
The result of applying a multiplication operation to two or more numbers or quantities.
An infinite series of the form
with p a positive real number. See the related article for details.
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.
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.
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.
A closed, plane figure with four straight sides.
The number that results from dividing one number by another.
Cf. division algorithm.