A regular polygon having four equal sides and four right angles.
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
(rare) An irrational root of a number, e.g., the square root of two.
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.
Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus.
The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.
Cf. intersection, union.
A tiling of the plane, i.e. the use of plane figures to completely cover the plane without overlaps or gaps. A regular tesselation uses only a finite number of distinct shapes. Most regular tesselations are periodic, but some are aperiodic.
Cf. polygon, Penrose tiles.
A polyhedron having four faces.
The faces of a regular tetrahedron are congruent, equilateral triangles.
Cf. Platonic solid.
Generally, topology is the study of those properties of a space which are invariant under continuous deformations, i.e., deformations which do not create tears or holes. More specifically, given a set X, a topology on X is a collection of subsets of X, called the open sets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersections and arbitrary (i.e., not necessarily finite or countable) unions. A set X with a topology defined upon it is called a topological space.
A function which is not an algebraic function, i.e., a function whose action on its argument(s) cannot be represented by the arithmetic and algebraic operations: addition and subtraction, multiplication and division, raising to a power, or extraction of roots. The exponential function, the logarithmic function, and the trigonometric functions are examples of transcendental functions.
A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well.
A relation ~ on a set X is called transitive if it is the case that for every x, y, and z in X, if x ~ y and y ~ z , then x ~ z. For example, the relation < (less-than) on the set of natural numbers is transitive.
Cf. reflexive relation, symmetric relation.
Geometry: A closed plane figure with three straight sides meeting at three vertices. If one side of a triangle is chosen as the base, then the height of the triangle is the perpendicular distance to the base from the vertex opposite the base.
Triangles are classified by their angle measures:
On a right triangle, the sides adjacent to the right angle are called the legs, and the side opposite is called the hypotenuse.
- Acute all angles less than 90°.
- Obtuse one angle greater than 90°.
- Right one angle exactly 90°.
- Scalene all angles and sides unequal.
- Isosceles two angles equal (equivalently, two sides equal).
- Equilateral all angles equal (equivalently, all sides equal).
Cf. Pythagorean Theorem.
Graph Theory: A cycle with three vertices.
The trig functions are transcendental functions defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions. See the related article for a complete description.
The mathematical subject concerned with trigonometric functions, which are defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions.
The smallest, simplest, and usually least interesting example of any object or construction. Every field has a specific definition of what is considered the trivial object of study in that field. The following entries provide examples.
Logic: A conclusion is trivial if it is so obvious that no proof or demonstration is required to establish its truth.