## On-line Mathematics Dictionary

This dictionary is far from complete. We are constantly adding definitions. Suggestions for terms that should be added may be sent to lexicographer@MathPro.com. See also our on-line glossary of technical notation for details about unfamiliar mathematical symbols.

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z

abundant number
A positive integer that is smaller than the sum of its proper divisors.
alphametic
A cryptarithm in which the letters, which represent distinct digits, form related words or meaningful phrases.
amicable numbers
Two numbers are said to be amicable if each is equal to the sum of the proper divisors of the other.
ball
A sphere together with its interior.
bijection
A one-to-one onto function.
Caliban puzzle
A logic puzzle in which one is asked to infer one or more facts from a set of given facts.
ceiling function
The ceiling function of x is the smallest integer greater than or equal to x.
centroid
The center of mass of a figure. The centroid of a triangle is the intersection of the medians.
cevian
A line segment extending from a vertex of a triangle to the opposite side.
Chebyshev polynomials
chord
The line joining two points on a curve is called a chord.
circumcenter
The circumcenter of a triangle is the center of the circumscribed circle.
circumcircle
The circle circumscribed about a figure.
coprime
Integers m and n are coprime if gcd(m,n)=1.
cryptarithm
A number puzzle in which an indicated arithmetical operation has some or all of its digits replaced by letters or symbols and where the restoration of the original digits is required. Each letter represents a unique digit.
cyclic polygon
A polygon whose vertices lie on a circle.
deficient number
A positive integer that is larger than the sum of its proper divisors.
digimetic
A cryptarithm in which digits represent other digits.
disc
A circle together with its interior.
divisor
The nonzero integer d is a divisor of the integer n if n/d is an integer.
Diophantine equation
An equation that is to be solved in integers.
domino
Two congruent squares joined along an edge.
Egyptian fraction
A number of the form 1/x where x is an integer is called an Egyptian fraction.
equichordal point
A point inside a closed convex curve in the plane is called an equichordal point if all chords through that point have the same length.
escribed circle
An escribed circle of a triangle is a circle tangent to one side of the triangle and to the extensions of the other sides.
excenter
The center of an excircle.
excircle
An escribed circle of a triangle.
An exradius of a triangle is the radius of an escribed circle.
Farey sequence
The sequence obtained by arranging in numerical order all the proper fractions having denominators not greater than a given integer.
Fermat number
A number of the form 2^{2^n}+1.
Fibonacci number
A member of the sequence 0, 1, 1, 2, 3, 5,... where each number is the sum of the previous two numbers.
floor function
The floor function of x is the greatest integer in x, i.e. the largest integer less than or equal to x.
A line segment from the focus of an ellipse to a point on the perimeter of the ellipse.
geoboard
A flat board into which nails have been driven in a regular rectangular pattern. These nails represent the lattice points in the plane.
Gergonne point
In a triangle, the lines from the vertices to the points of contact of the opposite sides with the inscribed circle meet in a point called the Gergonne point.
gnomon magic square
A 3 X 3 array in which the elements in each 2 X 2 corner have the same sum.
golden ratio
(1+Sqrt[5])/2.
golden rectangle
A rectangle whose sides are in the golden ratio.
graceful graph
A graph is said to be graceful if you can number the n vertices with the integers from 1 to n and then label each edge with the difference between the numbers at the vertices, in such a way that each edge receives a different label.
graph
A graph is a set of points (called vertices) and a set of lines (called edges) joinging these vertices.
Hankel matrix
A matrix in which all the elements are the same along any diagonal that slopes from northeast to southwest.
harmonic mean
The harmonic mean of two numbers a and b is 2ab/(a + b).
hexagonal number
A number of the form n(2n-1).
hexomino
A six-square polyomino.
Heronian triangle
A triangle with integer sides and integer area.
homeomorphism
A one-to-one continuous transformation that preserves open and closed sets.
homomorphism
A function that preserve the operators associated with the specified structure.
incenter
The incenter of a triangle is the center of its inscribed circle.
incircle
The circle inscribed in a given figure.
irrational number
A number that is not rational.
isogonal conjugate
Isogonal lines of a triangle are cevians that are symmetric with respect to the angle bisector. Two points are isogonal conjugates if the corresponding lines to the vertices are isogonal.
isotomic conjugate
Two points on the side of a triangle are isotomic if they are equidistant from the midpoint of that side. Two points inside a triangle are isotomic conjugates if the corresponding cevians through these points meet the opposite sides in isotomic points.
knight's tour
A knight's tour of a chessboard is a sequence of moves by a knight such that each square of the board is visited exactly once.
L-tetromino
A tetromino in the shape of the letter L.
lattice point
A point with integer coordinates.
Legendre polynomials
Lucas number
A member of the sequence 2, 1, 3, 4, 7,... where each number is the sum of the previous two numbers.
magic square
A square array of n numbers such that sum of the n numbers in any row, column, or main diagonal is a constant (known as the magic sum).
magic tour
If a chess piece visits each square of a chessboard in succession, this is called a tour of the chessboard. If the successive squares of a tour on an n X n chessboard are numbered from 1 to n^2, in order, the tour is called a magic tour if the resulting square is a magic square.
Malfatti circles
Three equal circles that are mutually tangent and each tangent to two sides of a given triangle.
medial triangle
The triangle whose vertices are the midpoints of the sides of a given triangle.
Mersenne number
A number of the form 2^p-1 where p is a prime.
Mersenne prime
A Mersenne number that is prime.
monic polynomial
A polynomial in which the coefficient of the term of highest degree is 1.
monochromatic triangle
A triangle whose vertices are all colored the same.
Nagel point
In a triangle, the lines from the vertices to the points of contact of the opposite sides with the excircles to those sides meet in a point called the Nagel point.
nine point center
In a triangle, the circumcenter of the medial triangle is called the nine point center.
nonagonal number
A number of the form n(7n-5)/2.
one to one
A function f is said to be one to one if f(x)=f(y) implies that x=y.
onto
A function f is said to map A onto B if for every b in B, there is some a in A such f(a)=b.
orthic triangle
The triangle whose vertices are the feet of the altitudes of a given triangle.
orthocenter
The point of intersection of the altitudes of a triangle.
palindrome
A positive integer whose digits read the same forward and backwards.
palindromic
A positive integer is said to be palindromic with respect to a base b if its representation in base b reads the same from left to right as from right to left.
pandiagonal magic square
A magic square in which all the broken diagonals as well as the main diagonals add up to the magic constant.
pandigital
A decimal integer is called pandigital if it contains each of the digits from 0 to 9.
Pascal's triangle
A triangular array of binomial coefficients.
pedal triangle
The pedal triangle of a point P with respect to a triangle ABC is the triangle whose vertices are the feet of the perpendiculars dropped from P to the sides of triangle ABC.
Pell number
The nth term in the sequence 0, 1, 2, 5, 12,... defined by the recurrence
P[0]=0, P[1]=1, and P[n]=2P[n-1]+P[n-2].
pentagonal number
A number of the form n(3n-1)/2.
pentomino
A five-square polyomino.
perfect cube perfect number
A positive integer that is equal to the sum of its proper divisors. For example, 28 is perfect because 28=1+2+4+7+14.
perfect power 1.
perfect square polyomino
A planar figure consisting of congruent squares joined edge-to-edge.
practical number
A practical number is a positive integer m such that every natural number n not exceeding m is a sum of distinct divisors of m.
prime
A prime number is an integer larger than 1 whose only positive divisors are 1 and itself.
primitive Pythagorean triangle
A right triangle whose sides are relatively prime integers.
primitive root of unity
The complex number z is a primitive nth root of unity if z^n=1 but z^k is not equal to 1 for any positive integer k less than n.
pronic number
A number of the form n(n+1).
proper divisor
The integer d is a proper divisor of the integer n if 0Pythagorean triangle
A right triangle whose sides are integers.
Pythagorean triple
An ordered set of three positive integers (a,b,c) such that a^2+b^2=c^2.
A geometric figure with four sides.
rational number
A rational number is a number that is the ratio of two integers. All other real numbers are said to be irrational.
reciprocal
The reciprocal of the number x is the number 1/x.
repdigit
An integer all of whose digits are the same.
repunit
An integer consisting only of 1's.
root of unity
A solution of the equation x^n=1, where n is a positive integer.
rusty compass
A pair of compasses that are fixed open in a given position.
semi-magic square
A square array of n numbers such that sum of the n numbers in any row or column is a constant (known as the magic sum).
skeleton division
A long division in which most or all of the digits have been replaced by asterisks to form a cryptarithm.
square free
An integer is said to be square free if it is not divisible by a perfect square, n^2, for n>1.
square number
A number of the form n^2.
Stirling numbers
symmedian
Reflection of a median of a triangle about the corresponding angle bisector.
tetromino
A four-square polyomino.
Toeplitz matrix
A matrix in which all the elements are the same along any diagonal that slopes from northwest to southeast.
trapezium
A quadrilateral in which no sides are parallel.
trapezoid
A quadrilateral in which two sides are parallel.
tree
A tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges.
triangle
A geometric figure with three sides.
triangular number
A number of the form n(n+1)/2.
tromino
A three-square polyomino.
unimodal
A finite sequence is unimodal if it first increases and then decreases.
unimodular
A square matrix is unimodular if it's determinant is 1.
unit circle
A unit circle is a circle with radius 1.
unit square
A unit square is a square of side length 1.
unitary divisor
A divisor d of c is called unitary if gcd(d,c/d) = 1.
X-pentomino
A pentomino in the shape of the letter X.
zeta function

Go to the Electronic publications page.