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.
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- 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.
- exradius
- 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.
- focal radius
- 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.
- quadrilateral
- 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
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Last modified March 24, 1995.