What are tessellations?
Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps. Remember the last puzzle you put together? Well, that was a tessellation! The shapes were just really weird.
Example:
We usually add a few more rules to make things interesting!
REGULAR TESSELLATIONS:
RULE #1: The tessellation must tile a floor (that goes on forever) with no overlapping or gaps.
RULE #2: The tiles must be regular polygons - and all the same.
RULE #3: Each vertex must look the same.
What's a vertex? where all the "corners" meet!
What can we tessellate using these rules?
Triangles? Yep!
Notice what happens at each vertex!
The interior angle of each equilateral triangle is
60 degrees.....
60 + 60 + 60 + 60 + 60 + 60 = 360 degrees
Squares? Yep!
What happens at each vertex?
90 + 90 + 90 + 90 = 360 degrees again!
So, we need to use regular polygons that add up to 360 degrees.
Will pentagons work?
The interior angle of a pentagon is 108 degrees. . .
108 + 108 + 108 = 324 degrees . . . Nope!
Hexagons?
120 + 120 + 120 = 360 degrees Yep!
Heptagons?
No way!! Now we are getting overlaps!
Octagons? Nope!
They'll overlap too. In fact, all polygons with more than six sides will overlap! So, the only regular polygons that tessellate are triangles, squares and hexagons!
SEMI-REGULAR TESSELLATIONS:
These tessellations are made by using two or more different regular polygons. The rules are still the same. Every vertex must have the exact same configuration.
Examples:
3, 6, 3, 6
3, 3, 3, 3, 6
These tessellations are both made up of hexagons and triangles, but their vertex configuration is different. That's why we've named them!
To name a tessellation, simply work your way around one vertex counting the number of sides of the polygons that form that vertex. The trick is to go around the vertex in order so that the smallest numbers possible appear first.
That's why we wouldn't call our 3, 3, 3, 3, 6 tessellation a 3, 3, 6, 3, 3!
Here's another tessellation made up of hexagons and triangles.
Can you see why this isn't an official semi-regular tessellation?
It breaks the vertex rule! Do you see where?
Here are some tessellations using squares and triangles:
3, 3, 3, 4, 4
3, 3, 4, 3, 4
Negative numbers
What are negative numbers?
Let's look at one example, the weather. On a cold December night you can watch the thermometer as the temperature drops, as the numbers go down:
2 degrees, 1 degree, zero degrees
But what happens to the numbers if it gets even colder? The temperature and the numbers keep going down!
The numbers below zero are called negative numbers
A negative number is less than zero.
We write negative numbers like this:
negative 2 is the same as -2
The dash is the negative sign. It is usually written slightly shorter and a little higher up than a normal minus signs. But on the internet and in newspapers you will see it written using a minus sign.
Sometimes negative numbers are called minus numbers, careful you don't confuse these with subtraction.
Positive numbers
You already use positive numbers all the time! But, unlike negative numbers you don't have to put a + sign in front of them. Here are some examples of positive numbers:
3, 46, 689, 1 982
A positive number is more than zero.
Why do negative numbers 'get bigger'?
As you extend a number line showing negative numbers, they seem to get 'bigger'.
The numbers seem to increase in value as they go down the number line.
But as the negative number gets bigger, the value gets lower. -10 is a larger number than -5, so it is further below zero. If you look at the number line you can see that -10 is less than -5.
If it helps you remember, think about the weather. As the temperature gets lower the negative numbers seem to get bigger.
Freezing point of water
On a weather forecast temperatures are given in degrees Celsius. Sometimes temperatures are called degrees centigrade, which is the same scale.
You can use a short hand for writing temperatures. You use the degree sign ° and C for Celsius. For example you write the temperature 8 degrees Celsius as 8°C.
When water turns to ice it freezes. The temperature water freezes at is zero degrees Celcius, that's 0°C. Here are some other common temperatures:
Water freezes at 0°C
Below and above freezing
Negative temperatures are called below freezing and positive temperatures are above freezing.
Reading a thermometer
A thermometer is something that is used to measure temperature. When you've been to the doctors they might have taken your temperature with a thermometer. Or you might have one at home showing how hot, or cold, it is inside your house.
Thermometers come in all shapes and sizes. The digital ones give the temperature in numbers, such as 5°C.
With other thermometers you have to read off the scale - the numbers along the side
Negative numbers glossary
Here are some words you'll come across when working with negative numbers
A negative number is less than zero.
-100, -25, -12 and -4 are all negative numbers.
A positive number is more than zero.
4, 12, 89 and 568 are all positive numbers.
The negative sign goes in front of a negative number.
Negative 4 is the same as -4.
° is the symbol for degree.
Celsius or C is a scale for measuring temperature.
6 degrees Celsius is the same as 6°C.
Centigrade is another name for Celsius.
A thermometer is something used to measure temperature.
The thermometer showed the temperature as -5°C
Water freezes, turns solid into ice, at 0°C
Negative temperatures are called below freezing.
Positive temperatures are called above freezing.
When there is a temperature increase it gets warmer.
When there is a temperature decrease it gets colder.
--------------------------------------------------------------------------------
Let's look at one example, the weather. On a cold December night you can watch the thermometer as the temperature drops, as the numbers go down:
2 degrees, 1 degree, zero degrees
But what happens to the numbers if it gets even colder? The temperature and the numbers keep going down!
The numbers below zero are called negative numbers
A negative number is less than zero.
We write negative numbers like this:
negative 2 is the same as -2
The dash is the negative sign. It is usually written slightly shorter and a little higher up than a normal minus signs. But on the internet and in newspapers you will see it written using a minus sign.
Sometimes negative numbers are called minus numbers, careful you don't confuse these with subtraction.
Positive numbers
You already use positive numbers all the time! But, unlike negative numbers you don't have to put a + sign in front of them. Here are some examples of positive numbers:
3, 46, 689, 1 982
A positive number is more than zero.
Why do negative numbers 'get bigger'?
As you extend a number line showing negative numbers, they seem to get 'bigger'.
The numbers seem to increase in value as they go down the number line.
But as the negative number gets bigger, the value gets lower. -10 is a larger number than -5, so it is further below zero. If you look at the number line you can see that -10 is less than -5.
If it helps you remember, think about the weather. As the temperature gets lower the negative numbers seem to get bigger.
Freezing point of water
On a weather forecast temperatures are given in degrees Celsius. Sometimes temperatures are called degrees centigrade, which is the same scale.
You can use a short hand for writing temperatures. You use the degree sign ° and C for Celsius. For example you write the temperature 8 degrees Celsius as 8°C.
When water turns to ice it freezes. The temperature water freezes at is zero degrees Celcius, that's 0°C. Here are some other common temperatures:
Water freezes at 0°C
Below and above freezing
Negative temperatures are called below freezing and positive temperatures are above freezing.
Reading a thermometer
A thermometer is something that is used to measure temperature. When you've been to the doctors they might have taken your temperature with a thermometer. Or you might have one at home showing how hot, or cold, it is inside your house.
Thermometers come in all shapes and sizes. The digital ones give the temperature in numbers, such as 5°C.
With other thermometers you have to read off the scale - the numbers along the side
Negative numbers glossary
Here are some words you'll come across when working with negative numbers
A negative number is less than zero.
-100, -25, -12 and -4 are all negative numbers.
A positive number is more than zero.
4, 12, 89 and 568 are all positive numbers.
The negative sign goes in front of a negative number.
Negative 4 is the same as -4.
° is the symbol for degree.
Celsius or C is a scale for measuring temperature.
6 degrees Celsius is the same as 6°C.
Centigrade is another name for Celsius.
A thermometer is something used to measure temperature.
The thermometer showed the temperature as -5°C
Water freezes, turns solid into ice, at 0°C
Negative temperatures are called below freezing.
Positive temperatures are called above freezing.
When there is a temperature increase it gets warmer.
When there is a temperature decrease it gets colder.
--------------------------------------------------------------------------------
Math Dictionary A-Z
abacus
An oriental counting device and calculator.
Abelian group
A group in which the binary operation is commutative, that is, ab=ba for all elements a and b in the group.
abscissa
The x-coordinate of a point in a 2-dimensional coordinate system.
absolute value
The positive value for a real number, disregarding the sign. Written x. For example, 3=3, -4=4, and 0=0.
abundant number
A positive integer that is smaller than the sum of its proper divisors.
acceleration
The rate of change of velocity with respect to time.
acute angle
An angle that is less than 90 degrees
addition
The process of adding two numbers to obtain their sum.
algebraic equation
An equation of the form f(x)=0 where f is a polynomial.
algebraic number
A number that is the root of an algebraic polynomial. For example, sqrt(2) is an algebraic number because it is a solution of the equation x2=2.
alphametic
A cryptarithm in which the letters, which represent distinct digits, form related words or meaningful phrases.
altitude
The altitude of a triangle is the line segment from one vertex that is perpendicular to the opposite side.
amicable numbers
Two numbers are said to be amicable if each is equal to the sum of the proper divisors of the other.
angle
The figure formed by two line segments or rays that extend from a given point.
annulus
The region enclosed by two concentric circles.
arc
A portion of a circle.
area
The amount of surface contained by a figure.
arithmetic
The type of mathematics that studies how to solve problems involving numbers (but no variables).
arithmetic mean
The arithmetic mean of n numbers is the sum of the numbers divided by n.
automorphism
An isomorphism from a set onto itslef.
average
Typically this refers to the arithmetic mean.
ball
A sphere together with its interior.
bar graph
A type of chart used to compare data in which the length of a bar represents the size of the data.
base
In the expression xy, x is called the base and y is the exponent.
Bayes's Rule
A rule for finding conditional probability.
binary number
A number written to base 2.
binary operation
A binary operation is an operation that involves two operands. For example, addition and subtraction are binary operations.
bijection
A one-to-one onto function.
binomial
An expression that is the sum of two terms.
binomial coefficient
The coefficients of x in the expansion of (x+1)n.
biquadratic equation
A polynomial equation of the 4th degree.
bisect
to cut in half.
bit
A binary digit.
braces
The symbols { and } used for grouping or to represent a set.
byte
The amount of memory needed to represent one character on a computer, typically 8 bits.
Calculator
A machine for performing arithemtical calculations.
Caliban puzzle
A logic puzzle in which one is asked to infer one or more facts from a set of given facts.
cardinal number
A number that indicates the quantity but not the order of things.
catenary
A curve whose equation is y = (a/2)(ex/a+e-x/a). A chain suspended from two points forms this curve.
ceiling function
The ceiling function of x is the smallest integer greater than or equal to x.
central angle
An angle between two radii of a circle.
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.
circle
The set of points equidistant from a given point (the center).
circular cone
A cone whose base is a circle.
circumcenter
The circumcenter of a triangle is the center of the circumscribed circle.
circumcircle
The circle circumscribed about a figure.
circumference
The boundary of a circle.
cissoid
A curve with equation y2(a-x)=x3.
coefficient
The constant multipliers of the indeterminate variable in a polynomial. For example, in the polynomial x2+3x+7, the coefficients are 1, 3, and 7.
common denominator
A multiple shared by the denominators of two or more fractions.
complementary angles
Two angles whose sum is 90o.
complex number
The sum of a real number and an imaginary number, for example 3+4i where i=sqrt(-1).
compute
To solve problems that use numbers.
concave
curved from the inside.
cone
A three-dimensional solid that rises froma circular base to a single point at the top.
congruent figures
two geometric figures that are identical in size and shape.
conic section
The cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections.
coordinates
Numbers that determine the position of a point.
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.
cube
A solid figure bounded by 6 congruent squares.
cubic equation
A polynomial equation of degree 3.
cyclic polygon
A polygon whose vertices lie on a circle.
cylinder
A rounded three-dimensional solid that has a flat circular face at each end.
data
Facts that have been collected but not yet interpreted.
decagon
A polygon with 10 sides.
decimal number
A number written to the base 10.
decimal point
The period in a deimal number separating the integer part from the fractional part.
deficient number
A positive integer that is larger than the sum of its proper divisors.
degree
The degree of a term in one variable is the exponent of that variable. For example, the degree of 7x5 is 5.
denominator
In the fraction x/y, x is called the numerator and y is called the denominator.
diagonal
In a polygon, the line segment joining a vertex with another (non-adjacent) vertex is called a diagonal.
diameter
The longest chord of a figure. In a circle, a diameter is a chord that passes through the center of the circle.
difference
The difference between two numbers is what you get when you subtract one from the other.
differential calculus
That part of calculus that deals with the opeation of differentiation of functions.
digimetic
A cryptarithm in which digits represent other digits.
digit
In the decimal system, one of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
dihedral angle
The angle formed by two planes meeting in space.
dimension
The indication of how far something extends in space.
disc
A circle together with its interior.
distributive law
The formula a(x+y)=ax+ay.
dividend
In the expression "a divided by b", a is the divident and b is the divisor.
division
A basic arithmetical operation determining how many times one quantity is contained within another.
divisor
In the expression "a divided by b", a is the divident and b is the divisor.
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.
dodecagon
A polygon with 12 sides.
dodecahedron
A solid figure with 12 faces A regular dodecahedron is a regular polyhedron with 12 faces. Each face is a rgular pentagon.
domain
The domain of a function f(x) is the set of x values for which the function is defined.
domino
Two congruent squares joined along an edge.
duodecimal number system
The system of numeration with base 12.
Egyptian fraction
A number of the form 1/x where x is an integer is called an Egyptian fraction.
eigenvalue
characteristic value
elementary function
one of the functions: rational functions, trigonometric functions, exponential functions, and logarithmic functions.
ellipse
A plane figure whose equation isx2/a2+y2/b2=1.
ellipsoid
A solid figure whose equation is x2/a2+y2/b2+z2/c2=1.
empty set
The set with no elements in it.
enumerable set
A countable set.
equation
A statement that two expressions are equal to each other.
equiangular polygon
A polygon all of whose interior angles are equal.
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.
equilateral polygon
A polygon all of whose sides are equal.
equilateral triangle
A triangle with three equal sides.
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.
estimate
A rough guess at the value of a number.
Euler line
The Euler line of a triangle is the line connecting the centroid and the circumcenter.
Euler's constant
The limit of the series 1/1+1/2+1/3+...+1/n-ln n as n goes to infinity. Its value is approximately 0.577216.
even function
A function f(x) is called an even function if f(x)=f(-x) for all x.
even number
An integer that is divisible by 2.
excenter
The center of an excircle.
excircle
An escribed circle of a triangle.
exponent
In the expression xy, x is called the base and y is called the exponent.
exponential function
The function f(x)=ex.
expoential function to base a
The function f(x)=ax.
exradius
An exradius of a triangle is the radius of an escribed circle.
face angle
The plane angle formed by adjacent edges of a polygonal angle in space.
factor (noun)
An exact divisor of a number. This 7 is a factor of 28.
factor (verb)
To find the factors of a number.
factorial
n! (read n factorial) is equal to the product of the integers from 1 to n.
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.
Fermat's spiral
A parabolic spiral.
Fibonacci number
A member of the sequence 0, 1, 1, 2, 3, 5,... where each number is the sum of the previous two numbers.
figurate numbers
polygonal numbers
finite group
A group containing a finite number of elements.
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 chord
A chord of a conic that passes through a focus.
focal radius
A line segment from the focus of an ellipse to a point on the perimeter of the ellipse.
foot of altitude
The intersection of an altitude of a triangle with the base to which it is drawn.
foot of line
The point of intersection of a line with a line or plane.
formula
A concise statement expressing the symbolic relationship between two or more quantities.
Fourier series
A periodic function with period 2 pi.
fraction
An expression of the form a/b.
frequency
The number of times a value occurs in some time interval.
frustum
For a given solid figure, a related figure formed by two parallel planes meeting the given solid. In particular, for a cone or pyramid, a frustum is determined by the plane of the base and a plane parallel to the base. NOTE: this word is frequently incorrectly misspelled as frustrum.
Gaussian curve
A normal curve.
geoboard
A flat board into which nails have been driven in a regular rectangular pattern. These nails represent the lattice points in the plane.
geodesic
The arc on a surface of shortest length joining two given points.
geodesy
A branch of mathematics dealing with the shape, size, and curvature of the Earth.
geometric mean
The geometric mean of n numbers is the nth root of the product of the numbers.
geometric progression
A sequence in which the ratio of each term to the preceding term is a given constant.
geometric series
A series in which the ratio of each term to the preceding term is a given constant.
geometric solid
The bounding surface of a 3-dimensional portion of space.
geometry
The branch of mathematics that deals with the nature of space and the size, shape, and other properties of figures as well as the transformations that preserve these properties.
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.
grad (or grade)
1/100th of a right angle
graph
A graph is a set of points (called vertices) and a set of lines (called edges) joinging these vertices.
great circle
A circle on the surface of a sphere whose center is the center of the sphere.
greatest common divisor
The greatest common divisor of a sequence of integers, is the largest integer that divides each of them exactly.
greatest common factor
Same as greatest common divisor.
greatest lower bound
The greatest lower bound of a set of real numbers, is the largest real number that is smaller than each of the numbers in the set.
group
A mathematical system consisting of elements from a set G and a binary operation * such that
x*y is a member of G whenever x and y are
(x*y)*z=x*(y*z) for all x, y, and z
there is an identity element e such that e*x=x*e=e for all x
each member x in G has an inverse element y such that x*y=y*x=e
half-line
A ray.
half-plane
The part of a plane that lies on one side of a given line.
Hankel matrix
A matrix in which all the elements are the same along any diagonal that slopes from northeast to southwest.
harmonic analysis
The study of the representation of functions by means of linear operations on characteristic sets of functions.
harmonic division
A line segment is divided harmonically by two points when it is divided externally and internally int he same ratio.
harmonic mean
The harmonic mean of two numbers a and b is 2ab/(a + b).
hectare
A unit of measurement in the metric system equal to 10,000 square meters (approximately 2.47 acres).
helix
The path followed by a point moving on the surface of a right circular cylinder that moves along the cylinder at a constant ratio as it moves around the cylinder. The parameteric equation for a helix is
x=a cos ty=a sin tz=bt
heptagon
A polygon with 7 sides.
hexagon
A polygoin with 6 sides.
hexagonal number
A number of the form n(2n-1).
hexagonal prism
A prism with a hexagonal base.
hexahedron
A polyhedron having 6 faces. The cube is a regular hexahedron.
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.
horizontal line
A line parallel to the earth's surface or the bottom of a page.
hyperbola
A curve with equation x2/a2-y2/b2=1.
hyperbolic spiral
The curve whose equation in polar coordinates is r*theta=a.
hyperboloid
A geometric solid whose equation is x2/a2+y2/b2-z2/c2=1 orx2/a2+y2/b2-z2/c2=-1.
hypotenuse
The longest side of a right triangle.
icosahedron
A polyhedron with 20 faces.
idempotent
The element x in some algebraic structure is called idempotent if x*x=x.
imaginary axis
The y-axis of an Argand diagram.
imaginary number
A complex number of the form xi where x is real and i=sqrt(-1).
imaginary part
The imaginary part of a complex number x+iy where x and y are real is y.
incenter
The incenter of a triangle is the center of its inscribed circle.
incircle
The circle inscribed in a given figure.
inequality
The statement that one quantity is less than (or greater than) another.
infinite
becoming large beyond bound.
infinitesimal
A variable that approaches 0 as a limit.
infinity
A reference to a quantity larger than any specific integer.
inflection
A point of inflection of a plane curve is a point where the curve has a stationary tangent, at which the tangent is changing from rotating in one direction to rotating in the oppostie direction.
injection
A one-to-one mapping.
inscribed angle
The angle formed by two chords of a curve that meet at the same point on the curve.
integer
One of the numbers ..., -3, -2, -1, 0, 1, 2, 3, ...
intersect
Two figures are said to intersect if they meet or cross each other.
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.
isometry
A length preserving map.
isosceles tetrahedron
A tetrahedron in which each pair of opposite sides have the same length.
isosceles triangle
A triangle with two equal sides.
isosceles trapezoid
Ain which the two non-parallel sides have the same length.
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.
joint probability function
A function that gives the probability that each of two or more random variables takes at a particular value.
joint variation
A variation in which the values of one variable depend upon those of 2 or more variables.
Jordan curve
A simple closed curve.
Jordan matrix
A matrix whose diagonal elements are all equal (and nonzero) and whose elements above the principal diagonal are equal to 1, but all other elements are 0.
joule
A unit of energy or work.
jump discontinuity
A discontinuity in a function where the left and righ-hand limits exist but are not equal to each other.
kilometer
A unit of length equal to 1,000 meters.
kinematics
A branch of mechanics dealing with the motion of rigid bodies without reference to their masses or the forces acting on the bodies.
kite
A quadrilateral which has two pairs of adjacent sides equal.
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.
knot
A curve in space formed by interlacing a piece of string and then joining the ends together.
knot
a unit of speed in navigation equal to one nautical mile per hour.
L-tetromino
A tetromino in the shape of the letter L.
latera recta
plural of lattice rectum.
latin square
An n X n array of numbers in which only n numbers appear. No number appears more than once in any row or column.
latitude
The angular distance of a point on the Earth from the equator, measured along the meridian through that point.
lattice point
A point with integer coordinates.
latus rectum
A chord of an ellipse passing through a focus and perpendicular to the major axis of the ellipse.Plural: latera recta.
least common multiple
The least common multiple of a set of integers is the smallest integer that is an exact multiple of every number in the set.
least upper bound
The least upper bound of a set of numbers is the smallest number that is larger than every member of the set.
lemata
plural of lemma.
lemma
A proposition that is useful mainly for the proof of some other theorem.
length
The straight line distance between two points.
Legendre polynomials
line
A geometrical figure that has length but no width.
linear function
A function of the form y=ax+b.
line graph
A chart that shows data by means of points connected by lines.
line segment
The part of a line between two given distinct points on that line (including the two points).
locus
The set of all points meeting some specified condition.
logic
The study of the formal laws of reasoning.
lowest common denominator
The smallest number that is exactly divisible by each denominator of a set of fractions.
loxodrome
On a sphere, a curve that cuts all parallels under the same angle.
lowest common denominator
The smallest multiple shared by the denominators of a set of fractions.
lowest terms
A fraction is said to be in lowest terms if its numerator and denominator have no common factor.
Lucas number
A member of the sequence 2, 1, 3, 4, 7,... where each number is the sum of the previous two numbers. L0=2, L1=1, Ln=Ln-1+Ln-2.
lune
The portion of a sphere between two great semicircles having common endpoints (including the semicircles).
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.
main diagonal
In the matrix [aij], the elements a11, a22, ..., ann.
major axis
The major axis of an ellipse is it's longest chord.
Malfatti circles
Three equal circles that are mutually tangent and each tangent to two sides of a given triangle.
maximum
The largest of a set of values.
matrix
A rectangular array of elements.
mean
Same as average.
medial triangle
The triangle whose vertices are the midpoints of the sides of a given triangle.
median
The median of a triangle is the line from a vertex to the midpoint of the opposite side.
median
When a set of numbers is ordered from smallest to largest, the median number is the one in the middle of the list.
Mersenne number
A number of the form 2p-1 where p is a prime.
Mersenne prime
A Mersenne number that is prime.
midpoint
The point M is the medpoint of line segment AB if AM=MB. That is, M is halfway between A and B.
minor axis
The minor axis of an ellipse is its smallest chord.
minimum
The smallest of a set of values.
mode
The most frequently occurring value in a sequence of numbers.
modulo
The integers a and b are said to be congruent modulo m if a-b is divisible by m.
monomial
An algebraic expression consisting of just one term.
monotone
A sequence is monotone if its terms are increasing or decreasing.
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.
multinomial
An algebraic expression consisting of 2 or more terms.
multiple
The integer b is a multiple of the integer a if there is an integer d such that b=da.
multiplication
The basic arithemtical operation of repeated addition.
nadir
The point on the celestial spehere in the direction downwards of the plumb-line.
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.
natural number
Any one of the numbers 1, 2, 3, 4, 5, ... .
negative number
A number smaller than 0.
nine point center
In a triangle, the circumcenter of the medial triangle is called the nine point center.
nine point circle
In a triangle, the circle that passes through the midpoints of the sides is called the nine point circle.
nomograph
A graphical device used for computation which uses a straight edge and several scales of numbers.
nonagonal number
A number of the form n(7n-5)/2.
nonary
associated with 9
normal
perpendicular
null hypothesis
The null hypothesis is the hypothesis that is being tested in a hypothesis-testing situation.
null set
the empty set
number line
A line on which each point represents a real number.
number theory
The study of integers.
numeral
A symbol that stands for a number.
numerator
In the fraction x/y, x is called the numerator and y is called the denominator.
numerical analysis
The study of methods for approximation of solutions of various classes of mathematical problems including error analysis.
oblate spheroid
An ellipsoid produced by rotating an ellipse through 360o about its minor axis.
oblique angle
an angle that is not 90o
oblique coordinates
A coordinate system in which the axes are not perpendicular.
oblique triangle
A triangle that is not a right triangle.
obtuse angle
an angle larger than 90o but smaller than 180o
obtuse triangle
A triangle that contains an obtuse angle.
octagon
A polygon with 8 sides.
octahedron
A polyhedron with 8 faces.
octant
any one of the 8 portions of space dtermined by the 3 coordinate planes.
odd function
A function f(x) is called an odd function if f(x)=-f(-x) for all x.
odd number
An integer that is not divisible by 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.
open interval
An interval that does not include its two endpoints.
ordered pair
A pair of numbers in which one number is distinguished as the first number and the other as the second number of the pair
ordinal number
A number indicating the order of a thing in a series
ordinate
The y-coordinate of a point in the plane.
origin
The point in a coordinate plane with coordinates (0,0).
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.
paraboloid
A paraboloid of revolution is a surface of revolution produced by rotating a parabola about its axis.
parallel
Two lines in the plane are said to be parallel if they do not meet.
parallelogram
A quadrilateral whose opposite sides are parallel.
parallelepiped
A prism whose bases are parallelograms.
parentheses
The symbols ( and ) used for grouping expressions.
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 recurrenceP0=0, P1=1, and Pn=2Pn-1+Pn-2.
pentagon
A polygon with 5 sides.
pentagonal number
A number of the form n(3n-1)/2.
pentomino
A five-square polyomino.
percent
A way of expressing a number as a fraction of 100.
perfect cube
An integer is a perfect cube if it is of the form m3 where m is an integer.
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
An integer is a perfect power if it is of the form mn where m and n are integers and n>1.
perfect square
An integer is a perfect square if it is of the form m2 where m is an integer.
perimeter
The distance around the edge of a multisided figure.
perpendicular
Two straight lines are said to be perpendicular if they meet at right angles.
pi
The ratio of the circumference of a circle to its diameter.
pie chart
A type of chart in which a circle is divided up into portions in which the area of each portion represents the size of the data.
place value
Within a number, each digit is given a place value depending on it's location within the number.
plane
A two-dimensional area in geometry.
point
In geometry, a point represents a position, but has no size.
polygon
A plane figure with many sides.
polyomino
A planar figure consisting of congruent squares joined edge-to-edge.
positive number
A number larger than 0.
power
A number multiplied by itself a specified number of times.
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 zn=1 but zk is not equal to 1 for any positive integer k less than n.
probability
The chance that a particular event will happen.
product
The result of multiplying two numbers.
pronic number
A number of the form n(n+1).
proper divisor
The integer d is a proper divisor of the integer n if 0 < d < n and d is a divisor of n.
proportion
A comparison of ratios.
pyramid
A three-dimensional solid whose base is a polygon and whose sides are triangles that come to a point at the top.
Pythagorean triangle
A right triangle whose sides are integers.
Pythagorean triple
An ordered set of three positive integers (a,b,c) such that a2+b2=c2.
QED
Abbreviation for quod erat demonstrandum, used to denote the end of a proof.
quadrangle
A closed broken line in the plane consisting of 4 line segments.
quadrangular prism
A prism whose base is a quadrilateral.
quadrangular pyramid
A pyramid whose base is a quadrilateral.
quadrant
Any one of the four portions of the plane into which the plane is divided by the coordinate axes.
quadratfrie
square free
quadratic equation
An equation of the form f(x)=0 where f(x) is a second degree polynomial. That is, ax2+bx+c=0.
quadrature
The quadrature of a geometric figure is the determination of its area.
quadric curve
The graph of a second degree equation in two variables.
quadric surface
The graph of a second degree equation in three variables.
quadrilateral
A geometric figure with four sides.
quadrinomial
An algebraic expression consisting of 4 terms.
quartic polynomial
A polynomial of degree 4.
quartile
The first quartile of a sequence of numbers is the number such that one quuarter of the numbers in the sequence are less than this number.
quintic polynomial
A polynomial of degree 5.
quotient
The result of a division.
radian
A unit of angular measurement such that there are 2 pi radians in a complete circle. One radian = 180/pi degrees. One radian is approximately 57.3o.
radical axis
the locus of points of equal power with respect to two circle.
radical center
The radical center of three circles is the common point of interesection of the radical axes of each pair of circles.
radii
Plural of radius.
radius
The length of a stright line drown from the center of a circle to a point on its circumference.
radix point
The generalization of decimal point to bases of numeration other than base 10.
range
The set of values taken on by a function.
rate
A way of comparing two quantities.
ratio
quotient of two numbers.
rational number
A rational number is a number that is the ratio of two integers. All other real numbers are said to be irrational.
real axis
The x-axis of an Argand diagram.
real part
The real number x is called ther eal part of the complex number x+iy where x and y are real and i=sqrt(-1).
real variable
A variable whose value ranges over the real numbers.
reciprocal
The reciprocal of the number x is the number 1/x.
rectangle
A quadrilateral with 4 right angles.
reflex angle
An angle between 180o and 360o.
remainder
The number left over when one number is divided by another.
repdigit
An integer all of whose digits are the same.
repeating decimal
A decimal whose digits eventually repeat.
repunit
An integer consisting only of 1's.
rhombus
A parallelogram with four equal sides.
right angle
an angle formed by two perpendicular lines; a 90o angle.
right triangle
A triangle that contains a right angle.
roman numerals
A system of numeration used by the ancient Romans.
root of unity
A solution of the equation xn=1, where n is a positive integer.
round-off error
The error accumulated during a calculation due to rounding intermediate results.
rounding
The process of approximating a number to a nearby one.
ruled surface
A surface formed by moving a straight line (called the generator).
rusty compass
A pair of compasses that are fixed open in a given position.
scalene triangle
A triangle with unequal sides.
secant
A straight lien that meets a curve in two or more points.
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).
sequence
A collection of numbers in a prescribed order: a1, a2, a3, a4, ...
series
The sum of a finite or infinite sequence
set
A collection of objects.
similar figures
Two geometric figures are similar if their sides are in proportion and all their angles are the same.
skeleton division
A long division in which most or all of the digits have been replaced by asterisks to form a cryptarithm.
slide rule
A calculating device consisting of two sliding logarithmic scales.
solid
A three-dimensional figure.
solid of revolution
A solid formed by rotation a plane figure about an axis in three-space.
solidus
The slanted line in a fraction such as a/b dividing the numerator from the denominator.
sphere
The locus of pointsin three-space that are a fixed distance froma given point (called the center).
spherical trigonometry
The branch of mathematics dealing with measurements on the sphere.
square
A quadrilateral with 4 equal sides and 4 right angles.
square free
An integer is said to be square free if it is not divisible by a perfect square, n2, for n>1.
square number
A number of the form n2.
square root
The number x is said to be a square root of y if x2 = y.
Stirling numbers
subtraction
A basic operation of arithemtic in which you take away one number from another.
sum
The result of adding two or more numbers.
supplementary
Two angels are supplementary of they add up to 180o.
surface area
The measure of a surface of a three-dimensional solid indicating how large it is.
symmedian
Reflection of a median of a triangle about the corresponding angle bisector.
tangent
A line that meets a smooth curve at a single point and does not cut across the curve.
tautology
A sentence that is true because of its logical structure.
tetrahedron
A polyhedron with four faces.
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.
torus
A geometric solid in the shape of a donut.
trace
The trace of a matrix is the sum of the terms along the principal diagonal.
transcendental number
A number that is not algebraic.
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.
trinomial
An algebraic expression consisting of 3 terms.
tromino
A three-square polyomino.
truncated pyramid
A section of a pyramid between its base and a plane parallel to the base.
twin primes
Two prime numbers that differ by 2. For example, 11 and 13 are twin primes.
unilateral surface
A surface with only one side, such as a Moebius strip.
unimodal
A finite sequence is unimodal if it first increases and then decreases.
unimodular
A square matrix is unimodular if its determinant is 1.
unit circle
A unit circle is a circle with radius 1.
unit cube
A cube with edge length 1.
unit fraction
A fraction whose numerator is 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.
unity
one
variable
A symbol whose value can change.
velocity
The rate of change of position.
vertical line
A line that runs up and down and is perpendicular to a horizontal line.
vigesimal
related to intervals of 20.
vinculum
The horizontal bar in a fraction separating the numerator from the denominator.
volume
The measure of spce occupied by a solid body.
vulgar fraction
A common fraction.
weak inequality
An inequality that permits the equality case. For example, a is less than or equal to b.
wff
A well-formed formula.
whole number
A natural number.
winding number
The number of times a closed curve in the plane passes around a given point in the counterclockwise direction.
witch of Agnesi
A curve whose equation is x2y=4a2(2a-y).
X
Roman numeral for 10.
x-axis
The horizontal axis in the plane.
x-intercept
The point at which a line crosses the x-axis.
X-pentomino
A pentomino in the shape of the letter X.
y-axis
The vertical axis in the plane.
y-intercept
The point at which a line crosses the y-axis.
yard
A measure of length equal to 3 feet.
year
A measure of time equal to the period of one revolution of the earth about the sun. Approximately equal to 365 days.
z-intercept
The point at which a line crosses the z-axis.
zero
0
zero divisors
Nonzero elements of a ring whose product is 0.
zero element
The element 0 is a zero element of a group if a+0=a and 0+a=a for all elements a.
zeta function
zone
The portion of a sphere between two parallel planes.
An oriental counting device and calculator.
Abelian group
A group in which the binary operation is commutative, that is, ab=ba for all elements a and b in the group.
abscissa
The x-coordinate of a point in a 2-dimensional coordinate system.
absolute value
The positive value for a real number, disregarding the sign. Written x. For example, 3=3, -4=4, and 0=0.
abundant number
A positive integer that is smaller than the sum of its proper divisors.
acceleration
The rate of change of velocity with respect to time.
acute angle
An angle that is less than 90 degrees
addition
The process of adding two numbers to obtain their sum.
algebraic equation
An equation of the form f(x)=0 where f is a polynomial.
algebraic number
A number that is the root of an algebraic polynomial. For example, sqrt(2) is an algebraic number because it is a solution of the equation x2=2.
alphametic
A cryptarithm in which the letters, which represent distinct digits, form related words or meaningful phrases.
altitude
The altitude of a triangle is the line segment from one vertex that is perpendicular to the opposite side.
amicable numbers
Two numbers are said to be amicable if each is equal to the sum of the proper divisors of the other.
angle
The figure formed by two line segments or rays that extend from a given point.
annulus
The region enclosed by two concentric circles.
arc
A portion of a circle.
area
The amount of surface contained by a figure.
arithmetic
The type of mathematics that studies how to solve problems involving numbers (but no variables).
arithmetic mean
The arithmetic mean of n numbers is the sum of the numbers divided by n.
automorphism
An isomorphism from a set onto itslef.
average
Typically this refers to the arithmetic mean.
ball
A sphere together with its interior.
bar graph
A type of chart used to compare data in which the length of a bar represents the size of the data.
base
In the expression xy, x is called the base and y is the exponent.
Bayes's Rule
A rule for finding conditional probability.
binary number
A number written to base 2.
binary operation
A binary operation is an operation that involves two operands. For example, addition and subtraction are binary operations.
bijection
A one-to-one onto function.
binomial
An expression that is the sum of two terms.
binomial coefficient
The coefficients of x in the expansion of (x+1)n.
biquadratic equation
A polynomial equation of the 4th degree.
bisect
to cut in half.
bit
A binary digit.
braces
The symbols { and } used for grouping or to represent a set.
byte
The amount of memory needed to represent one character on a computer, typically 8 bits.
Calculator
A machine for performing arithemtical calculations.
Caliban puzzle
A logic puzzle in which one is asked to infer one or more facts from a set of given facts.
cardinal number
A number that indicates the quantity but not the order of things.
catenary
A curve whose equation is y = (a/2)(ex/a+e-x/a). A chain suspended from two points forms this curve.
ceiling function
The ceiling function of x is the smallest integer greater than or equal to x.
central angle
An angle between two radii of a circle.
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.
circle
The set of points equidistant from a given point (the center).
circular cone
A cone whose base is a circle.
circumcenter
The circumcenter of a triangle is the center of the circumscribed circle.
circumcircle
The circle circumscribed about a figure.
circumference
The boundary of a circle.
cissoid
A curve with equation y2(a-x)=x3.
coefficient
The constant multipliers of the indeterminate variable in a polynomial. For example, in the polynomial x2+3x+7, the coefficients are 1, 3, and 7.
common denominator
A multiple shared by the denominators of two or more fractions.
complementary angles
Two angles whose sum is 90o.
complex number
The sum of a real number and an imaginary number, for example 3+4i where i=sqrt(-1).
compute
To solve problems that use numbers.
concave
curved from the inside.
cone
A three-dimensional solid that rises froma circular base to a single point at the top.
congruent figures
two geometric figures that are identical in size and shape.
conic section
The cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections.
coordinates
Numbers that determine the position of a point.
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.
cube
A solid figure bounded by 6 congruent squares.
cubic equation
A polynomial equation of degree 3.
cyclic polygon
A polygon whose vertices lie on a circle.
cylinder
A rounded three-dimensional solid that has a flat circular face at each end.
data
Facts that have been collected but not yet interpreted.
decagon
A polygon with 10 sides.
decimal number
A number written to the base 10.
decimal point
The period in a deimal number separating the integer part from the fractional part.
deficient number
A positive integer that is larger than the sum of its proper divisors.
degree
The degree of a term in one variable is the exponent of that variable. For example, the degree of 7x5 is 5.
denominator
In the fraction x/y, x is called the numerator and y is called the denominator.
diagonal
In a polygon, the line segment joining a vertex with another (non-adjacent) vertex is called a diagonal.
diameter
The longest chord of a figure. In a circle, a diameter is a chord that passes through the center of the circle.
difference
The difference between two numbers is what you get when you subtract one from the other.
differential calculus
That part of calculus that deals with the opeation of differentiation of functions.
digimetic
A cryptarithm in which digits represent other digits.
digit
In the decimal system, one of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
dihedral angle
The angle formed by two planes meeting in space.
dimension
The indication of how far something extends in space.
disc
A circle together with its interior.
distributive law
The formula a(x+y)=ax+ay.
dividend
In the expression "a divided by b", a is the divident and b is the divisor.
division
A basic arithmetical operation determining how many times one quantity is contained within another.
divisor
In the expression "a divided by b", a is the divident and b is the divisor.
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.
dodecagon
A polygon with 12 sides.
dodecahedron
A solid figure with 12 faces A regular dodecahedron is a regular polyhedron with 12 faces. Each face is a rgular pentagon.
domain
The domain of a function f(x) is the set of x values for which the function is defined.
domino
Two congruent squares joined along an edge.
duodecimal number system
The system of numeration with base 12.
Egyptian fraction
A number of the form 1/x where x is an integer is called an Egyptian fraction.
eigenvalue
characteristic value
elementary function
one of the functions: rational functions, trigonometric functions, exponential functions, and logarithmic functions.
ellipse
A plane figure whose equation isx2/a2+y2/b2=1.
ellipsoid
A solid figure whose equation is x2/a2+y2/b2+z2/c2=1.
empty set
The set with no elements in it.
enumerable set
A countable set.
equation
A statement that two expressions are equal to each other.
equiangular polygon
A polygon all of whose interior angles are equal.
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.
equilateral polygon
A polygon all of whose sides are equal.
equilateral triangle
A triangle with three equal sides.
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.
estimate
A rough guess at the value of a number.
Euler line
The Euler line of a triangle is the line connecting the centroid and the circumcenter.
Euler's constant
The limit of the series 1/1+1/2+1/3+...+1/n-ln n as n goes to infinity. Its value is approximately 0.577216.
even function
A function f(x) is called an even function if f(x)=f(-x) for all x.
even number
An integer that is divisible by 2.
excenter
The center of an excircle.
excircle
An escribed circle of a triangle.
exponent
In the expression xy, x is called the base and y is called the exponent.
exponential function
The function f(x)=ex.
expoential function to base a
The function f(x)=ax.
exradius
An exradius of a triangle is the radius of an escribed circle.
face angle
The plane angle formed by adjacent edges of a polygonal angle in space.
factor (noun)
An exact divisor of a number. This 7 is a factor of 28.
factor (verb)
To find the factors of a number.
factorial
n! (read n factorial) is equal to the product of the integers from 1 to n.
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.
Fermat's spiral
A parabolic spiral.
Fibonacci number
A member of the sequence 0, 1, 1, 2, 3, 5,... where each number is the sum of the previous two numbers.
figurate numbers
polygonal numbers
finite group
A group containing a finite number of elements.
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 chord
A chord of a conic that passes through a focus.
focal radius
A line segment from the focus of an ellipse to a point on the perimeter of the ellipse.
foot of altitude
The intersection of an altitude of a triangle with the base to which it is drawn.
foot of line
The point of intersection of a line with a line or plane.
formula
A concise statement expressing the symbolic relationship between two or more quantities.
Fourier series
A periodic function with period 2 pi.
fraction
An expression of the form a/b.
frequency
The number of times a value occurs in some time interval.
frustum
For a given solid figure, a related figure formed by two parallel planes meeting the given solid. In particular, for a cone or pyramid, a frustum is determined by the plane of the base and a plane parallel to the base. NOTE: this word is frequently incorrectly misspelled as frustrum.
Gaussian curve
A normal curve.
geoboard
A flat board into which nails have been driven in a regular rectangular pattern. These nails represent the lattice points in the plane.
geodesic
The arc on a surface of shortest length joining two given points.
geodesy
A branch of mathematics dealing with the shape, size, and curvature of the Earth.
geometric mean
The geometric mean of n numbers is the nth root of the product of the numbers.
geometric progression
A sequence in which the ratio of each term to the preceding term is a given constant.
geometric series
A series in which the ratio of each term to the preceding term is a given constant.
geometric solid
The bounding surface of a 3-dimensional portion of space.
geometry
The branch of mathematics that deals with the nature of space and the size, shape, and other properties of figures as well as the transformations that preserve these properties.
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.
grad (or grade)
1/100th of a right angle
graph
A graph is a set of points (called vertices) and a set of lines (called edges) joinging these vertices.
great circle
A circle on the surface of a sphere whose center is the center of the sphere.
greatest common divisor
The greatest common divisor of a sequence of integers, is the largest integer that divides each of them exactly.
greatest common factor
Same as greatest common divisor.
greatest lower bound
The greatest lower bound of a set of real numbers, is the largest real number that is smaller than each of the numbers in the set.
group
A mathematical system consisting of elements from a set G and a binary operation * such that
x*y is a member of G whenever x and y are
(x*y)*z=x*(y*z) for all x, y, and z
there is an identity element e such that e*x=x*e=e for all x
each member x in G has an inverse element y such that x*y=y*x=e
half-line
A ray.
half-plane
The part of a plane that lies on one side of a given line.
Hankel matrix
A matrix in which all the elements are the same along any diagonal that slopes from northeast to southwest.
harmonic analysis
The study of the representation of functions by means of linear operations on characteristic sets of functions.
harmonic division
A line segment is divided harmonically by two points when it is divided externally and internally int he same ratio.
harmonic mean
The harmonic mean of two numbers a and b is 2ab/(a + b).
hectare
A unit of measurement in the metric system equal to 10,000 square meters (approximately 2.47 acres).
helix
The path followed by a point moving on the surface of a right circular cylinder that moves along the cylinder at a constant ratio as it moves around the cylinder. The parameteric equation for a helix is
x=a cos ty=a sin tz=bt
heptagon
A polygon with 7 sides.
hexagon
A polygoin with 6 sides.
hexagonal number
A number of the form n(2n-1).
hexagonal prism
A prism with a hexagonal base.
hexahedron
A polyhedron having 6 faces. The cube is a regular hexahedron.
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.
horizontal line
A line parallel to the earth's surface or the bottom of a page.
hyperbola
A curve with equation x2/a2-y2/b2=1.
hyperbolic spiral
The curve whose equation in polar coordinates is r*theta=a.
hyperboloid
A geometric solid whose equation is x2/a2+y2/b2-z2/c2=1 orx2/a2+y2/b2-z2/c2=-1.
hypotenuse
The longest side of a right triangle.
icosahedron
A polyhedron with 20 faces.
idempotent
The element x in some algebraic structure is called idempotent if x*x=x.
imaginary axis
The y-axis of an Argand diagram.
imaginary number
A complex number of the form xi where x is real and i=sqrt(-1).
imaginary part
The imaginary part of a complex number x+iy where x and y are real is y.
incenter
The incenter of a triangle is the center of its inscribed circle.
incircle
The circle inscribed in a given figure.
inequality
The statement that one quantity is less than (or greater than) another.
infinite
becoming large beyond bound.
infinitesimal
A variable that approaches 0 as a limit.
infinity
A reference to a quantity larger than any specific integer.
inflection
A point of inflection of a plane curve is a point where the curve has a stationary tangent, at which the tangent is changing from rotating in one direction to rotating in the oppostie direction.
injection
A one-to-one mapping.
inscribed angle
The angle formed by two chords of a curve that meet at the same point on the curve.
integer
One of the numbers ..., -3, -2, -1, 0, 1, 2, 3, ...
intersect
Two figures are said to intersect if they meet or cross each other.
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.
isometry
A length preserving map.
isosceles tetrahedron
A tetrahedron in which each pair of opposite sides have the same length.
isosceles triangle
A triangle with two equal sides.
isosceles trapezoid
Ain which the two non-parallel sides have the same length.
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.
joint probability function
A function that gives the probability that each of two or more random variables takes at a particular value.
joint variation
A variation in which the values of one variable depend upon those of 2 or more variables.
Jordan curve
A simple closed curve.
Jordan matrix
A matrix whose diagonal elements are all equal (and nonzero) and whose elements above the principal diagonal are equal to 1, but all other elements are 0.
joule
A unit of energy or work.
jump discontinuity
A discontinuity in a function where the left and righ-hand limits exist but are not equal to each other.
kilometer
A unit of length equal to 1,000 meters.
kinematics
A branch of mechanics dealing with the motion of rigid bodies without reference to their masses or the forces acting on the bodies.
kite
A quadrilateral which has two pairs of adjacent sides equal.
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.
knot
A curve in space formed by interlacing a piece of string and then joining the ends together.
knot
a unit of speed in navigation equal to one nautical mile per hour.
L-tetromino
A tetromino in the shape of the letter L.
latera recta
plural of lattice rectum.
latin square
An n X n array of numbers in which only n numbers appear. No number appears more than once in any row or column.
latitude
The angular distance of a point on the Earth from the equator, measured along the meridian through that point.
lattice point
A point with integer coordinates.
latus rectum
A chord of an ellipse passing through a focus and perpendicular to the major axis of the ellipse.Plural: latera recta.
least common multiple
The least common multiple of a set of integers is the smallest integer that is an exact multiple of every number in the set.
least upper bound
The least upper bound of a set of numbers is the smallest number that is larger than every member of the set.
lemata
plural of lemma.
lemma
A proposition that is useful mainly for the proof of some other theorem.
length
The straight line distance between two points.
Legendre polynomials
line
A geometrical figure that has length but no width.
linear function
A function of the form y=ax+b.
line graph
A chart that shows data by means of points connected by lines.
line segment
The part of a line between two given distinct points on that line (including the two points).
locus
The set of all points meeting some specified condition.
logic
The study of the formal laws of reasoning.
lowest common denominator
The smallest number that is exactly divisible by each denominator of a set of fractions.
loxodrome
On a sphere, a curve that cuts all parallels under the same angle.
lowest common denominator
The smallest multiple shared by the denominators of a set of fractions.
lowest terms
A fraction is said to be in lowest terms if its numerator and denominator have no common factor.
Lucas number
A member of the sequence 2, 1, 3, 4, 7,... where each number is the sum of the previous two numbers. L0=2, L1=1, Ln=Ln-1+Ln-2.
lune
The portion of a sphere between two great semicircles having common endpoints (including the semicircles).
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.
main diagonal
In the matrix [aij], the elements a11, a22, ..., ann.
major axis
The major axis of an ellipse is it's longest chord.
Malfatti circles
Three equal circles that are mutually tangent and each tangent to two sides of a given triangle.
maximum
The largest of a set of values.
matrix
A rectangular array of elements.
mean
Same as average.
medial triangle
The triangle whose vertices are the midpoints of the sides of a given triangle.
median
The median of a triangle is the line from a vertex to the midpoint of the opposite side.
median
When a set of numbers is ordered from smallest to largest, the median number is the one in the middle of the list.
Mersenne number
A number of the form 2p-1 where p is a prime.
Mersenne prime
A Mersenne number that is prime.
midpoint
The point M is the medpoint of line segment AB if AM=MB. That is, M is halfway between A and B.
minor axis
The minor axis of an ellipse is its smallest chord.
minimum
The smallest of a set of values.
mode
The most frequently occurring value in a sequence of numbers.
modulo
The integers a and b are said to be congruent modulo m if a-b is divisible by m.
monomial
An algebraic expression consisting of just one term.
monotone
A sequence is monotone if its terms are increasing or decreasing.
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.
multinomial
An algebraic expression consisting of 2 or more terms.
multiple
The integer b is a multiple of the integer a if there is an integer d such that b=da.
multiplication
The basic arithemtical operation of repeated addition.
nadir
The point on the celestial spehere in the direction downwards of the plumb-line.
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.
natural number
Any one of the numbers 1, 2, 3, 4, 5, ... .
negative number
A number smaller than 0.
nine point center
In a triangle, the circumcenter of the medial triangle is called the nine point center.
nine point circle
In a triangle, the circle that passes through the midpoints of the sides is called the nine point circle.
nomograph
A graphical device used for computation which uses a straight edge and several scales of numbers.
nonagonal number
A number of the form n(7n-5)/2.
nonary
associated with 9
normal
perpendicular
null hypothesis
The null hypothesis is the hypothesis that is being tested in a hypothesis-testing situation.
null set
the empty set
number line
A line on which each point represents a real number.
number theory
The study of integers.
numeral
A symbol that stands for a number.
numerator
In the fraction x/y, x is called the numerator and y is called the denominator.
numerical analysis
The study of methods for approximation of solutions of various classes of mathematical problems including error analysis.
oblate spheroid
An ellipsoid produced by rotating an ellipse through 360o about its minor axis.
oblique angle
an angle that is not 90o
oblique coordinates
A coordinate system in which the axes are not perpendicular.
oblique triangle
A triangle that is not a right triangle.
obtuse angle
an angle larger than 90o but smaller than 180o
obtuse triangle
A triangle that contains an obtuse angle.
octagon
A polygon with 8 sides.
octahedron
A polyhedron with 8 faces.
octant
any one of the 8 portions of space dtermined by the 3 coordinate planes.
odd function
A function f(x) is called an odd function if f(x)=-f(-x) for all x.
odd number
An integer that is not divisible by 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.
open interval
An interval that does not include its two endpoints.
ordered pair
A pair of numbers in which one number is distinguished as the first number and the other as the second number of the pair
ordinal number
A number indicating the order of a thing in a series
ordinate
The y-coordinate of a point in the plane.
origin
The point in a coordinate plane with coordinates (0,0).
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.
paraboloid
A paraboloid of revolution is a surface of revolution produced by rotating a parabola about its axis.
parallel
Two lines in the plane are said to be parallel if they do not meet.
parallelogram
A quadrilateral whose opposite sides are parallel.
parallelepiped
A prism whose bases are parallelograms.
parentheses
The symbols ( and ) used for grouping expressions.
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 recurrenceP0=0, P1=1, and Pn=2Pn-1+Pn-2.
pentagon
A polygon with 5 sides.
pentagonal number
A number of the form n(3n-1)/2.
pentomino
A five-square polyomino.
percent
A way of expressing a number as a fraction of 100.
perfect cube
An integer is a perfect cube if it is of the form m3 where m is an integer.
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
An integer is a perfect power if it is of the form mn where m and n are integers and n>1.
perfect square
An integer is a perfect square if it is of the form m2 where m is an integer.
perimeter
The distance around the edge of a multisided figure.
perpendicular
Two straight lines are said to be perpendicular if they meet at right angles.
pi
The ratio of the circumference of a circle to its diameter.
pie chart
A type of chart in which a circle is divided up into portions in which the area of each portion represents the size of the data.
place value
Within a number, each digit is given a place value depending on it's location within the number.
plane
A two-dimensional area in geometry.
point
In geometry, a point represents a position, but has no size.
polygon
A plane figure with many sides.
polyomino
A planar figure consisting of congruent squares joined edge-to-edge.
positive number
A number larger than 0.
power
A number multiplied by itself a specified number of times.
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 zn=1 but zk is not equal to 1 for any positive integer k less than n.
probability
The chance that a particular event will happen.
product
The result of multiplying two numbers.
pronic number
A number of the form n(n+1).
proper divisor
The integer d is a proper divisor of the integer n if 0 < d < n and d is a divisor of n.
proportion
A comparison of ratios.
pyramid
A three-dimensional solid whose base is a polygon and whose sides are triangles that come to a point at the top.
Pythagorean triangle
A right triangle whose sides are integers.
Pythagorean triple
An ordered set of three positive integers (a,b,c) such that a2+b2=c2.
QED
Abbreviation for quod erat demonstrandum, used to denote the end of a proof.
quadrangle
A closed broken line in the plane consisting of 4 line segments.
quadrangular prism
A prism whose base is a quadrilateral.
quadrangular pyramid
A pyramid whose base is a quadrilateral.
quadrant
Any one of the four portions of the plane into which the plane is divided by the coordinate axes.
quadratfrie
square free
quadratic equation
An equation of the form f(x)=0 where f(x) is a second degree polynomial. That is, ax2+bx+c=0.
quadrature
The quadrature of a geometric figure is the determination of its area.
quadric curve
The graph of a second degree equation in two variables.
quadric surface
The graph of a second degree equation in three variables.
quadrilateral
A geometric figure with four sides.
quadrinomial
An algebraic expression consisting of 4 terms.
quartic polynomial
A polynomial of degree 4.
quartile
The first quartile of a sequence of numbers is the number such that one quuarter of the numbers in the sequence are less than this number.
quintic polynomial
A polynomial of degree 5.
quotient
The result of a division.
radian
A unit of angular measurement such that there are 2 pi radians in a complete circle. One radian = 180/pi degrees. One radian is approximately 57.3o.
radical axis
the locus of points of equal power with respect to two circle.
radical center
The radical center of three circles is the common point of interesection of the radical axes of each pair of circles.
radii
Plural of radius.
radius
The length of a stright line drown from the center of a circle to a point on its circumference.
radix point
The generalization of decimal point to bases of numeration other than base 10.
range
The set of values taken on by a function.
rate
A way of comparing two quantities.
ratio
quotient of two numbers.
rational number
A rational number is a number that is the ratio of two integers. All other real numbers are said to be irrational.
real axis
The x-axis of an Argand diagram.
real part
The real number x is called ther eal part of the complex number x+iy where x and y are real and i=sqrt(-1).
real variable
A variable whose value ranges over the real numbers.
reciprocal
The reciprocal of the number x is the number 1/x.
rectangle
A quadrilateral with 4 right angles.
reflex angle
An angle between 180o and 360o.
remainder
The number left over when one number is divided by another.
repdigit
An integer all of whose digits are the same.
repeating decimal
A decimal whose digits eventually repeat.
repunit
An integer consisting only of 1's.
rhombus
A parallelogram with four equal sides.
right angle
an angle formed by two perpendicular lines; a 90o angle.
right triangle
A triangle that contains a right angle.
roman numerals
A system of numeration used by the ancient Romans.
root of unity
A solution of the equation xn=1, where n is a positive integer.
round-off error
The error accumulated during a calculation due to rounding intermediate results.
rounding
The process of approximating a number to a nearby one.
ruled surface
A surface formed by moving a straight line (called the generator).
rusty compass
A pair of compasses that are fixed open in a given position.
scalene triangle
A triangle with unequal sides.
secant
A straight lien that meets a curve in two or more points.
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).
sequence
A collection of numbers in a prescribed order: a1, a2, a3, a4, ...
series
The sum of a finite or infinite sequence
set
A collection of objects.
similar figures
Two geometric figures are similar if their sides are in proportion and all their angles are the same.
skeleton division
A long division in which most or all of the digits have been replaced by asterisks to form a cryptarithm.
slide rule
A calculating device consisting of two sliding logarithmic scales.
solid
A three-dimensional figure.
solid of revolution
A solid formed by rotation a plane figure about an axis in three-space.
solidus
The slanted line in a fraction such as a/b dividing the numerator from the denominator.
sphere
The locus of pointsin three-space that are a fixed distance froma given point (called the center).
spherical trigonometry
The branch of mathematics dealing with measurements on the sphere.
square
A quadrilateral with 4 equal sides and 4 right angles.
square free
An integer is said to be square free if it is not divisible by a perfect square, n2, for n>1.
square number
A number of the form n2.
square root
The number x is said to be a square root of y if x2 = y.
Stirling numbers
subtraction
A basic operation of arithemtic in which you take away one number from another.
sum
The result of adding two or more numbers.
supplementary
Two angels are supplementary of they add up to 180o.
surface area
The measure of a surface of a three-dimensional solid indicating how large it is.
symmedian
Reflection of a median of a triangle about the corresponding angle bisector.
tangent
A line that meets a smooth curve at a single point and does not cut across the curve.
tautology
A sentence that is true because of its logical structure.
tetrahedron
A polyhedron with four faces.
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.
torus
A geometric solid in the shape of a donut.
trace
The trace of a matrix is the sum of the terms along the principal diagonal.
transcendental number
A number that is not algebraic.
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.
trinomial
An algebraic expression consisting of 3 terms.
tromino
A three-square polyomino.
truncated pyramid
A section of a pyramid between its base and a plane parallel to the base.
twin primes
Two prime numbers that differ by 2. For example, 11 and 13 are twin primes.
unilateral surface
A surface with only one side, such as a Moebius strip.
unimodal
A finite sequence is unimodal if it first increases and then decreases.
unimodular
A square matrix is unimodular if its determinant is 1.
unit circle
A unit circle is a circle with radius 1.
unit cube
A cube with edge length 1.
unit fraction
A fraction whose numerator is 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.
unity
one
variable
A symbol whose value can change.
velocity
The rate of change of position.
vertical line
A line that runs up and down and is perpendicular to a horizontal line.
vigesimal
related to intervals of 20.
vinculum
The horizontal bar in a fraction separating the numerator from the denominator.
volume
The measure of spce occupied by a solid body.
vulgar fraction
A common fraction.
weak inequality
An inequality that permits the equality case. For example, a is less than or equal to b.
wff
A well-formed formula.
whole number
A natural number.
winding number
The number of times a closed curve in the plane passes around a given point in the counterclockwise direction.
witch of Agnesi
A curve whose equation is x2y=4a2(2a-y).
X
Roman numeral for 10.
x-axis
The horizontal axis in the plane.
x-intercept
The point at which a line crosses the x-axis.
X-pentomino
A pentomino in the shape of the letter X.
y-axis
The vertical axis in the plane.
y-intercept
The point at which a line crosses the y-axis.
yard
A measure of length equal to 3 feet.
year
A measure of time equal to the period of one revolution of the earth about the sun. Approximately equal to 365 days.
z-intercept
The point at which a line crosses the z-axis.
zero
0
zero divisors
Nonzero elements of a ring whose product is 0.
zero element
The element 0 is a zero element of a group if a+0=a and 0+a=a for all elements a.
zeta function
zone
The portion of a sphere between two parallel planes.
Numbers A-Z
Abundant Numbers
An abundant number has the sum of its proper divisors greater than the number itself. As an example 24 is an abundant number because its proper divisors are {1, 2, 3, 4, 6, 8, 12} and 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 .
Amicable Pairs
An amicable pair are 2 numbers, if the sum of the proper divisors of the first number is the second number and if the sum of the proper divisors of the second number, is the first number.
An example of amicable pair is 2620 and 2924.
The proper divisors of 2620 are {1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310}.
The sum of proper divisorsof 2620 is 1 + 2 + 4 + 5 + 10 + 20 + 131 + 262 + 524 + 655 + 1310 = 2924.
The proper divisors of 2924 are {1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462}.
The sum pf proper divisors of 2924 is 1 + 2 + 4 + 17 + 34 + 43 + 68 + 86 + 172 + 731 + 1462 = 2620.
Thus 2620 and 2924 are amicable.
Cube Numbers
Cube numbers are the result of multiplying a number by itself twice:
1 to the 3rd power = 1,
2 to the 3rd power = 8,
3 to the 3rd power = 27,
4 to the 3rd power =64 and so on...
The cube of 4 is 64, so the cube root of 64 is 4.
To illustrate, a cube number is the number of small cube-shaped blocks we need to build a larger cube.
As an example in order to build a 10-inch cube we would need 1000 small 1-inch cubeblocks, the So 1000 is a cube of 10.
Cute Numbers
If a square can be cut into n squares of at the most two different sizes, then n is called a cute number. For example, 4 and 10 are cute numbers.
Deficient Numbers
If the sum of a number's proper divisors is less than the original number, it is called a deficient number. For instance, 16 is deficient. The proper divisors of 16 are {1, 2, 4, 8}, but 1 + 2 + 4 + 8 = 15.
Figurate Numbers
The number of dots in an arrangement of equally spaced points is a figurate number. Here's an example: 1 3 5 7
* * * *
** * *
*** *
****
The points can be arranged in one, two, three, or even more dimensions. There are many different kinds of figurate numbers, such as polygonal and tetrahedral numbers.
Happy Numbers
A happy number is a number for which the sum of the squares of the digits eventually equals 1. For instance, 203 is happy:
2^2 + 0^2 + 3^2 = 13
1^2 + 3^2 = 10
1^2 + 0^2 = 1.
Numbers that are not happy, such as 16, are called unhappy numbers.
Narcissistic Numbers
A narcissistic person is only interested in himself; a narcissistic number might seem a little self-centered, too. A narcissistic number is an integer equal to an expression that uses the same digits. For example, 36 = 3! * 6. Sometimes a narcissistic number is defined as a number equal to the sum of its digits raised to a certain power, or, more specifically, as an n-digit number equal to the sum of its digits raised to the nth power. For instance, 371 is narcissistic because 3^3 + 7^3 + 1^3 = 371, and 9474 is narcissistic because 9^4 + 4^4 + 7^4 + 4^4 = 9474.
Palindromic Numbers
A palindrome is a word that's the same read either forward or backward, such as noon or kayak. Palindromic numbers, like 88 and 1540451, have the same digits forward and backward.
There's a simple way to turn most numbers into palindromic numbers:
Pick a number:Reverse its digits:Add them together:Repeat the process until you get a palindromic number.
19+ 91 110+011 121
Nobody knows whether or not this works for every number. People have used computers to try the flip-and-add process on 196 nearly ten million times, without finding a palindrome-- but it might still be possible. We do know that it won't work for every number written in every base: try 10110 in base 2.
At 8:02 P.M. on Wednesday, February 20th, 2002, time (for sixty seconds only) read in perfect symmetry:
20:02, 20/02, 2002 (200,220,022,002)
It will happen again at 9:12 P.M. on Dec. 21, 2112: 21:12, 21/12, 2112 (211,221,122,112).
Perfect Numbers
The numbers that divide evenly into an integer are called its divisors. For example, the divisors of 6 are {1, 2, 3, 6}. Proper divisors are the divisors less than the integer you started with: the proper divisors of 6 are {1, 2, 3}. A number is perfect if it is equal to the sum of its proper divisors. 6 is perfect, because 1 + 2 + 3 = 6.
Polygonal Numbers
A polygonal number is the number of equally spaced dots needed to draw a polygon. (A polygonal number is a special type of figurate number.) Sequences of polygonal numbers are based on nested polygons. Here's one example:1 6 15
** ***
* * * ** *
** * * *
** *
***
There are many different kinds of polygonal numbers, beginning with square and triangular numbers.
Proper Divisors
The divisors of an integer are the numbers that it can be divided by without leaving a remainder. For instance, the divisors of 12 are {1, 2, 3, 4, 6, 12}. (Divisors are also called factors.) The proper divisors of a positive integer are all of the divisors less than the integer you started with. Thus, the proper divisors of 12 are {1, 2, 3, 4, 6}.
Semiperfect Numbers
A semiperfect number is the sum of some of its proper divisors. For instance, 18 is semiperfect because its proper divisors are {1, 2, 3, 6, 9} and 3 + 6 + 9 = 18. If a semiperfect number is the sum of all of its proper divisors, it is called a perfect number.
Sociable Numbers
Sociable numbers are like amicable numbers, but they come in larger groups. The proper divisors of the first number in the group add up to the second number, the proper divisors of the second number add up to the third number, and so on. The sum of the proper divisors of the last number in the group is equal to the first number. Sociable numbers tend to be quite large, so they are hard to find without using a computer. One example of a sociable group is 12496, 14288, 15472, 14536, and 14264.
Square Numbers
Square numbers are the result of multiplying a number by itself once. These are the same as the "perfect squares": 12 = 1, 22 = 4, 32 = 9, and so on. (The small 2 means 'squared' and in e-mail we write it ^2, so that 2^2 is 'two squared'.)
The square of 5 is 25, and working backward, we say the square root of 25 is 5.
The number of evenly spaced dots needed to make a square is a square number. This is just one kind of polygonal number. Here are some pictures of the first few square numbers:* * * * * * * * * * * * * * *
1 * * * * * * * * * * * * * *
4 * * * * * * * * * * * *
9 * * * * * * * * *
16 * * * * *
25
Tetrahedral Numbers
Tetrahedral numbers are one kind of figurate number. They are found by counting the number of evenly spaced points needed to build a tetrahedron. Tetrahedra are pyramids with triangular bases.
Triangular Numbers
A triangular number is the number of dots needed to draw a triangle. This is one kind of polygonal number. Here is a picture of the first few triangular numbers: * * * * *
1 * * * * * * * *
3 * * * * * * * * *
6 * * * * * * * *
10 * * * * *
15
The formula for the nth triangular number is T(n) = n (n+1)/2.
Weird Numbers
A number is weird if it is abundant without being semiperfect; 70 is the first weird number.
An abundant number has the sum of its proper divisors greater than the number itself. As an example 24 is an abundant number because its proper divisors are {1, 2, 3, 4, 6, 8, 12} and 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 .
Amicable Pairs
An amicable pair are 2 numbers, if the sum of the proper divisors of the first number is the second number and if the sum of the proper divisors of the second number, is the first number.
An example of amicable pair is 2620 and 2924.
The proper divisors of 2620 are {1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310}.
The sum of proper divisorsof 2620 is 1 + 2 + 4 + 5 + 10 + 20 + 131 + 262 + 524 + 655 + 1310 = 2924.
The proper divisors of 2924 are {1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462}.
The sum pf proper divisors of 2924 is 1 + 2 + 4 + 17 + 34 + 43 + 68 + 86 + 172 + 731 + 1462 = 2620.
Thus 2620 and 2924 are amicable.
Cube Numbers
Cube numbers are the result of multiplying a number by itself twice:
1 to the 3rd power = 1,
2 to the 3rd power = 8,
3 to the 3rd power = 27,
4 to the 3rd power =64 and so on...
The cube of 4 is 64, so the cube root of 64 is 4.
To illustrate, a cube number is the number of small cube-shaped blocks we need to build a larger cube.
As an example in order to build a 10-inch cube we would need 1000 small 1-inch cubeblocks, the So 1000 is a cube of 10.
Cute Numbers
If a square can be cut into n squares of at the most two different sizes, then n is called a cute number. For example, 4 and 10 are cute numbers.
Deficient Numbers
If the sum of a number's proper divisors is less than the original number, it is called a deficient number. For instance, 16 is deficient. The proper divisors of 16 are {1, 2, 4, 8}, but 1 + 2 + 4 + 8 = 15.
Figurate Numbers
The number of dots in an arrangement of equally spaced points is a figurate number. Here's an example: 1 3 5 7
* * * *
** * *
*** *
****
The points can be arranged in one, two, three, or even more dimensions. There are many different kinds of figurate numbers, such as polygonal and tetrahedral numbers.
Happy Numbers
A happy number is a number for which the sum of the squares of the digits eventually equals 1. For instance, 203 is happy:
2^2 + 0^2 + 3^2 = 13
1^2 + 3^2 = 10
1^2 + 0^2 = 1.
Numbers that are not happy, such as 16, are called unhappy numbers.
Narcissistic Numbers
A narcissistic person is only interested in himself; a narcissistic number might seem a little self-centered, too. A narcissistic number is an integer equal to an expression that uses the same digits. For example, 36 = 3! * 6. Sometimes a narcissistic number is defined as a number equal to the sum of its digits raised to a certain power, or, more specifically, as an n-digit number equal to the sum of its digits raised to the nth power. For instance, 371 is narcissistic because 3^3 + 7^3 + 1^3 = 371, and 9474 is narcissistic because 9^4 + 4^4 + 7^4 + 4^4 = 9474.
Palindromic Numbers
A palindrome is a word that's the same read either forward or backward, such as noon or kayak. Palindromic numbers, like 88 and 1540451, have the same digits forward and backward.
There's a simple way to turn most numbers into palindromic numbers:
Pick a number:Reverse its digits:Add them together:Repeat the process until you get a palindromic number.
19+ 91 110+011 121
Nobody knows whether or not this works for every number. People have used computers to try the flip-and-add process on 196 nearly ten million times, without finding a palindrome-- but it might still be possible. We do know that it won't work for every number written in every base: try 10110 in base 2.
At 8:02 P.M. on Wednesday, February 20th, 2002, time (for sixty seconds only) read in perfect symmetry:
20:02, 20/02, 2002 (200,220,022,002)
It will happen again at 9:12 P.M. on Dec. 21, 2112: 21:12, 21/12, 2112 (211,221,122,112).
Perfect Numbers
The numbers that divide evenly into an integer are called its divisors. For example, the divisors of 6 are {1, 2, 3, 6}. Proper divisors are the divisors less than the integer you started with: the proper divisors of 6 are {1, 2, 3}. A number is perfect if it is equal to the sum of its proper divisors. 6 is perfect, because 1 + 2 + 3 = 6.
Polygonal Numbers
A polygonal number is the number of equally spaced dots needed to draw a polygon. (A polygonal number is a special type of figurate number.) Sequences of polygonal numbers are based on nested polygons. Here's one example:1 6 15
** ***
* * * ** *
** * * *
** *
***
There are many different kinds of polygonal numbers, beginning with square and triangular numbers.
Proper Divisors
The divisors of an integer are the numbers that it can be divided by without leaving a remainder. For instance, the divisors of 12 are {1, 2, 3, 4, 6, 12}. (Divisors are also called factors.) The proper divisors of a positive integer are all of the divisors less than the integer you started with. Thus, the proper divisors of 12 are {1, 2, 3, 4, 6}.
Semiperfect Numbers
A semiperfect number is the sum of some of its proper divisors. For instance, 18 is semiperfect because its proper divisors are {1, 2, 3, 6, 9} and 3 + 6 + 9 = 18. If a semiperfect number is the sum of all of its proper divisors, it is called a perfect number.
Sociable Numbers
Sociable numbers are like amicable numbers, but they come in larger groups. The proper divisors of the first number in the group add up to the second number, the proper divisors of the second number add up to the third number, and so on. The sum of the proper divisors of the last number in the group is equal to the first number. Sociable numbers tend to be quite large, so they are hard to find without using a computer. One example of a sociable group is 12496, 14288, 15472, 14536, and 14264.
Square Numbers
Square numbers are the result of multiplying a number by itself once. These are the same as the "perfect squares": 12 = 1, 22 = 4, 32 = 9, and so on. (The small 2 means 'squared' and in e-mail we write it ^2, so that 2^2 is 'two squared'.)
The square of 5 is 25, and working backward, we say the square root of 25 is 5.
The number of evenly spaced dots needed to make a square is a square number. This is just one kind of polygonal number. Here are some pictures of the first few square numbers:* * * * * * * * * * * * * * *
1 * * * * * * * * * * * * * *
4 * * * * * * * * * * * *
9 * * * * * * * * *
16 * * * * *
25
Tetrahedral Numbers
Tetrahedral numbers are one kind of figurate number. They are found by counting the number of evenly spaced points needed to build a tetrahedron. Tetrahedra are pyramids with triangular bases.
Triangular Numbers
A triangular number is the number of dots needed to draw a triangle. This is one kind of polygonal number. Here is a picture of the first few triangular numbers: * * * * *
1 * * * * * * * *
3 * * * * * * * * *
6 * * * * * * * *
10 * * * * *
15
The formula for the nth triangular number is T(n) = n (n+1)/2.
Weird Numbers
A number is weird if it is abundant without being semiperfect; 70 is the first weird number.
Roman Numerals
How do I read and write Roman numerals?
A numeral is a symbol used to represent a number. (Our digits 0-9 are often called Arabic numerals.) Each letter used in Roman numerals stands for a different number:
Roman Numeral
Number
I
1
V
5
X
10
L
50
C
100
D
500
M
1000
A string of letters means that their values should be added together. For example, XXX = 10 + 10 + 10 = 30, and LXI = 50 + 10 + 1 = 61. If a smaller value is placed before a larger one, we subtract instead of adding. For instance, IV = 5 - 1 = 4.
You can use these rules to write a number in Roman numerals. Convert one digit at a time. Let's try 982:
982
= 900 + 80 + 2
= CM + LXXX + II
= CMLXXXII.
Here are the official rules for subtracting letters:
Subtract only powers of ten, such as I, X, or C. Writing VL for 45 is not allowed: write XLV instead.
Subtract only a single letter from a single numeral. Write VIII for 8, not IIX; 19 is XIX, not IXX.
Don't subtract a letter from another letter more than ten times greater. This means that you can only subtract I from V or X, and X from L or C, so MIM is illegal.
These rules only became official in the Middle Ages. Even today, not everybody follows them: you might notice that some clocks say IIII instead of IV.
The biggest Roman numeral is M, for 1000, so one easy way to write large numbers is to line up the Ms: MMMMMMM would be 7000, for instance. This system gets cumbersome quickly. When they needed to work with many large numbers, the Romans often wrote a bar above a numeral. The bar meant to multiply by 1000. Using this method, 7000 would be VII with a line on top.
A numeral is a symbol used to represent a number. (Our digits 0-9 are often called Arabic numerals.) Each letter used in Roman numerals stands for a different number:
Roman Numeral
Number
I
1
V
5
X
10
L
50
C
100
D
500
M
1000
A string of letters means that their values should be added together. For example, XXX = 10 + 10 + 10 = 30, and LXI = 50 + 10 + 1 = 61. If a smaller value is placed before a larger one, we subtract instead of adding. For instance, IV = 5 - 1 = 4.
You can use these rules to write a number in Roman numerals. Convert one digit at a time. Let's try 982:
982
= 900 + 80 + 2
= CM + LXXX + II
= CMLXXXII.
Here are the official rules for subtracting letters:
Subtract only powers of ten, such as I, X, or C. Writing VL for 45 is not allowed: write XLV instead.
Subtract only a single letter from a single numeral. Write VIII for 8, not IIX; 19 is XIX, not IXX.
Don't subtract a letter from another letter more than ten times greater. This means that you can only subtract I from V or X, and X from L or C, so MIM is illegal.
These rules only became official in the Middle Ages. Even today, not everybody follows them: you might notice that some clocks say IIII instead of IV.
The biggest Roman numeral is M, for 1000, so one easy way to write large numbers is to line up the Ms: MMMMMMM would be 7000, for instance. This system gets cumbersome quickly. When they needed to work with many large numbers, the Romans often wrote a bar above a numeral. The bar meant to multiply by 1000. Using this method, 7000 would be VII with a line on top.
Subscribe to:
Posts (Atom)
