addend
In the expression a+b, a and b are addends.
aleph null
Aleph null (À 0) represents the cardinality of the counting numbers.
aleph one
Aleph one (À 1) is the next transfinite number after aleph null.
There are no transfinite numbers between aleph null and aleph one.
algebraic number
A number is algebraic if it is a solution to a polynomial
equation: anxn + an-1xn-1 + an-2xn-2... + a0 = 0
where all of the coefficients ai are integers.
Learn more about algebraic and transcendental numbers at Cut the Knot.
anomaly
An anomaly is an inconsistency, something contrary to expectation. In Kuhn's theory,
observation of anomalies can help to bring about a scientific revolution.
base of a power
In the expression kn, k is the base.
bounded sequence
A sequence is bounded if it has both an upper bound and a lower bound
cardinality
The cardinality of a set measures how many members it contains.
chaos
A dynamical system exhibits chaotic behavior if it satisfies these three requirements:
closed interval
A closed interval from a to b includes all real numbers greater than or equal to a and less than or equal to b.
The notation for a closed interval uses square brackets: [a,b]. Compare to open interval.
complex number
A complex number is two-dimensional and can be placed on a number plane
defined by a real number axis (horizontal) and an imaginary number axis (vertical).
It can also be written in the form a + bi, where a represents the position along the real
axis, while b represents the position along the imaginary axis.
Learn more about real and complex numbers at Cut the Knot.
composite number
A counting number is composite if it has more than two factors.
confidence level
In sampling, the confidence level (usually expressed as a percentage) indicates
how often a sample can be expected to be within the margin of error.
convergent sequence
A sequence converges if there exists some number L, such that the terms of the sequence get arbitrarily close to L and stay within that tolerance.
continuum
The continuum is a transfinite number representing the cardinality of the real numbers.
If Cantor's Continuum Hypothesis is true, then the continuum is equivalent to aleph one.
countable set
A set is countable if it is possible to list the members of the set in a one-to-one correspondence with the counting numbers.
Countably infinite sets have a cardinality of aleph null.
counting number
The set of counting numbers starts at 1 and increases in increments of 1:
1,2,3,4,5,6,7...
degree of a polynomial
A polynomial written in the form
anxn + an-1xn-1 + an-2xn-2... + a0,
where n is a nonnegative integer, has degree n.
density
A subset of the real numbers is called dense in the real numbers if a member of that subset can be found within
any open interval of the real numbers. Rational numbers are dense in the real numbers, but integers are not.
A consequence of the definition of density is that given any two members of a dense set, a and b, with a < b, there exists yet another member of the set, q,
such that a < q < b. In other words, between any two numbers, you can always find another number.
difference
A difference is the result of a subtraction problem.
digit
A digit is one of the characters (also known as "squiggles") used to represent a number. In base ten, there are
exactly ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Combinations of digits
make numerals, such as "195." Our number system has a finite number of digits, but an
infinite number of numerals.
To make a number system with a base higher than ten, you need additional digits. In base 16, for example, the digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, and f.
disjoint sets
Set A is disjoint from Set B if there are no common members between the two sets.
dividend
In the division problem a/b, a is the dividend.
divisor
In the division problem a/b, b is the divisor.
dynamical system
In a dynamical system, the current state of the system depends on the previous state of the
system. This dependence is typically described by a recursion rule.
e
The number e is the result of this infinite series:
| ¥ | |
| S | 1
n! |
| n=0 |
equilibrium value of a dynamical system
Given a dynamical system where Pn+1 is some function of Pn, the equilibrium value (P*), is the value of Pn such that Pn+1= Pn.
The equilibrium value is also known as a "fixed point" for that dynamical system.
escape set
In a dynamical system, the escape set includes all seeds whose orbit is an unbounded sequence.
Euclidean Geometry
Euclidean Geometry is one of many different types of geometries. It is the most
familiar one, typically studied in high school (and never again). All of the theorems
and conclusions of Euclidean Geometry can be derived from these five basic postulates:
1. A straight line segment can be drawn joining any two points.The fifth postulate is equivalent to what is known as "The Parallel Postulate." Another way to state The Parallel Postulate is to say: Given a line(l) and a point(P) not on that line, there is exactly one line which can be drawn through P that is parallel to l. Non-Euclidean geometries are created by failing to accept The Parallel Postulate. These include Hyperbolic Geometry and Spherical Geometry, among others.
2. Any straight line segment can be extgended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
eventually fixed
In a dynamical system, if a seed leads to a sequence that hits the equilibrium value at some point,
we say that the orbit of that seed is eventually fixed.
eventually periodic
In a dynamical system, if a seed leads to a sequence that falls into a cycle at some point,
we say that the orbit of that seed is eventually periodic.
exponent
In the expression kn, n is the exponent.
factorial
If n is an integer greater than 0, n factorial (n!) is the product: n* (n-1) * (n-2) * ( n-3)... * 1.
By convention, 0! = 1.
fractal
A fractal is a shape where self-similarity dimension is greater than topological dimension.
googol
A googol is equal to 10 to the 100th power.
greatest lower bound
If a sequence is bounded, it has not just one lower bound, but many. (See definition of lower bound.)
Of all the possible lower bounds, the greatest lower bound is the one with the tightest fit, i.e. it
is greater than or equal to all other lower bounds for that sequence.
half-life
In a model of exponential decay, the period of time required for half of the substance to decay is known as the half-life of that substance.
higher
Higher numbers are farther to the right on the number line.
histogram
A histogram is a bar graph, typically used to compare the frequency of several events.
infinite product
An infinite product is exactly what you would think it would be, a multiplication of
the terms of a sequence of infinite length.
infinite series
On behalf of mathematicians everywhere, I have to apologize for this word choice.
Knowing what "series" means in normal life, one might infer that it is another word
for "sequence," i.e. a list of terms. Unfortunately, that is not true. An infinite series
should be called an infinite sum, because it refers specifically to adding up the terms
in a sequence of infinite length.
infinite set
A set is infinite if its members can be placed in a one-to-one correspondence with the members of a proper subset.
interval
An interval is a continuous subset of the real numbers. Intervals can be open or closed.
irrational number
An irrational number is a real number that cannot be written in the form of a rational number.
Learn more about rational and irrational numbers at Cut the Knot.
iteration
An iteration is one "pass" through a recursive rule.
Julia Set
In a complex dynamical system, the Julia Set includes all seed values which are not members of the escape set.
In some books, the Julia Set is defined more strictly as the boundary between the escape set and the trapped set.
larger
Larger numbers are farther from zero.
least upper bound
If a sequence is bounded, it has not just one upper bound, but many. (See definition of upper bound.)
Of all the possible lower bounds, the least upper bound is the one with the tightest fit, i.e. it
is less than or equal to all other upper bounds for that sequence.
limit of a sequence
I hesitate to even tell you this word. It should win a prize as the worst word choice ever
for a mathematical term. The limit of a sequence is the number that the sequence converges to.
It is not the same thing as a bound, even though it sounds like it would be.
logistic model
In dynamical systems, the logistic model is described by this quadratic recursion rule:
Pn+1 = a*Pn(1 - b*Pn)
lower
Lower numbers are farther to the left on the number line.
lower bound of a sequence
A sequence has a lower bound, B, if all of the terms of the sequence are greater than or equal to B.
Mandelbrot Set
In the complex dynamical system with recursion rule, z ---> z2 + c, the
Mandelbrot Set includes all values of c for which the Julia Set includes the origin.
margin of error
In sampling, the margin of error (usually expressed in percentage points) indicates how
far the sample's results can stray from the true value in the entire population. For example, if a poll reports that
78% of Americans eat peanut butter and the margin of error is stated to be 3%, we can expect that the
true value of peanut butter eaters is somewhere between 75% and 81%. The sample is not
guaranteed to be within that range, but likely to be. (See confidence level).
minuend
In the subtraction problem a-b, a is the minuend.
mixing
In a dynamical system, mixing occurs if you can take any size interval, however small you want it,
and find a seed in there whose orbit eventually visits every other interval (again, the intervals can be arbitrarily small).
Mixing is also known as "transitivity," so you may see it referred to by that term.
modulus
In the expression a mod b, b is the modulus.
monotonic
A sequence increases monontonically if, for every n, Pn+1 is greater than or equal to Pn.
Similarly, a sequence decreases monontonically if, for every n, Pn+1 is less than or equal to Pn.
multiple
If n is an integer, k*n is a multiple of k.
natural number
The set of natural numbers is the same as the set of counting numbers.
normal science
In Kuhn's theory, normal science takes place in the context of a shared paradigm.
It consists of three main scientific activities: "determination of significant fact,
matching fact with theory, and articulation of theory.... Work under the paradigm can be
conducted in no other way, and to desert the paradigm is to cease practicing the science
it defines." (p. 34)
open interval
An open interval from a to b includes all real numbers greater than a and less than b.
Compare to closed interval.
The notation for an open interval uses parentheses: (a,b). Please don't confuse it with
the (x,y) pairs from graphing on the coordinate plane. Intervals live strictly on the number line.
It is unfortunate notation, in my opinion.
orbit
The sequence of values produced by plugging a seed into a recursion rule and iterating repeatedly is known as the orbit of that seed.
paradigm
Introduced by Thomas Kuhn in his 1962 work,The Structure of Scientific Revolutions,
the concept of paradigm is linked to a "coherent tradition of scientific research (p.11)."
Examples include Newtonian mechanics or Copernican astronomy.
To say that a group of scientists shares a certain paradigm means that they have a
common "way of seeing the world and of practicing science in it (p. 4)."
perfect number
A number is perfect if it is the sum of its proper factors. Learn more about perfect numbers
at Cut the Knot.
period
In a cyclic orbit of a dynamical system, the number of iterations it takes to complete a cycle is known as the period of that orbit.
periodic points
In a dynamical system, the members of a cyclic orbit are known as periodic points of that system.
permutation
Given an ordered set of elements, a permutation is a reordering of that
set where each element appears exactly once. For example, "egam" is a permutation
of "game", or "2431" is a permutation of "1234".
pi
Pi (p ) is the ratio of a circle's circumference to its diameter.
See also information about pi at Math World.
polynomial
A polynomial is an expression that can be written in the form
anxn + an-1xn-1 + an-2xn-2... + a0
where n is an integer greater than or equal to zero.
power
If n is an integer, kn is a power of k.
pre-image
Given a function f(x), if f(a) = b, we say that a is the pre-image of b.
prime number
A counting number is prime if it is divisible only by 1 and itself. By convention, the number 1 is excluded from this definition. 1 is neither prime nor composite.
product
A product is the result of a multiplication problem.
proper factor
The proper factors of some integer k include all of k's factors except for k itself.
proper subset
A subset is proper if it does not contain all of the elements of its superset.
quotient
A quotient is the result of a division problem.
random sequence
In a random sequence of elements, two criteria must be satisfied:
rational number
A rational number can be written in the form a/b, where a and b are both integers and b is not equal to zero.
Learn more about rational and irrational numbers at Cut the Knot.
real number
A real number is one-dimensional and can be placed somewhere on the number line.
The set of real numbers includes all rational and all irrational numbers.
relatively prime
Two numbers are relatively prime if they have no common factors.
retrograde motion
Retrograde motion occurs when planets appear to be moving "backwards" in their orbits.
For more information about the geocentric model of the universe and a demonstration of
epicycles, go to this link
for an astronomy course taught at the University of Tennessee.
seed of a dynamical system
The seed is the first input value put into the recursion rule of a dynamical system.
self-similarity
A geometric object exhibits self-similarity if, at increasing levels of magnification, it is possible to find a replica of the object.
sensitivity
In a sensitive system, small changes in initial conditions have a dramatic effect on outcome.
sigma notation
The Greek letter sigma (S) indicates summation.
| b | |
| S | f(n) |
| n=a |
smaller
Smaller numbers are closer to zero.
stability
In a stable system, small changes in initial conditions have little effect on outcome.
strictly monotonic
A sequence increases in a strictly monotonic fashion if, for every n,
Pn+1 is greater than Pn. Similarly,a sequence decreases in a strictly monotonic fashion if, for every n,
Pn+1 is less than Pn.
subset
Set A is a subset of Set B if all of the members of A are also memebers of B.
subtrahend
In the subtraction problem a-b, b is the subtrahend.
sum
A sum is the result of an addition problem.
superset
If Set A is a subset of Set B, then Set B is a superset of Set A.
time series
A time series graph shows the growth of a sequence as a function of time. The word
"series" in this definition has no connection to the term "inifinite series." Yet another
example of confusing mathematical vocabulary, I'm afraid.
tolerance
Similar to margin of error, tolerance indicates the amount of allowable discrepancy
between a target number and an actual result. If a value is within the tolerance, we
consider it to be "close enough" for what we are trying to achieve.
transcendental number
A real number is said to be transcendental if it is not an algebraic number.
Learn more about algebraic and transcendental numbers at Cut the Knot.
transfinite number
A transfinite number is greater than any counting number. Aleph null and aleph one are transfinite numbers.
transient phase
The transient phase of a dynamical system occurs at the beginning of the sequence, while
you are waiting for the orbit to settle in to its ultimate fate. Sometimes you have to wait
quite a while for the "noise" to go away.
ultimate fate
In a dynamical system, the behavior of a seed's orbit after many, many iterations is known as the ultimate fate of a seed.
This ultimate fate might be convergence to a particular value, growth without bound, cyclic behavior, or chaos.
uncountable set
A set is uncountable if its cardinality is greater than aleph null. This means that it is impossible
to place the elements of this set in a one-to-one correspondence with the counting numbers.
upper bound of a sequence
A sequence has an upper bound, B, if all of the terms of the sequence are less than or equal to B.