Increasing sequences have the pleasing property
that their convergence (or divergence) is easy to identify:
An increasing sequence converges if, and only
if, it is bounded above.
(See BOUND and CONVERGENT SEQUENCE.) This seems
clear geometrically. If on the real number line the
sequence of numbers {a n } moves steadily to the right yet
does not penetrate a barrier at position B, then the
numbers must “pile up” at some limit value L ≤ B.
Conversely, if an increasing sequence converges to a
value L, then all the terms of the sequence lie to the left
of L on the number line and hence are bounded above
by L. (These statements can be proved rigorously by
making use of the “completeness” of the real number
line via the notion of a DEDEKIND CUT.)
A sequence {a n } is said to be decreasing if a 1 ≥ a 2 ≥
a 3 ≥ …, that is, each term in the sequence is less than or
equal to the one that precedes it. It is called strictly
decreasing if a 1 > a 2 > a 3 >…. Similarly one can show
that a decreasing sequence converges if, and only if, it
is bounded below. A sequence that is either increasing
or decreasing is called monotonic.
A real-valued function f is said to be increasing
over an interval if, over that interval, greater input
values produce greater (or possibly equal) output values, that is, if a and b are two values in the interval
with a < b, then f(a) ≤ f(b). If it is always the case
that f(a) < f(b), then the function is said to be strictly
increasing on the interval. The MEAN-VALUE THEOREM shows that a differentiable function f is increasing precisely on those intervals where the DERIVATIVE
is nonnegative, f ′ (x) ≥ 0, and strictly increasing if
f ′ (x) > 0.
A real-valued function f is said to be decreasing
over an interval if greater input values produces smaller
(or possibly equal) output values of the function, that is,
if a < b, then f(a) ≥ f(b). If it is always the case that f(a)
> f(b), then the function is said to be strictly decreasing
on the interval. The MEAN-VALUE THEOREM shows that
a differentiable function f is decreasing precisely on
those intervals for which f ′ (x) ≤ 0, and strictly decreasing if f ′ (x) < 0.
A function that is either increasing or decreasing
on an interval is called monotonic on that interval.
See also DIFFERENTIAL CALCULUS; GRAPH OF A
FUNCTION; MAXIMUM/MINIMUM.
increment A small finite change in the value of a variable or in the value of a function is called an increment.
If the variable is x, then an increment of x is usually
denoted ∆x or δx. If f is a function of x, then the corresponding increment of f, denoted ∆f, equals ∆f = f(x +
∆x) – f(x). The LIMIT of the ratio ∆f/∆x, as the increment ∆x decreases to zero, is called the DERIVATIVE of f.
A negative increment is sometimes called a decrement.
See also DIFFERENTIAL.
indefinite integral See ANTIDIFFERENTIATION; INTEGRAL CALCULUS.
independent axiom An AXIOM of a mathematical
theory is said to be independent if it cannot be derived
from the remaining axioms of a theory as a logical consequence. For instance, the existence of NONEUCLIDEAN GEOMETRIES establishes that Euclid’s
famous PARALLEL POSTULATE is independent of his four
remaining postulates. In general, this is the approach
mathematicians take to show that an axiom A in a
mathematical system is independent of the remaining
axioms: present an example of another consistent
mathematical theory proved to be free of contradictions in which all axioms except A hold and for which
A is false. (Consequently, it cannot be the case that
these axioms do imply that A is true, for then one has a
system in which both A and not A hold.)
See also EUCLID’S POSTULATES.
independent events Two experiments run in succession are deemed independent if the outcomes obtained
from one experiment do not affect the outcomes
obtained in the second. For example, the results from
casting a die do not influence the results obtained in
later tossing a coin. These two actions are independent.
In PROBABILITY theory we say two EVENTs A and B
are independent if knowledge of A having occurred has
no influence on the likelihood of B next occurring. For
example, in two rolls of die casting, an even number on
the first roll and casting a 6 on the second are independent events—the chances of rolling a six are 1/6, no
matter the result of the first roll.
Two events that are not independent are called
dependent. For example, the event “Sally is wearing a
262 increment
that their convergence (or divergence) is easy to identify:
An increasing sequence converges if, and only
if, it is bounded above.
(See BOUND and CONVERGENT SEQUENCE.) This seems
clear geometrically. If on the real number line the
sequence of numbers {a n } moves steadily to the right yet
does not penetrate a barrier at position B, then the
numbers must “pile up” at some limit value L ≤ B.
Conversely, if an increasing sequence converges to a
value L, then all the terms of the sequence lie to the left
of L on the number line and hence are bounded above
by L. (These statements can be proved rigorously by
making use of the “completeness” of the real number
line via the notion of a DEDEKIND CUT.)
A sequence {a n } is said to be decreasing if a 1 ≥ a 2 ≥
a 3 ≥ …, that is, each term in the sequence is less than or
equal to the one that precedes it. It is called strictly
decreasing if a 1 > a 2 > a 3 >…. Similarly one can show
that a decreasing sequence converges if, and only if, it
is bounded below. A sequence that is either increasing
or decreasing is called monotonic.
A real-valued function f is said to be increasing
over an interval if, over that interval, greater input
values produce greater (or possibly equal) output values, that is, if a and b are two values in the interval
with a < b, then f(a) ≤ f(b). If it is always the case
that f(a) < f(b), then the function is said to be strictly
increasing on the interval. The MEAN-VALUE THEOREM shows that a differentiable function f is increasing precisely on those intervals where the DERIVATIVE
is nonnegative, f ′ (x) ≥ 0, and strictly increasing if
f ′ (x) > 0.
A real-valued function f is said to be decreasing
over an interval if greater input values produces smaller
(or possibly equal) output values of the function, that is,
if a < b, then f(a) ≥ f(b). If it is always the case that f(a)
> f(b), then the function is said to be strictly decreasing
on the interval. The MEAN-VALUE THEOREM shows that
a differentiable function f is decreasing precisely on
those intervals for which f ′ (x) ≤ 0, and strictly decreasing if f ′ (x) < 0.
A function that is either increasing or decreasing
on an interval is called monotonic on that interval.
See also DIFFERENTIAL CALCULUS; GRAPH OF A
FUNCTION; MAXIMUM/MINIMUM.
increment A small finite change in the value of a variable or in the value of a function is called an increment.
If the variable is x, then an increment of x is usually
denoted ∆x or δx. If f is a function of x, then the corresponding increment of f, denoted ∆f, equals ∆f = f(x +
∆x) – f(x). The LIMIT of the ratio ∆f/∆x, as the increment ∆x decreases to zero, is called the DERIVATIVE of f.
A negative increment is sometimes called a decrement.
See also DIFFERENTIAL.
indefinite integral See ANTIDIFFERENTIATION; INTEGRAL CALCULUS.
independent axiom An AXIOM of a mathematical
theory is said to be independent if it cannot be derived
from the remaining axioms of a theory as a logical consequence. For instance, the existence of NONEUCLIDEAN GEOMETRIES establishes that Euclid’s
famous PARALLEL POSTULATE is independent of his four
remaining postulates. In general, this is the approach
mathematicians take to show that an axiom A in a
mathematical system is independent of the remaining
axioms: present an example of another consistent
mathematical theory proved to be free of contradictions in which all axioms except A hold and for which
A is false. (Consequently, it cannot be the case that
these axioms do imply that A is true, for then one has a
system in which both A and not A hold.)
See also EUCLID’S POSTULATES.
independent events Two experiments run in succession are deemed independent if the outcomes obtained
from one experiment do not affect the outcomes
obtained in the second. For example, the results from
casting a die do not influence the results obtained in
later tossing a coin. These two actions are independent.
In PROBABILITY theory we say two EVENTs A and B
are independent if knowledge of A having occurred has
no influence on the likelihood of B next occurring. For
example, in two rolls of die casting, an even number on
the first roll and casting a 6 on the second are independent events—the chances of rolling a six are 1/6, no
matter the result of the first roll.
Two events that are not independent are called
dependent. For example, the event “Sally is wearing a
262 increment