A finite continued fraction, where is a non-negative integer, < sub > 0 </ sub > is an integer, and is a positive integer, for = 1 ,…,.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

The numerical value of an infinite continued fraction will be irrational ; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions.

Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers.

In particular, it must terminate and produce a finite continued fraction representation of the number.

* The continued fraction representation for a rational number is finite and only rational numbers have finite representations.

In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, coefficients a < sub > i </ sub > of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x and is known as Khinchin's constant.

The global state approach was continued in automata theory for finite state machines and push down stack machines, including their nondeterministic versions.

The set of points z such that the function can be analytically continued to the interior of the disk with diameter 0z is a polygon when the function has only a finite number of singularities, called the Borel polygon.

It is clear that for each finite continued fraction expression one can repeatedly move to its parent, and reach the root < nowiki ></ nowiki >= of the tree in finitely many steps ( in steps to be precise ).

Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.

If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, and has a unique eventually repeating infinite decimal expansion.

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

* approximating a fraction with periodic decimal expansion by a finite decimal fraction, e. g., 5 / 3 by 1. 6667 ;

The increment m is normally a finite fraction in whatever number system that is used to represent the numbers.

Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.

In cases where the sampling fraction exceeds 5 %, analysts can adjust the margin of error using a " finite population correction ", ( FPC ) to account for the added precision gained by sampling close to a larger percentage of the population.

When the sampling fraction is large ( approximately at 5 % or more ), the estimate of the error must be corrected by multiplying by a " finite population correction "

It says that finite collections of points fall into one of two extremes ; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.

* Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator

Beck's theorem says that finite collections of points in the plane fall into one of two extremes ; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.

A transmission line of finite length ( lossless or lossy ) that is terminated at one end with a resistor equal to the characteristic impedance appears to the source like an infinitely long transmission line.

Starting from an initial state and initial input ( perhaps empty ), the instructions describe a computation that, when executed, will proceed through a finite number of well-defined successive states, eventually producing " output " and terminating at a final ending state.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables ( see more at finite state machine, state transition table and control table ), as flowcharts ( see more at state diagram ), or as a form of rudimentary machine code or assembly code called " sets of quadruples " ( see more at Turing machine ).

These regarded issues such as whether the universe is eternal or non-eternal ( or whether it is finite or infinite ), the unity or separation of the body and the self, the complete inexistence of a person after Nirvana and death, and others.

Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.

This sequence starts with the natural numbers including zero ( finite cardinals ), which are followed by the aleph numbers ( infinite cardinals of well-ordered sets ).

Still, in the absence of naked singularities, the universe is deterministic — it's possible to predict the entire evolution of the universe ( possibly excluding some finite regions of space hidden inside event horizons of singularities ), knowing only its condition at a certain moment of time ( more precisely, everywhere on a spacelike 3-dimensional hypersurface, called the Cauchy surface ).

Unlike the discrete-time Fourier transform ( DTFT ), the DFT only evaluates enough frequency components to reconstruct the finite segment that was analyzed.

The input to the DFT is a finite sequence of real or complex numbers ( with more abstract generalizations discussed below ), making the DFT ideal for processing information stored in computers.

Modern school textbooks often define separate figures called lines ( infinite ), rays ( semi-infinite ), and line segments ( of finite length ).

Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space ( as in elliptic geometry ), and all five axioms are consistent with a variety of topologies ( e. g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry ).

He also introduced the concept of a finite field ( also known as a Galois field in his honor ), in essentially the same form as it is understood today.

Another advantage of FIR filters is that their impulse response can be made symmetric, which implies a response in the frequency domain which has zero phase at all frequencies ( not considering a finite delay ), which is absolutely impossible with any IIR filter.

This ring is a field because it contains a multiplicative inverse for each element N other than zero ( an integer that, multiplied by the element modulo p yields 1 ), and it has a finite number of elements ( p ), making it a finite field.

The number of elements of a finite set is a natural number ( non-negative integer ), and is called the cardinality of the set.

* The nominative case ( subjective pronouns such as I, he, she, we ), used for the subject of a finite verb and sometimes for the complement of a copula.

Which is to say: as much as we might try to order our world with a certain set of norms and goals ( which we consider our real world ), the paradox of a finite consciousness in an infinite universe creates a zone of irreality (" that which is beyond the real ") that offsets, opposes, or threatens the real world of the human subject.

* Lattice ( discrete subgroup ), a discrete subgroup of a topological group with finite covolume

In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections ( sets ), but not when it is used in discourse over infinite sets ( e. g. the natural numbers ).

* The group of smooth maps from a manifold to a finite dimensional Lie group is an example of a gauge group ( with operation of pointwise multiplication ), and is used in quantum field theory and Donaldson theory.

In the smectic C * phase ( an asterisk denotes a chiral phase ), the molecules have positional ordering in a layered structure ( as in the other smectic phases ), with the molecules tilted by a finite angle with respect to the layer normal.