" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.

The name algebraic integer comes from the fact that the only rational numbers which are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers.

Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.

The most commonly used grading systems are the Fontainebleau system which ranges from 1 to 8c +, and the John Sherman V-grade system, beginning at V0 and increasing by integers to a current top grade of proposed V16 ( The Wheel of Life by Dai Koyamada in the Grampians, Australia ; The Game, by Daniel Woods, Boulder Canyon, CO ; Lucid Dreaming, by Paul Robinson.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

Instead of listing every single element, sometimes an ellipsis ("...") is used, if the writer believes that the reader can easily guess what is missing ; for example, presumably denotes the set of integers from 1 to 100.

Büttner, gave him a task: add a list of integers in arithmetic progression ; as the story is most often told, these were the numbers from 1 to 100.

The two integers a and b are coprime if and only if the point with coordinates ( a, b ) in a Cartesian coordinate system is " visible " from the origin ( 0, 0 ), in the sense that there is no point with integer coordinates between the origin and ( a, b ).

This huge improvement made many DFT-based algorithms practical ; FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers.

For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class ( known as a congruence class ) as a single entity.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

A hash function that maps names to integers from 0 to 15 ..

A perfect hash function for n keys is said to be minimal if its range consists of n consecutive integers, usually from 0 to n − 1.

Typically all integers from the minimum through the maximum can be represented.

< li > Consider the group ( Z < sub > 6 </ sub >, +), the integers from 0 to 5 with addition modulo 6.

Ancient Romans used a decimal system for integers, but switched to duodecimal for fractions, and correspondingly Latin developed a rich vocabulary for duodecimal-based fractions ( see Roman numerals ).

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

An example is the " divides " relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p ( and not with any integer that is not a multiple of p ).

For given nonzero integers a and b there is a nonzero integer of minimal absolute value among all those of the form ax + by with x and y integers ; one can assume d > 0 by changing the signs of both s and t if necessary.

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

In effect, integer size sets a hardware limit on the range of integers the software run by the CPU can utilize.

For instance, can hold an unboxed integer in a range supported by the hardware and implementation, permitting more efficient arithmetic than on big integers or arbitrary precision types.

An example of a problem which is known to be in NP and in co-NP is integer factorization: given positive integers m and n determine if m has a factor less than n and greater than one.

In number theory, a branch of mathematics, two integers a and b are said to be coprime ( also spelled co-prime ) or relatively prime if the only positive integer that evenly divides both of them is 1.

The numbers 1 and − 1 are coprime to every integer, and they are the only integers to be coprime with 0.

The number of integers coprime to a positive integer n, between 1 and n, is given by Euler's totient function ( or Euler's phi function ) φ ( n ).

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

Similar to the view of Leopold Kronecker that " God made the integers ; all else is the work of man ," musicians drawn to the alphorn and other instruments that sound the natural harmonics, such as the natural horn, consider the notes of the natural harmonic series — particularly the 7th and 11th harmonics — to be God's Notes, the remainder of the chromatic scale enabled by keys, valves, slides and other methods of changing the qualities of the simple open pipe being an artifact of mere mortals.

Set blk =+$ Piece ( input ," ", 4 ) ; blk-size of blocks of integers is optional fourth piece

1968 ( with Hilary Putnam ), " Degrees of unsolvability of constructible sets of integers ," Journal of Symbolic Logic 33: 497-513.

* 1944, " Recursively enumerable sets of positive integers and their decision problems ," Bulletin of the American Mathematical Society 50: 284 – 316.

He is said to have developed the partially tongue-in-cheek " Strong Law of Small Numbers ," which says there are not enough small integers available for the many tasks assigned to them — thus explaining many coincidences and patterns found among numerous cultures.

Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words.

For example, there are infinitely many odd integers, infinitely many even integers, and ( hence ) infinitely many integers overall.

Initially e is assumed to be a rational number of the form a / b. We then analyze a blown-up difference x of the series representing e and its strictly smaller partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction a / b and the partial sum are turned into integers, hence x must be a positive integer.

Thus, the ring of integers of Q (√ d ) is a principal ideal domain ( and hence a unique factorization domain ) if and only if the class number of Q (√ d ) is equal to 1.

As a consequence of the Fermat's little theorem, in the ring of p-adic integers we have p − 1 roots of unity, each of which is congruent mod p to some number in the range 1 to p − 1 ; we can therefore define a Dirichlet character ω ( the Teichmüller character ) with values in by requiring that for n relatively prime to p, ω ( n ) be congruent to n modulo p. The p part of the class group is a-module ( since it is p-primary ), hence a module over the group ring.

The integers are an example of an integral domain which does not allow all divisions as, again, whole numbers are needed.

If one contemplates actual reaction mechanisms, stoichiometric coefficients will always be integers, since elementary reactions always involve whole molecules.

Square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers ( that is to say they cannot be written exactly as m / n, where m and n are integers ).

However, one of the main concepts of the MMX instruction set is the concept of packed data types, which means instead of using the whole register for a single 64-bit integer ( quadword ), one may use it to contain two 32-bit integers ( doubleword ), four 16-bit integers ( word ) or eight 8-bit integers ( byte ).

This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole.

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

The main usage of the MMX instruction set is based on the concept of packed data types, which means that instead of using the whole register for a single 64-bit integer, two 32-bit integers, four 16-bit integers, or eight 8-bit integers may be processed concurrently.

The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers ( together with zero ) are referred to as integers.

* Spec &# 8484 ;, the spectrum of the integers has a closed point for every prime number p corresponding to the maximal ideal ( p ) ⊂ &# 8484 ;, and one non-closed generic point ( i. e., whose closure is the whole space ) corresponding to the zero ideal ( 0 ).

The real numbers consist of irrational numbers and rational numbers, as well as the integers, whole numbers, and the natural numbers ( the counting numbers ).

The subtraction linked list also does not require casting C pointers to integers, provided the whole list structure is inside a single contiguous block of memory.

Here " whole real numbers " are ordinary integers and " whole complex numbers " are Gaussian integers ; one should similarly interpret " real and complex prime numbers ".