integermathematics

...1910. The theory of rings (structures in which it is possible to add, subtract, and multiply but not necessarily divide) was much harder to formalize. It is important for two reasons: the theory of algebraic integers forms part of it, because algebraic integers naturally form into rings; and (as Kronecker and Hilbert had argued) algebraic geometry forms another part. The rings that arise there...

In another direction, important progress in number theory by German mathematicians such as Ernst Kummer, Richard Dedekind, and Leopold Kronecker used rings of algebraic integers. (An algebraic integer is a complex number satisfying an algebraic equation of the form...

...can be handled arithmetically. These expressions have many properties akin to those of whole numbers, and mathematicians have even defined prime numbers of this form; therefore, they are called algebraic integers. In this case they are obtained by grafting onto the rational numbers a solution of the polynomial equation x² − 2 = 0. In general an...

...numbers. Using the concept of field and some other derivative ideas, Dedekind identified the precise subset of the complex numbers for which the theorem could be extended. He named that subset the algebraic...

A partition of a positive integer n is a representation of n as a sum of positive integers n = x₁ + x₂ + · · · + x_k, x_i ⋜ 1, i = 1, 2, · · · , k. The numbers x_i are called the parts of the partition. The ...

A related concept, central to the mathematical topics of combinatorics and number theory, is the partition of a positive integer—that is, the number of ways that an integer n can be expressed as the sum of k smaller integers. For example, the number of ways of representing the number 7 as the sum of 3 smaller whole numbers (n = 7, k = 3) is 4 (5 + 1 + 1, 4 + 2...

any mathematical function defined for integers (…, −3, −2, −1, 0, 1, 2, 3, …) and dependent upon those properties of the integer itself as a number, in contrast to functions that are defined for other values (real numbers, complex numbers, or even other functions) and that involve various operations from algebra and calculus. Examples of arithmetic...

...that is, the number −1, as well as all products of the form −1 × n, in which n is a whole number. The extended collection of numbers is called the integers, of which the positive integers are the same as the natural numbers. The numbers that are newly introduced in this way are called negative integers.

The geometers immediately following Pythagoras (c. 580–c. 500 bc) shared the unsound intuition that any two lengths are “commensurable” (that is, measurable) by integer multiples of some common unit. To put it another way, they believed that the whole (or counting) numbers, and their ratios (rational numbers or fractions), were sufficient to describe any quantity. Geometry...

For example, in the usual construction of the ring of integers, an integer is defined as an equivalence class of pairs (m,n) of natural numbers, where (m,n) is equivalent to (m′,n′) if and only if m + n′ = m′ + n. The idea is that the equivalence class of (m,n) is to be viewed as m...

Continuing his investigations into the properties and relationships of integers—that is, the idea of number—Dedekind published Über die Theorie der ganzen algebraischen Zahlen (1879;...

...led to the factorization properties of numbers of the type a + ib (a and b integers and i = √(−1) ), sometimes called Gaussian integers. In doing so, Gauss not only used complex numbers to solve a problem involving ordinary integers, a fact remarkable in itself, but he also opened the way to the detailed...

...in which L is an integer. The possible values of L depend on the individual l values and the orientations of their orbits for all the electrons composing the atom. The total spin momentum has magnitude √(S(S + 1)) (ℏ), in which S is an integer or half an odd integer, depending on whether the number of electrons is even or...