Def/Integer

Logical Connections

This definition logically relies on the following definitions and statements: Def/Natural number, Def/Equivalence class

The following statements and definitions logically rely on the material of this page: Def/Binary quadratic form, Def/Diophantine equation, Def/Divides, Def/Division with remainder, Def/Even, Def/Gaussian integer, Def/Greatest common divisor, Def/Lax vector, Def/Least common multiple, Def/Odd, Def/Rational number, Def/Residue, Def/Square root, State/Classification of subgroups of the integers, State/Greatest common divisors can be found with the Euclidean algorithm, and State/Systems of two linear Diophantine equations

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Set theoretically, every integer can be viewed as an equivalence class of subtraction problems involving natural numbers. If $a,b \in \NN$, we write "$a - b$" (in quotation marks) as a formal subtraction problem, abbreviating the ordered pair $(a,b)$.

We say that two formal subtraction problems "$a-b$" and "$c-d$" are equivalent, if $a+d = b+c$. This defines an equivalence relation $\sim$ on the set $\NN \times \NN$ of formal subtraction problems. We simply write $a-b$ for the equivalence class of the formal subtraction problem "$a-b$".

The set $\ZZ$ is defined as the set $(\NN \times \NN) / \sim$ of equivalence classes of subtraction problems. Furthermore, if $n \in \NN$, we write $-n$ for the equivalence class of the subtraction problem $0-n$.

It can be proven that every element $z \in \ZZ$ is equal to $n$ or $-n$ (or both, if $n = 0$), for a unique natural number $n$.

This article has been written by: User:Marty