\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Integer Numbers
Natural Sums
Needed by:
Integer Arithmetic and Order
Natural Integer Isomorphism
Rational Order
Sheet PDF
Graph PDF

Integer Order


We want to order the integers.


Consider $\eqc{(a, b)}, \eqc{(b, c)} \in \Z $. If $a + d < b + c$, then we say that $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$.1 If $\eqc{(a, b)}$ is less than $\eqc{(b, c)}$ or equal, then we say that $\eqc{(a, b)}$ is less than or equal to $\eqc{(b, c)}$.


If $x, y \in \Z $ and $x$ is less than $y$, then we write $x < y$. If $x$ is less than or equal to $y$, we write $x \leq y$.

Positive and negative integers

We call an integer $z$ positive if $z > 0$ and we call $z$ negative if $z < 0$.2 We call an integer $z$ nonnegative if $z > 0$ or $z = 0$ and nonpositive if $z < 0$ or $z = 0$.


We denote the set $\Set{z \in \Z }{z \geq 0_{Z}}$ by $\Z _{++}$.

  1. One needs to show that this is well-defined. The account will appear in future editions. ↩︎
  2. Some authors use the term positive for the case when $z > 0$ or $z = 0$. We use the term nonnegative in this case. ↩︎
Copyright © 2023 The Bourbaki Authors — All rights reserved — Version 13a6779cc About Show the old page view