Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1: Introduction

 

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Integers are the building blocks of the theory of numbers. This chapter contains somewhat very simple and obvious observations starting with properties of integers and yet the proofs behind those observations are not as simple. In this chapter we introduce basic operations on integers and some algebraic definitions that will be necessary to understand basic concepts in this book. We then introduce the Well ordering principle which states basically that every set of positive integers has a smallest element. Proof by induction is also presented as an efficient method for proving several theorems throughout the book. We proceed to define the concept of divisibility and the division algorithm. We then introduce the elementary but fundamental concept of a greatest common divisor (gcd) of two integers, and the Euclidean algorithm for finding the gcd of two integers. We end this chapter with Lame’s Lemma on an estimate of the number of steps in the Euclidean algorithm needed to find the gcd of two integers.

Contributors

  • Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.