# Definition:Integer

## Contents

## Definition

The numbers $\set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ are called the **integers**.

This set is usually denoted $\Z$.

An individual element of $\Z$ is called **an integer**.

### Formal Definition

Let $\struct {\N, +}$ be the commutative semigroup of natural numbers under addition.

From Inverse Completion of Natural Numbers, we can create $\struct {\N', +'}$, an inverse completion of $\struct {\N, +}$.

From Construction of Inverse Completion, this is done as follows:

Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:

- $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

From Cross-Relation is Congruence Relation, $\boxtimes$ is a congruence relation.

Let $\struct {\N \times \N, \oplus}$ be the external direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$:

- $\tuple {x_1, y_1} \oplus \tuple {x_2, y_2} = \tuple {x_1 + x_2, y_1 + y_2}$

Let the quotient structure defined by $\boxtimes$ be $\struct {\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}$

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$.

Let us use $\N'$ to denote the quotient set $\dfrac {\N \times \N} \boxtimes$.

Let us use $+'$ to denote the operation $\oplus_\boxtimes$.

Thus $\struct {\N', +'}$ is the Inverse Completion of Natural Numbers.

As the Inverse Completion is Unique up to isomorphism, it follows that we can *define* the structure $\struct {\Z, +}$ which is isomorphic to $\struct {\N', +'}$.

An element of $\N'$ is therefore an equivalence class of the congruence relation $\boxtimes$.

So an element of $\Z$ is the isomorphic image of an element $\eqclass {\tuple {a, b} } \boxtimes$ of $\dfrac {\N \times \N} \boxtimes$.

The set of elements $\Z$ is called **the integers**.

### Natural Number Difference

In the context of the natural numbers, the difference is defined as:

- $n - m = p \iff m + p = n$

from which it can be seen that the above congruence can be understood as:

- $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1 \iff x_1 - y_1 = x_2 - y_2$

Thus this congruence defines an equivalence between pairs of elements which have the same difference.

## Also known as

The **integers** are also referred to as **whole numbers**, so as to distinguish them from fractions. However, use of this term is discouraged because it is ambiguous: it can refer to the **integers**, the positive integers, or the negative integers, depending on the preference of the author.

Some sources refer to the **integers** as **rational integers**, to clearly distinguish them from the algebraic integers.

Some sources use the term **directed numbers** or **signed numbers** for **integers**, so as to distinguish them from the natural numbers which are, by definition defined without a sign.

Variants on $\Z$ are often seen, for example $\mathbf Z$ and $\mathcal Z$, or even just $Z$.

Some sources use $I$, while others use $\mathbf J$ or a variant.

## Also see

- Results about
**integers, in an abstract algebraic context,**can be found here.

## Historical Note

The use of $Z$ to denote the set of **integers** appears to have originated with Nicolas Bourbaki.

Earlier, the symbol $\overline{\mathfrak Z}$ had been used by Edmund Landau.

## Linguistic Note

The word integer is pronounced with the stress on the first syllable, and the **g** is soft (i.e. sounds like **j**): ** in-te-jer**.

This is inconsistent with the pronunciation of the related term **integral** where the **g** is hard (as in **get**): ** in-te-gral**.

Also note the use of the word **integral** as an adjective, meaning **necessary** or **inherent**, usually encountered in rhetoric. For further confusion, this is pronounced **in- teg-ral**, the stress being on the second syllable.

The symbol $\Z$ is for **Zahlen**, which is German for **whole numbers**, with overtones of **unbroken**.

This is reflected in English in the word **integrity**, which means **wholeness** in the sense of **unbrokenness**. The word also has a similarly applicable definition in the context of moral philosophy.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets: Example $1$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Special Symbols - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 1$. Introduction - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 1$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): Notation for Some Important Sets - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.8$: Collections of Sets: Definition $8.4$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational integers - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $2$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(b)}$ - 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $2$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*: Entry:**Integer** - 1990: H.A. Priestley:
*Introduction to Complex Analysis*(revised ed.) ... (next): Notation and terminology - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.14$: Exercise $19 \ \text{(b)}$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.1$: Sets - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Countable Sets - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers: Negative numbers