Integer
The integer hides inside a single Latin word that means untouched. It comes from in, meaning not, joined with tangere, meaning to touch. The same root gives us the word entire, by way of the French entier, which means both entire and whole. For most of history, the only integers anyone counted were the positive ones. Zero and the negatives arrived later, admitted one by one as people found uses for them. How did a number meaning untouched come to include debts below zero and a set stretching infinitely in both directions? And why does a single bold letter Z stand for the whole collection? The answers move from a Latin verb to a German noun, from elementary school chalkboards to the abstract machinery of rings and fields.
Leonhard Euler, in his 1765 Elements of Algebra, defined integers to include both positive and negative numbers. Before that expansion, the term described a number that was a multiple of 1, or the whole part of a mixed number. Only positive integers were considered, which made the word a synonym for the natural numbers. The negations of those positive naturals carry their own name. They are the negative integers, the additive inverses written as -1, -2, -3, and onward. Zero sits among them as neither positive nor negative, a value defined as standing apart from both. The set itself grew along with the definition, and naming that set would take until the very end of the 19th century.
Georg Cantor introduced the concept of infinite sets and set theory near the end of the 19th century, which is when the phrase the set of the integers entered use. The familiar letter Z comes from the German word Zahlen, meaning numbers, and the choice has been attributed to David Hilbert. Its earliest known appearance in a textbook is in Algèbre by the collective Nicolas Bourbaki, dated 1947. Adoption was slow and inconsistent. One textbook used the letter J instead, and a 1960 paper used Z to denote the non-negative integers rather than the full set. By 1961, modern algebra texts had settled on Z for the positive and negative integers together. The symbol still carries annotations that vary by author, marking positive integers, non-negative integers, non-zero integers, the set {-1, 1}, integers modulo a number, or the p-adic integers. That ambiguity has a cousin in the classroom, where the word whole stopped meaning what it once did.
Whole numbers were synonymous with the integers up until the early 1950s. The split came in the late 1950s with the New Math movement, when American elementary school teachers began drawing a line. They taught that whole numbers meant the natural numbers and excluded negatives, while integer kept the negative numbers inside. The result was confusion that never fully resolved. Whole numbers remain ambiguous to the present day, their meaning depending on who is speaking and when they learned the term. The naturals themselves sit at the bottom of a nested chain: the natural numbers form a subset of the integers, which are a subset of the rationals, which are a subset of the reals. An integer is a real number that can be written without a fractional component, so 21, 4, 0, and -2048 qualify, while 9.75, 5/4, and the square root of 2 do not.
Closure under subtraction is what sets the integers apart from the natural numbers. Add or multiply any two integers and you get an integer, just as with the naturals, but the inclusion of the negatives means subtraction stays inside the set too. Division breaks this pattern, since 1 divided by 2 is not an integer, and exponentiation fails when a negative exponent produces a fraction. The integers form the smallest group and the smallest ring containing the natural numbers. Under addition they make an abelian group that is also cyclic, because every non-zero integer is a finite sum of 1s or of -1s. They are the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to them. Under multiplication they form a commutative monoid but not a group, since the number 2 has no multiplicative inverse. The only invertible integers, called units, are -1 and 1.
An integral domain is what the integers become once you note they have no zero divisors. With addition and multiplication together they form a commutative ring with unity, the prototype of all such structures. They are not a field, because the missing multiplicative inverses leave them not closed under division. The smallest field that contains them as a subring is the field of rational numbers, built by mimicking the field of fractions of any integral domain. Ordinary division may be absent, but division with remainder survives as Euclidean division. Given two integers and a non-zero divisor, there exist a unique quotient and a unique remainder, with the remainder smaller in absolute value than the divisor. The Euclidean algorithm for greatest common divisors runs on a sequence of these divisions. This makes the integers a Euclidean domain, and therefore a principal ideal domain, which is why any positive integer factors into primes in an essentially unique way, the fundamental theorem of arithmetic.
Ordered pairs of natural numbers can construct the integers without a single special case. The traditional route defines them as the union of the positive naturals, zero, and the negations of the naturals, formalized with the Peano axioms and a one-to-one copy of the naturals. That route is tedious, because each arithmetic operation has to be defined separately for positive, negative, and zero arguments. The modern alternative treats an integer as an equivalence class of ordered pairs, where a pair stands for the result of subtracting one natural from the other. Under this rule 1 - 2 and 4 - 5 fall into the same class. Negation reverses the order of the pair, and subtraction becomes addition of the additive inverse. The proof assistant Isabelle uses exactly this technique, where the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2). There exist at least ten such constructions of signed integers, differing in their basic operations and in whether those operations are free constructors. Many other tools prefer the free-constructor approach, which is simpler and runs more efficiently.
Aleph-null is the size of the set of integers, the same cardinality as the natural numbers. The set is countably infinite, which means every integer can be paired with a unique natural number through a bijection. That same infinity meets a hard limit inside a computer. An integer is often a primitive data type, denoted int or Integer in languages such as Algol68, C, Java, and Delphi, but a fixed-length type can only hold a subset of all integers. The common two's complement representation distinguishes negative from non-negative rather than negative, positive, and zero. Fixed-size types usually run to a number of bits that is a power of 2, like 4, 8, or 16, or a memorable count of decimal digits such as 9 or 10. To escape the ceiling, variable-length representations called bignums can store any integer that fits in the computer's memory, the closest a finite machine comes to the boundless set the bijection describes.
Common questions
What is an integer in mathematics?
An integer is the number zero, a positive natural number such as 1, 2, or 3, or the negation of a positive natural number such as -1, -2, or -3. An integer is a real number that can be written without a fractional component, so 21, 4, 0, and -2048 are integers while 9.75, 5/4, and the square root of 2 are not.
Where does the word integer come from?
The word integer comes from the Latin integer, meaning whole or literally untouched, from in meaning not plus tangere meaning to touch. The word entire derives from the same origin through the French word entier, which means both entire and integer.
Why is the letter Z used to denote integers?
The letter Z comes from the German word Zahlen, meaning numbers, and its choice has been attributed to David Hilbert. Its earliest known textbook use appears in Algèbre by the collective Nicolas Bourbaki, dated 1947, and by 1961 modern algebra texts generally used Z for the positive and negative integers.
Who first defined integers to include negative numbers?
Leonhard Euler defined integers to include both positive and negative numbers in his 1765 Elements of Algebra. Before that, the term applied only to positive integers and was synonymous with the natural numbers.
What is the difference between whole numbers and integers?
Whole numbers were synonymous with the integers until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers meant the natural numbers and excluded negatives, while integer included the negative numbers, and the term whole numbers remains ambiguous today.
Why are the integers not a field?
The integers are not a field because they lack multiplicative inverses, which means they are not closed under division. They form an integral domain, and the smallest field that contains them as a subring is the field of rational numbers.
Are the integers countable?
Yes, the set of integers is countably infinite, meaning each integer can be paired with a unique natural number through a bijection. Its cardinality equals aleph-null, the same as the natural numbers.