Concept
Integer
Overview
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold
Z
{\displaystyle \mathbb {Z} }
.
The set of natural numbers
N
{\displaystyle \mathbb {N} }
is a subset of
Z
{\displaystyle \mathbb {Z} }
, which in turn is a subset of the set of all rational numbers
Q
{\displaystyle \mathbb {Q} }
, itself a subset of the real numbers
R
{\displaystyle \mathbb {R} }
. Like the set of natural numbers, the set of integers
Z
{\displaystyle \mathbb {Z} }
is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, 5/4, and the square root of 2 are not.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Integer is discussed.
M. Couturat begins by stating Peano's five assumptions, which are independent, as has been proved by Peano and Padoa. 1. Zero is an integer. 2. Zero is not the successor of any integer. 3. The successor of an integer is an integer. To this it would be proper to add, Every integer has a successor. 4. Two integers are equal if their successors are. The fifth assumption is the principle of complete induction. M.
Henri Poincaré, The Foundations of Science
names the idea; states what it is
This improvement also is due to Cantor, and it is one which has revolutionised mathematics. Only order is now relevant to limits. Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it. The study of different types of series is a general subject of which the study of ordinal numbers (mentioned above) is a special…
Bertrand Russell, Mysticism and Logic and Other Essays
states what it is; names the idea
Take such a case as the following: We know that any two numbers can be multiplied together, and will give a third called their _product_. We know that all pairs of integers the product of which is less than 100 have been actually multiplied together, and the value of the product recorded in the multiplication table. But we also know that the number of integers is infinite, and that only a finite number of pairs of…
Bertrand Russell, The Problems of Philosophy
names the idea
But this very paradox leads to the real principle of generalization concerning the properties of numbers. Two different numbers can not be formed in the same manner from the same numbers; but they may be formed in the same manner from different numbers; as nine is formed from three by multiplying it into itself, and sixteen is formed from four by the same process. Thus there arises a classification of modes of…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
states what it is
This usage was first employed in equations, which are methods of asking complicated arithmetical questions. In this connection, the letters representing numbers were termed ‘unknowns.’ But equations soon suggested a new idea, that, namely, of a function of one or more general symbols, these symbols being letters representing any numbers. In this employment the algebraic letters are called the ‘arguments’ of the…
Alfred North Whitehead, Science and the modern world
states what it is
Philosophers who discussed Integer
Related ideas
- French language
- German language
- Georg Cantor
- Proof assistant
- Infinite set
- If and only if
- Negation
- Latin
- Rational number
- Real number
- Natural number
- Number
- Ordered pair
- Peano axioms
- Automated theorem proving
- Set (mathematics)
- Set theory
- Subset
- Total order
- Square root of 2
- 0
- Bijection
- Countable set
- Cardinality
Place in the tradition
Part of First-order logic.