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An integer (from the Latin integer meaning "whole")[note ១] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 12, and 2 are not.

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers,[១][២] and their additive inverses (the negative integers, i.e., −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold '"`UNIQ--postMath-00000001-QINU`"' (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[៣][៤]

ℤ is a subset of the sets of rational numbers ℚ, in turn a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.



The symbol ℤ can be annotated to indicate various subsets, with varying usage amongst different authors: ℤ++ or ℤ> for the positive integers, ℤ for non-negative integers, ℤ for non-zero integers, Some authors use ℤ* for non-zero integers, others use it for non-negative integers. ℤn to mean the set of integers modulo n: {0, 1, 2, .. n-1}. [៥] [៦] [៧]

Algebraic properties[កែប្រែ]





Properties of addition and multiplication on integers
Addition Multiplication




The lack of zero-divisors in the integers (last property in the table) means that the commutative ring Z is an integral domain.


Moreover, Z is a principal ideal.[៨]



Order-theoretic properties[កែប្រែ]





The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.[៩] This is equivalent to the statement that any Noetherian valuation ring is either a field or a discrete valuation ring.


Representation of equivalence classes for the numbers −5 to 5

In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and the negations of the natural numbers. However, this style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that these operations obey the laws of arithmetic.[១០] Therefore, in modern set-theoretic mathematics a more abstract construction,[១១] which allows one to define the arithmetical operations without any case distinction, is often used instead.[១២] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).[១៣]

The intuition is that (a,b) stands for the result of subtracting b from a.[១៣] To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:


precisely when


Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[១៣] denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:






The standard ordering on the integers is given by:

'"`UNIQ--postMath-00000008-QINU`"' iff '"`UNIQ--postMath-00000009-QINU`"'



Thus, [(a,b)] is denoted by




Some examples are:


Computer science[កែប្រែ]











See also[កែប្រែ]

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  1. Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier, which means both entire and integer (see: Evans, Nick (1995)។ "A-Quantifiers and Scope"។ ជា Bach, Emmon W។ Quantification in Natural Languages។ Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers។ ទំ. 262។ ល.ស.ប.អ. 0-7923-3352-7https://books.google.com/?id=NlQL97qBSZkC )


  1. Weisstein, Eric W., "Counting Number", MathWorld.
  2. Weisstein, Eric W., "Whole Number", MathWorld.
  3. Miller, Jeff (2010-08-29)។ "Earliest Uses of Symbols of Number Theory"http://jeff560.tripod.com/nth.html។ បានយកមក 2010-09-20 
  4. Peter Jephson Cameron (1998)។ Introduction to Algebra។ Oxford University Press។ ទំ. 4។ ល.ស.ប.អ. 978-0-19-850195-4https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 
  6. LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.
  7. http://mathworld.wolfram.com/Z-Star.html
  8. Serge, Lang (1993), [Expression error: Unrecognized punctuation character "១". Algebra] (3rd រ.រ.), Addison-Wesley, ល.ស.ប.អ. 0-201-55540-9, Expression error: Unrecognized punctuation character "១". 
  9. Warner, Seth (2012), Modern Algebra, Dover Books on Mathematics, Courier Corporation, Theorem 20.14, p. 185, ល.ស.ប.អ. 9780486137094, https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185 .
  10. Mendelson, Elliott (2008), Number Systems and the Foundations of Analysis, Dover Books on Mathematics, Courier Dover Publications, ទ. 86, ល.ស.ប.អ. 9780486457925, https://books.google.com/books?id=3domViIV7HMC&pg=PA86 .
  11. Ivorra Castillo: Álgebra
  12. Frobisher, Len (1999), Learning to Teach Number: A Handbook for Students and Teachers in the Primary School, The Stanley Thornes Teaching Primary Maths Series, Nelson Thornes, ទ. 126, ល.ស.ប.អ. 9780748735150, https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126 .
  13. ១៣,០ ១៣,១ ១៣,២ Campbell, Howard E. (1970)។ The structure of arithmetic។ Appleton-Century-Crofts។ ទំ. 83។ ល.ស.ប.អ. 0-390-16895-5 


External links[កែប្រែ]


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ទំព័រគំរូ:Number Systems ទំព័រគំរូ:Rational numbers ទំព័រគំរូ:Integers