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69-481: An integer n {\displaystyle n} is divisible by a nonzero integer m {\displaystyle m} if there exists an integer k {\displaystyle k} such that n = k m . {\displaystyle n=km.} This is written as This may be read as that m {\displaystyle m} divides n , {\displaystyle n,} m {\displaystyle m}

138-517: A {\displaystyle p\mid a} or p ∣ b . {\displaystyle p\mid b.} A positive divisor of n {\displaystyle n} that is different from n {\displaystyle n} is called a proper divisor or an aliquot part of n {\displaystyle n} (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves

207-466: A ∣ b c , {\displaystyle a\mid bc,} and gcd ( a , b ) = 1 , {\displaystyle \gcd(a,b)=1,} then a ∣ c . {\displaystyle a\mid c.} This is called Euclid's lemma . If p {\displaystyle p} is a prime number and p ∣ a b {\displaystyle p\mid ab} then p ∣

276-445: A . 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; by using [( a , b )] to denote the equivalence class having ( a , b ) as a member, one has: The negation (or additive inverse) of an integer

345-549: A negative number is the opposite (mathematics) of a positive real number . Equivalently, a negative number is a real number that is less than zero . Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative . Negative numbers are used to describe values on

414-473: A few basic operations (e.g., zero , succ , pred ) and using natural numbers , which are assumed to be already constructed (using the Peano approach ). There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2), and the types of arguments accepted by these operations;

483-458: A finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1) . In fact, Z {\displaystyle \mathbb {Z} } under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to Z {\displaystyle \mathbb {Z} } . The first four properties listed above for multiplication say that Z {\displaystyle \mathbb {Z} } under multiplication

552-453: A larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example, since 8 − 5 = 3 . The minus sign "−" signifies the operator for both the binary (two- operand ) operation of subtraction (as in y − z ) and the unary (one-operand) operation of negation (as in − x , or twice in −(− x ) ). A special case of unary negation occurs when it operates on

621-403: A negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that losing a debt is the same thing as gaining a credit.) Thus and When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules: Thus and The reason behind the first example

690-458: A negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". Conversely, a number that is greater than zero is called positive ; zero is usually ( but not always ) thought of as neither positive nor negative . The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign . Every real number other than zero

759-456: A positive number, in which case the result is a negative number (as in −5 ). The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize

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828-414: A remainder is sometimes called an aliquant part of n . {\displaystyle n.} An integer n > 1 {\displaystyle n>1} whose only proper divisor is 1 is called a prime number . Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself. Any positive divisor of n {\displaystyle n}

897-454: A scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, − ‍ (−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents

966-400: A set P − {\displaystyle P^{-}} which is disjoint from P {\displaystyle P} and in one-to-one correspondence with P {\displaystyle P} via a function ψ {\displaystyle \psi } . For example, take P − {\displaystyle P^{-}} to be

1035-409: Is Euler–Mascheroni constant . One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about ln ⁡ n . {\displaystyle \ln n.} However, this is a result from the contributions of numbers with "abnormally many" divisors . In definitions that allow the divisor to be 0, the relation of divisibility turns

1104-462: 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 √ 2 are not. The integers form the smallest group and the smallest ring containing the natural numbers . In algebraic number theory , the integers are sometimes qualified as rational integers to distinguish them from

1173-433: Is a commutative monoid . However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that Z {\displaystyle \mathbb {Z} } under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that Z {\displaystyle \mathbb {Z} } together with addition and multiplication

1242-422: Is a commutative ring with unity . It is the prototype of all objects of such algebraic structure . Only those equalities of expressions are true in  Z {\displaystyle \mathbb {Z} } for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings. The lack of zero divisors in the integers (last property in

1311-680: Is a multiplicative function d ( n ) , {\displaystyle d(n),} meaning that when two numbers m {\displaystyle m} and n {\displaystyle n} are relatively prime , then d ( m n ) = d ( m ) × d ( n ) . {\displaystyle d(mn)=d(m)\times d(n).} For instance, d ( 42 ) = 8 = 2 × 2 × 2 = d ( 2 ) × d ( 3 ) × d ( 7 ) {\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)} ;

1380-410: 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} }

1449-420: Is a divisor of n , {\displaystyle n,} m {\displaystyle m} is a factor of n , {\displaystyle n,} or n {\displaystyle n} is a multiple of m . {\displaystyle m.} If m {\displaystyle m} does not divide n , {\displaystyle n,} then

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1518-574: Is a product of prime divisors of n {\displaystyle n} raised to some power. This is a consequence of the fundamental theorem of arithmetic . A number n {\displaystyle n} is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than n , {\displaystyle n,} and abundant if this sum exceeds n . {\displaystyle n.} The total number of positive divisors of n {\displaystyle n}

1587-469: Is called the quotient and r is called the remainder of the division of a by b . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. The above says that Z {\displaystyle \mathbb {Z} } is a Euclidean domain . This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain , and any positive integer can be written as

1656-424: Is considered to be less than negative 5 : In the context of negative numbers, a number that is greater than zero is referred to as positive . Thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative,

1725-465: Is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers . (Some definitions of the natural numbers exclude zero.) In bookkeeping , amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers were used in

1794-475: Is equivalent to the statement that any Noetherian valuation ring is either a field —or a discrete valuation ring . In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero , and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms , call this P {\displaystyle P} . Then construct

1863-468: Is greater than zero , and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: Thus it follows that Z {\displaystyle \mathbb {Z} } together with the above ordering is an ordered ring . The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered . This

1932-401: Is identified with the class [( n ,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [( n ,0)] ), and the class [(0, n )] is denoted − n (this covers all remaining classes, and gives the class [(0,0)] a second time since –0 = 0. Thus, [( a , b )] is denoted by If the natural numbers are identified with the corresponding integers (using

2001-404: Is not a trivial divisor is known as a non-trivial divisor (or strict divisor). A nonzero integer with at least one non-trivial divisor is known as a composite number , while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits. There are some elementary rules: If

2070-437: Is not defined on Z {\displaystyle \mathbb {Z} } , the division "with remainder" is defined on them. It is called Euclidean division , and possesses the following important property: given two integers a and b with b ≠ 0 , there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b | , where | b | denotes the absolute value of b . The integer q

2139-404: Is not free since the integer 0 can be written pair (0,0), or pair (1,1), or pair (2,2), etc.. This technique of construction is used by the proof assistant Isabelle ; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers. Negative number In mathematics ,

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2208-431: Is obtained by reversing the order of the pair: Hence subtraction can be defined as the addition of the additive inverse: The standard ordering on the integers is given by: It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form ( n ,0) or (0, n ) (or both at once). The natural number n

2277-450: Is often expressed in the form of a number line : Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are lesser. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example, even though (positive) 8 is greater than (positive) 5 , written negative 8

2346-468: Is simple: adding three −2 's together yields −6 : The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six: The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law . In this case, we know that Since 2 × (−3) = −6 ,

2415-508: Is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p -adic integers . The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that whole numbers referred to the natural numbers , excluding negative numbers, while integer included

2484-421: Is very similar to addition of two positive numbers. For example, The idea is that two debts can be combined into a single debt of greater magnitude. When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example: In the first example, a credit of 8 is combined with a debt of 3 , which yields a total credit of 5 . If

2553-616: The Nine Chapters on the Mathematical Art , which in its present form dates from the period of the Chinese Han dynasty (202 BC – AD 220), but may well contain much older material. Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed

2622-584: The lattice of subgroups of the infinite cyclic group Z. Integer 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} }

2691-405: The natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers ( a , b ). We can extend addition and multiplication to these pairs with the following rules: We define an equivalence relation ~ upon these pairs with the following rule: This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be

2760-426: The ordered pairs ( 1 , n ) {\displaystyle (1,n)} with the mapping ψ = n ↦ ( 1 , n ) {\displaystyle \psi =n\mapsto (1,n)} . Finally let 0 be some object not in P {\displaystyle P} or P − {\displaystyle P^{-}} , for example the ordered pair (0,0). Then

2829-736: The prime factorization of n {\displaystyle n} is given by then the number of positive divisors of n {\displaystyle n} is and each of the divisors has the form where 0 ≤ μ i ≤ ν i {\displaystyle 0\leq \mu _{i}\leq \nu _{i}} for each 1 ≤ i ≤ k . {\displaystyle 1\leq i\leq k.} For every natural n , {\displaystyle n,} d ( n ) < 2 n . {\displaystyle d(n)<2{\sqrt {n}}.} Also, where γ {\displaystyle \gamma }

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2898-414: The definition of negation to include zero and negative numbers. Specifically: For example, the negation of −3 is +3 . In general, The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3 , and the absolute value of 0 is 0 . In a similar manner to rational numbers , we can extend

2967-472: The eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbers m {\displaystyle m} and n {\displaystyle n} share a common divisor, then it might not be true that d ( m n ) = d ( m ) × d ( n ) . {\displaystyle d(mn)=d(m)\times d(n).} The sum of

3036-407: The embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} . Some examples are: In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines . Integers are represented as algebraic terms built using

3105-1179: The integers are defined to be the union P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}} . The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows: − x = { ψ ( x ) , if  x ∈ P ψ − 1 ( x ) , if  x ∈ P − 0 , if  x = 0 {\displaystyle -x={\begin{cases}\psi (x),&{\text{if }}x\in P\\\psi ^{-1}(x),&{\text{if }}x\in P^{-}\\0,&{\text{if }}x=0\end{cases}}} The traditional 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 integers obey

3174-458: The integers are not (since the result can be a fraction when the exponent is negative). The following table lists some of the basic properties of addition and multiplication for any integers a , b , and c : The first five properties listed above for addition say that Z {\displaystyle \mathbb {Z} } , under addition, is an abelian group . It is also a cyclic group , since every non-zero integer can be written as

3243-448: The integers as a subring is the field of rational numbers . The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z {\displaystyle \mathbb {Z} } as its subring . Although ordinary division

3312-447: The integers into this ring. This universal property , namely to be an initial object in the category of rings , characterizes the ring  Z {\displaystyle \mathbb {Z} } . Z {\displaystyle \mathbb {Z} } is not closed under division , since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation ,

3381-510: The more general algebraic integers . In fact, (rational) integers are algebraic integers that are also rational numbers . The word integer comes from the Latin integer meaning "whole" or (literally) "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 . Historically the term

3450-425: The negative number has greater magnitude, then the result is negative: Here the credit is less than the debt, so the net result is a debt. As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer: In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus and On the other hand, subtracting

3519-437: The negative numbers. The whole numbers remain ambiguous to the present day. Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra Like the natural numbers , Z {\displaystyle \mathbb {Z} } is closed under the operations of addition and multiplication , that is,

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3588-415: The notation is m ∤ n . {\displaystyle m\not \mid n.} There are two conventions, distinguished by whether m {\displaystyle m} is permitted to be zero: Divisors can be negative as well as positive, although often the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only

3657-686: The positive divisors of n {\displaystyle n} is another multiplicative function σ ( n ) {\displaystyle \sigma (n)} (for example, σ ( 42 ) = 96 = 3 × 4 × 8 = σ ( 2 ) × σ ( 3 ) × σ ( 7 ) = 1 + 2 + 3 + 6 + 7 + 14 + 21 + 42 {\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42} ). Both of these functions are examples of divisor functions . If

3726-702: The positive integers, Z 0 + {\displaystyle \mathbb {Z} ^{0+}} or Z ≥ {\displaystyle \mathbb {Z} ^{\geq }} for non-negative integers, and Z ≠ {\displaystyle \mathbb {Z} ^{\neq }} for non-zero integers. Some authors use Z ∗ {\displaystyle \mathbb {Z} ^{*}} for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the group of units of Z {\displaystyle \mathbb {Z} } ). Additionally, Z p {\displaystyle \mathbb {Z} _{p}}

3795-534: The positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even , and integers not divisible by 2 are called odd . 1, −1, n {\displaystyle n} and − n {\displaystyle -n} are known as the trivial divisors of n . {\displaystyle n.} A divisor of n {\displaystyle n} that

3864-727: The presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms. The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair ( x , y ) {\displaystyle (x,y)} that takes as arguments two natural numbers x {\displaystyle x} and y {\displaystyle y} , and returns an integer (equal to x − y {\displaystyle x-y} ). This operation

3933-420: The product (−2) × (−3) must equal 6 . These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a as follows: The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers . The sign rules for division are the same as for multiplication. For example, and If dividend and divisor have

4002-412: The products of primes in an essentially unique way. This is the fundamental theorem of arithmetic . Z {\displaystyle \mathbb {Z} } is a totally ordered set without upper or lower bound . The ordering of Z {\displaystyle \mathbb {Z} } is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... . An integer is positive if it

4071-496: The rules of subtracting and multiplying negative numbers and solved problems with negative coefficients . Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz held that negative numbers were invalid, but still used them in calculations. The relationship between negative numbers, positive numbers, and zero

4140-560: The same sign, the result is positive, if they have different signs the result is negative. The negative version of a positive number is referred to as its negation . For example, −3 is the negation of the positive number 3 . The sum of a number and its negation is equal to zero: That is, the negation of a positive number is the additive inverse of the number. Using algebra , we may write this principle as an algebraic identity : This identity holds for any positive number x . It can be made to hold for all real numbers by extending

4209-401: The set N {\displaystyle \mathbb {N} } of non-negative integers into a partially ordered set that is a complete distributive lattice . The largest element of this lattice is 0 and the smallest is 1. The meet operation ∧ is given by the greatest common divisor and the join operation ∨ by the least common multiple . This lattice is isomorphic to the dual of

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4278-562: The set of the integers was not used before the end of the 19th century, when Georg Cantor introduced the concept of infinite sets and set theory . The use of the letter Z to denote the set of integers comes from the German word Zahlen ("numbers") and has been attributed to David Hilbert . The earliest known use of the notation in a textbook occurs in Algèbre written by the collective Nicolas Bourbaki , dating to 1947. The notation

4347-404: The sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0 ), Z {\displaystyle \mathbb {Z} } , unlike the natural numbers, is also closed under subtraction . The integers form a ring which is the most basic one, in the following sense: for any ring, there is a unique ring homomorphism from

4416-422: The table) means that the commutative ring  Z {\displaystyle \mathbb {Z} } is an integral domain . The lack of multiplicative inverses, which is equivalent to the fact that Z {\displaystyle \mathbb {Z} } is not closed under division, means that Z {\displaystyle \mathbb {Z} } is not a field . The smallest field containing

4485-417: The term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: In general, the subtraction of

4554-565: The unary "−" along with its operand. For example, the expression 7 + −5 may be clearer if written 7 + (−5) (even though they mean exactly the same thing formally). The subtraction expression 7 – 5 is a different expression that doesn't represent the same operations, but it evaluates to the same result. Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in Addition of two negative numbers

4623-416: The various laws of arithmetic. In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers ( a , b ) . The intuition is that ( a , b ) stands for the result of subtracting b from

4692-657: Was not adopted immediately. For example, another textbook used the letter J, and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers. The symbol Z {\displaystyle \mathbb {Z} } is often annotated to denote various sets, with varying usage amongst different authors: Z + {\displaystyle \mathbb {Z} ^{+}} , Z + {\displaystyle \mathbb {Z} _{+}} , or Z > {\displaystyle \mathbb {Z} ^{>}} for

4761-445: Was used for a number that was a multiple of 1, or to the whole part of a mixed number . Only positive integers were considered, making the term synonymous with the natural numbers . The definition of integer expanded over time to include negative numbers as their usefulness was recognized. For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers. The phrase

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