Integers

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Integers include positive numbers, negative numbers, and zero. 'Integer' is a Latin word which means 'whole' or 'intact'. This means integers do not include fractions or decimals. Let us learn more about integers, the definition of integers, and the properties of integers in this article.

An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero. A few examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043. A set of integers, which is represented as Z, includes:

A number line is a visual representation of numbers on a straight line. This line is used for the comparison of numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally. Just like other numbers, the set of integers can also be represented on a number line.

Positive and negative integers can be visually represented on a number line. Integers on a number line help in performing arithmetic operations. The basic points to keep in mind while placing integers on a number line are as follows:

Adding integers is the process of finding the sum of two or more integers where the value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the addition of integers are given in the following section.

Subtracting integers is the process of finding the difference between two or more integers where the final value might increase or decrease depending on the integer being positive or negative. The different rules and the possible cases for the subtraction of integers are given in the following section.

Division of integers means equal grouping or dividing an integer into a specific number of groups. For the division of integers, we use the rules given in the following table. The different rules and the possible cases for the division of integers are given in the following section

The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b:

According to the associative property, changing the grouping of two integers does not change the result of the operation. The associative property applies to the addition and multiplication of two integers.

The given expression, 3 + (7 + 2) = (3 + 7) + 2, shows the associative property of integers, which says that changing the grouping of two integers does not change the result of the operation because 3 + (7 + 2) = (3 + 7) + 2 = 12.

According to the rules of integers in addition, when both the integers have the same signs, we add the absolute values of integers and give the same sign as that of the given integers to the result. The absolute values of the given integers is 9 and 5. So we will add 9 + 5 = 14 and the sign of the sum will be negative. Therefore, (-9) + (-5) = -14

According to the rules of integers in addition, when one integer is positive and the other is negative, we find the difference of the absolute values of the numbers and then give the sign of the larger number to the result. For example, if we need to add 6 + (-4), we will find the difference of the absolute values of the given integers, that is, the difference between 6 and 4 is 2, and the result will have a positive sign because the larger number (6) has a positive sign.

The application of positive and negative numbers in the real world is different. While positive numbers are commonly used everywhere, the negative integers are used in measuring temperature which can also have a negative value, that is, the temperature of a city can be -4C or -10C. The negative and positive numbers and zero in the scale denote different temperature readings. Bank credit and debit statements also use integers to represent the negative or positive values in transactions.

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 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.[10] Historically the term was used for a number that was a multiple of 1,[11][12] or to the whole part of a mixed number.[13][14] 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.[15] For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers.[16] However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.[15]

The use of the letter Z to denote the set of integers comes from the German word Zahlen (\"number\")[4][5] and has been attributed to David Hilbert.[17] The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947.[4][18] The notation was not adopted immediately, for example another textbook used the letter J[19] and a 1960 paper used Z to denote the non-negative integers.[20] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.[21]

The whole numbers were synonymous with the integers up until the early 1950s.[24][25][26] In the late 1950s, as part of the New Math movement,[27] American elementary school teachers began teaching that \"whole numbers\" referred to the natural numbers, excluding negative numbers, while \"integer\" included the negative numbers.[28][29] \"Whole number\" remains ambiguous to the present day.[30]

Like the natural numbers, Z {\\displaystyle \\mathbb {Z} } is closed under the operations of addition and multiplication, that is, 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.[31]

The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from 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, the integers are not (since the result can be a fraction when the exponent is negative).

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 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 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 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.

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 the various laws of arithmetic.[35]

In modern set-theoretic mathematics, a more abstract construction[36][37] allowing one to define arithmetical operations without any case distinction is often used instead.[38] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).[39]

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[39] by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has:

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 a few basic operations (e.g., zero, succ, pred) and, possibly, using natural numbers, which are assumed to be already constructed (using, say, the Peano approach).

There exist at least ten such constructions of signed integers.[40] 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; 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.

An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between \"negative\" and \"non-negative\" rather than \"negative, positive, and 0\". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). 153554b96e

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