Integer

An integer is a whole number that can be either positive, negative, or zero. It is a mathematical concept used to represent discrete quantities, such as the number of objects in a collection, the score in a game, or the value of a variable in an equation.

Integers are represented by the symbol "Z" and can be written as {...,-3,-2,-1,0,1,2,3,...}. They are a subset of the real numbers and are used in various mathematical operations, such as addition, subtraction, multiplication, and division.

Integers are a fundamental concept in mathematics, and they are used in many areas, including number theory, algebra, geometry, and calculus. They are a type of number that does not have any fractional or decimal parts, and they can be classified as positive, negative, or zero.

Positive integers are greater than zero, and negative integers are less than zero. Zero is considered to be neither positive nor negative, and it is sometimes referred to as a "neutral" integer. Integers can be represented on a number line, where positive integers are located to the right of zero, and negative integers are located to the left of zero.

Integers are closed under addition, subtraction, and multiplication, meaning that if you add, subtract, or multiply two integers, you will always get an integer as a result. However, they are not closed under division, meaning that dividing two integers may result in a non-integer value, such as a fraction or decimal.

Integers have many important properties that make them useful in various mathematical operations. Here are some of the key properties of integers:

1. Closure: Integers are closed under addition, subtraction, and multiplication, which means that the result of these operations is always an integer.

2. Commutativity: Addition and multiplication of integers are commutative, which means that the order of the operands does not affect the result. For example, a + b = b + a and a x b = b x a.

3. Associativity: Addition and multiplication of integers are associative, which means that the grouping of the operands does not affect the result. For example, (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).

4. Distributivity: Multiplication distributes over addition, which means that a x (b + c) = a x b + a x c.

5. Identity: Integers have additive and multiplicative identities, which are 0 and 1, respectively. This means that adding 0 to any integer does not change its value, and multiplying any integer by 1 does not change its value.

6. Inverses: Integers have additive inverses, which means that for every integer a, there exists an integer -a such that a + (-a) = 0. However, not all integers have multiplicative inverses, meaning that not all integers have another integer that can be multiplied by them to give 1.

7. Order: Integers have a natural order, which means that they can be arranged from least to greatest or from greatest to least. This order is preserved under addition and multiplication, which means that if a < b and c > 0, then a + c < b + c and a x c < b x c.