Integer
Definition and Properties
An integer is a number that can be written without a fractional or decimal component. It includes the counting numbers {1, 2, 3, ...}, zero {0}, and the negative of the counting numbers {-1, -2, -3, ...}. You can think of the integers as the whole numbers, but this number system is expanded to include negative numbers as well.
In the field of mathematics, integers have several important properties that distinguish them from other number sets. These properties include the following:
- Closure: The set of integers is closed under the operations of addition, subtraction, and multiplication. This means that if you add, subtract, or multiply any two integers, the result is always an integer.
- Associativity: The operations of addition and multiplication are associative for integers. That is, for any integers a, b, and c, the equation (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c) hold true.
- Commutativity: The operations of addition and multiplication are commutative for integers. That is, for any integers a and b, the equation a + b = b + a and a * b = b * a hold true.
- Existence of Identity: For addition, the identity element is 0, because adding 0 to any integer does not change its value. For multiplication, the identity element is 1, because multiplying any integer by 1 does not change its value.
- Existence of Inverse: Every integer has an additive inverse. The additive inverse of an integer a is the integer -a, such that a + (-a) = 0. However, integers do not have multiplicative inverses (except for 1 and -1).
- Distributivity: The operation of multiplication is distributive over addition for integers. That is, for any integers a, b, and c, the equation a * (b + c) = a * b + a * c holds true.
Classification of Integers
Integers can be classified into different types based on their properties. The main types of integers are:
- Positive Integers: These are the numbers greater than zero. They are the numbers that we use most often for counting and ordering.
- Negative Integers: These are the numbers less than zero. They are used to represent the opposite of something, such as debt, loss, or decrease in value.
- Zero: Zero is neither positive nor negative. It is used to represent the absence of value or a neutral position in a scale.
- Even and Odd Integers: An integer is even if it is divisible by 2, and odd if it is not. This classification is used in various mathematical theorems and computations.
- Prime and Composite Integers: A prime integer is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. A composite integer is a positive integer greater than 1 that has more than two positive integer divisors.
Operations on Integers
The basic operations that can be performed on integers are addition, subtraction, multiplication, and division. However, division of integers does not always result in an integer. When an integer is divided by another, the result is a rational number if the division does not result in a whole number.
- Addition: The sum of two integers is an integer. If both integers are positive, the sum is positive. If both integers are negative, the sum is negative. If one integer is positive and the other is negative, the sum is the integer with the larger absolute value, with the sign of the integer with the larger absolute value.
- Subtraction: The difference of two integers is an integer. Subtraction can be thought of as the addition of the minuend and the additive inverse of the subtrahend.
- Multiplication: The product of two integers is an integer. If the signs of the integers are the same, the product is positive. If the signs of the integers are different, the product is negative.
- Division: The quotient of two integers is a rational number. If the signs of the integers are the same, the quotient is positive. If the signs of the integers are different, the quotient is negative.
Applications of Integers
Integers have many applications in various fields of study and in everyday life. Here are some of the main applications:
- Mathematics: Integers are fundamental to many areas of mathematics, including number theory, algebra, and combinatorics.
- Computer Science: In computer science, integers are used in data structures and algorithms. They are also used in the representation of data in computer systems.
- Physics: In physics, integers are used in the quantum theory of particles, where they represent quantum numbers.
- Economics: In economics, integers are used in the modeling of discrete variables, such as the number of goods produced or consumed.
- Everyday Life: In everyday life, integers are used to count, to represent the time, to measure temperature, and to represent the score in games, among other things.