täisarvuga

Täisarvuga, also known as integer in English, refers to a whole number that can be positive, negative, or zero. In mathematics, integers are a fundamental part of the number system and are used extensively in various fields such as algebra, number theory, and computer science. They are distinct from natural numbers, which are typically considered to be the positive integers (1, 2, 3, ...), as integers include zero and negative numbers as well (..., -2, -1, 0, 1, 2, ...).

Integers are closed under addition, subtraction, and multiplication, meaning that performing these operations on any two integers will always yield another integer. However, division of integers is not always closed, as the result may not be an integer (e.g., 1 divided by 2 equals 0.5, which is not an integer). This property is why integers are not closed under division.

Integers are represented in various numeral systems, including the decimal system (base 10), which is the most commonly used in everyday life. They are also used in binary (base 2) and other bases in computer science, where they serve as the foundation for representing data and performing computations. In programming, integers are typically represented using data types such as `int` or `long`, which store whole numbers within specific ranges.

In number theory, integers play a crucial role in studying properties such as divisibility, prime numbers, and modular arithmetic. They are also essential in cryptography, where operations on integers underpin many encryption algorithms. The set of integers is often denoted by the symbol **ℤ**, derived from the German word *Zahlen*, meaning "numbers."

Integers are also used in real-world applications, such as counting objects, measuring discrete quantities, and indexing elements in sequences or arrays. Their simplicity and well-defined properties make them indispensable in both theoretical and applied mathematics.