Definition of Integers
Integers are a fundamental set of numbers in mathematics that consist of all whole numbers, including positive values, negative values, and zero. They are denoted by the symbol ℤ and include numbers like ..., -3, -2, -1, 0, 1, 2, 3, .... Unlike fractions or decimals, integers have no fractional or decimal parts.
Key Properties of Integers
Integers exhibit several important properties that make them closed under basic operations: addition and subtraction of integers always yield another integer, while multiplication and division (when defined) follow specific rules. They are ordered, with negative integers less than zero and positive integers greater than zero. The absolute value of an integer gives its distance from zero on the number line.
Practical Examples of Integers
Consider temperatures: 5°C represents a positive integer for mild weather, -3°C a negative integer for freezing conditions, and 0°C exactly at the freezing point. In everyday counting, integers appear in scores like +10 points gained or -2 points deducted in a game, illustrating their use in tracking changes.
Importance and Applications of Integers
Integers form the basis for arithmetic operations and are crucial in fields like algebra, computer science, and physics for modeling discrete quantities. They enable precise calculations in budgeting (positive and negative balances), coordinate geometry (integer coordinates), and programming (loop counters), providing a reliable framework for quantitative reasoning.