An integer is a whole number that can be positive, negative, or zero. These fundamental units, distinct from Fractions and decimal numbers, form the backbone of Number Theory and arithmetic. The set of all integers is commonly denoted by the symbol $\mathbb{Z}$.
Definition and Properties
Integers are numbers that can be written without a fractional component. The set of integers $\mathbb{Z}$ includes: $\mathbb{Z} = {..., -3, -2, -1, 0, 1, 2, 3, ...}$
Key properties of integers include:
- No fractional parts: Unlike Rational Number or Real Number, integers do not have decimal or fractional components.
- Closure: The set of integers is closed under Addition, Subtraction, and Multiplication. This means that if you add, subtract, or multiply any two integers, the result will always be an integer.
- Ordering: Integers can be ordered on a Number Line, extending infinitely in both positive and negative directions.
- Absolute Value: For any integer $x$, its Absolute Value, denoted $|x|$, is its distance from zero on the number line. For example, $|-5| = 5$ and $|5| = 5$.
Classification of Integers
Integers are typically classified into three categories:
- Positive Integers: These are integers greater than zero (1, 2, 3, ...). They are also known as [Natural Number]s (or counting numbers, depending on whether zero is included in natural numbers).
- Negative Integers: These are integers less than zero (-1, -2, -3, ...). They are the additive inverses of the positive integers.
- Zero: The integer 0 is unique in that it is neither positive nor negative. It is the additive identity, meaning that adding 0 to any integer leaves the integer unchanged.
Operations on Integers
Basic arithmetic operations can be performed on integers:
- Addition: Combining two integers (e.g., $5 + (-3) = 2$).
- Subtraction: Finding the difference between two integers (e.g., $2 - 7 = -5$).
- Multiplication: Repeated addition (e.g., $(-4) \times 2 = -8$).
- Division: While the division of two integers does not always result in an integer (e.g., $7 \div 2 = 3.5$), integer division yields a quotient and a remainder, both of which are integers.
Historical Context
The concept of positive integers for counting has existed since ancient times. However, the formal introduction and acceptance of negative numbers, and the concept of zero as a number, developed later. Early ideas of negative numbers can be traced back to ancient India and China, primarily in the context of debt and credit. European mathematicians fully integrated negative numbers into mainstream mathematics much later, by the 17th century, paving the way for modern Algebra and analysis.
Applications
Integers are fundamental in virtually all areas of mathematics and everyday life:
- Counting: Used for counting discrete objects.
- Measurement: Representing quantities like temperature (e.g., -10°C), elevation (e.g., -50 feet below sea level), or financial balances (e.g., -$20 debt).
- Computer Science: Integers are a fundamental data type in programming languages, used for indexing arrays, loop counts, and representing whole number values.
- Cryptography: Many cryptographic algorithms rely heavily on integer properties and modular arithmetic.
