Integers are governed by specific rules when performing arithmetic operations. Here's a breakdown of those rules, using information from the provided references:
Basic Properties of Integers
Integers are whole numbers, which can be positive, negative, or zero. Here are key rules that apply to integer operations:
Addition
- Sum of two positive integers: The result is always a positive integer.
- Example: 2 + 3 = 5
- Sum of two negative integers: The result is always a negative integer.
- Example: (-2) + (-3) = -5
Multiplication
- Product of two positive integers: The result is always a positive integer.
- Example: 2 * 3 = 6
- Product of two negative integers: The result is always a positive integer.
- Example: (-2) * (-3) = 6
Additive and Multiplicative Inverses
- Sum of an integer and its inverse (additive inverse): This equals zero.
- Example: For the integer 5, its additive inverse is -5, and 5 + (-5) = 0
- Product of an integer and its reciprocal (multiplicative inverse): This equals 1.
- Example: For the integer 2, its reciprocal is 1/2, and 2 * (1/2) = 1
Additional Rules and Considerations:
- Zero: Zero is an integer and acts as the additive identity. Adding zero to any integer doesn't change the integer's value.
- Subtraction: Subtraction of integers is defined as adding the additive inverse.
- Example: 5 - 3 is the same as 5 + (-3), which equals 2
- Division: Division of integers can result in either an integer or a non-integer.
- Example: 6 / 3 = 2 (integer), while 5 / 2 = 2.5 (not an integer)
- Order of Operations: When performing a series of arithmetic operations with integers, the order of operations must be followed (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction - PEMDAS).
- Closure: Under addition, subtraction, and multiplication integers are closed, meaning the result of these operations on integers will always be another integer.
Conclusion
Understanding these rules is fundamental to mastering integer operations. These rules ensure consistency and predictability when working with integers in mathematical computations.
