Properties of Integers

Definition of Integers

• Integers are whole numbers, either positive, negative, or zero, without a fractional part.
• They are denoted by the symbol "Z" and include the set of numbers {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Properties of Integers

• Closure: The result of adding or multiplying two integers is always an integer.
• Commutative Property: The order of integers in an operation does not change the result. (e.g., a + b = b + a, a × b = b × a)
• Associative Property: The order in which integers are grouped in an operation does not change the result. (e.g., (a + b) + c = a + (b + c), (a × b) × c = a × (b × c))
• Distributive Property: Integers can be distributed across addition and subtraction operations. (e.g., a × (b + c) = a × b + a × c)

Types of Integers

• Positive Integers: Integers greater than zero (e.g., 1, 2, 3, ...).
• Negative Integers: Integers less than zero (e.g., -1, -2, -3, ...).
• Zero: The additive identity, which is neither positive nor negative.

Operations on Integers

• Addition: The result of adding two integers is always an integer.
• Subtraction: The result of subtracting one integer from another is always an integer.
• Multiplication: The result of multiplying two integers is always an integer.
• Division: The result of dividing one integer by another is not always an integer, but may be a fraction or a decimal.

Important Concepts

• Absolute Value: The distance of an integer from zero, always a positive value.
• Opposite: The integer with the same magnitude but opposite sign.
• Reciprocal: The multiplicative inverse of an integer, except for zero.

Definition and Properties of Integers

• Integers are whole numbers, either positive, negative, or zero, without a fractional part, denoted by the symbol "Z".
• The set of integers includes {..., -3, -2, -1, 0, 1, 2, 3,...}.
• Integers exhibit closure under addition and multiplication.
• Integers follow the commutative property of addition and multiplication.
• Integers follow the associative property of addition and multiplication.
• Integers follow the distributive property across addition and subtraction.

Types of Integers

• Positive integers are integers greater than zero (e.g., 1, 2, 3,...).
• Negative integers are integers less than zero (e.g., -1, -2, -3,...).
• Zero is the additive identity, neither positive nor negative.

Operations on Integers

• The result of adding two integers is always an integer.
• The result of subtracting one integer from another is always an integer.
• The result of multiplying two integers is always an integer.
• The result of dividing one integer by another is not always an integer, but may be a fraction or a decimal.

Important Concepts

• The absolute value of an integer is its distance from zero, always a positive value.
• The opposite of an integer has the same magnitude but opposite sign.
• The reciprocal of an integer is its multiplicative inverse, except for zero.