Integers - Properties of Addition and Subtraction of Integers

Closure Property of Addition: For any two integers $a$ and $b$, their sum $a + b$ is always an integer. Visually, if you pick any two marked points on an infinite number line and add their values, the result will always land exactly on another marked integer point, never in the gaps between them.

Closure Property of Subtraction: For any two integers $a$ and $b$, the result of $a - b$ is always an integer. This ensures that the set of integers is complete under the operation of subtraction, meaning you can subtract any integer from another and stay within the same number system.

Commutative Property of Addition: The sum of two integers remains the same regardless of the order in which they are added, expressed as $a + b = b + a$. Imagine two physical stacks of weights; whether you place a $2kg$ weight on top of a $5kg$ weight or vice versa, the total pressure on the scale remains exactly the same.

Associative Property of Addition: When three or more integers are added, the result is the same regardless of how the numbers are grouped, shown as $(a + b) + c = a + (b + c)$. Visually, if you have three groups of dots, you will get the same total count whether you combine the first two groups first and then the third, or combine the last two first and then add the first.

Additive Identity: The integer $0$ is called the additive identity because adding $0$ to any integer $a$ results in the same integer $a$, such that $a + 0 = a$. On a number line, adding zero represents a 'zero-length' jump, meaning you remain at your original coordinate without moving left or right.

Non-commutativity of Subtraction: Subtraction does not follow the commutative property, meaning $a - b \neq b - a$ for any distinct integers $a$ and $b$. Visually, taking 3 steps forward and 5 steps back puts you at a different final location than taking 5 steps forward and 3 steps back.

Non-associativity of Subtraction: The way integers are grouped in subtraction changes the final result, so $(a - b) - c \neq a - (b - c)$. This means the parentheses are crucial in subtraction problems to define which operation happens first.