Whole Numbers - Properties of Whole Numbers (Closure, Commutativity, Associativity, Distributivity)

The Closure Property states that the sum or product of any two whole numbers is always a whole number. For example, if you take two points on a number line representing whole numbers $a$ and $b$, their sum $a + b$ will always land on another whole number point. Note that whole numbers are not closed under subtraction or division.

The Commutative Property means that changing the order of numbers in addition or multiplication does not change the result. Visually, if you have an array of dots with $3$ rows and $4$ columns, it contains the same number of dots as an array with $4$ rows and $3$ columns ($3 \times 4 = 4 \times 3$).

The Associative Property allows us to group numbers in different ways when adding or multiplying three or more numbers without changing the outcome. Imagine three boxes of marbles; whether you group the first two boxes together first or the last two boxes together first, the total number of marbles remains the same.

The Distributive Property of Multiplication over Addition connects multiplication and addition. It states that $a \times (b + c) = (a \times b) + (a \times c)$. Geometrically, this can be seen as calculating the area of a large rectangle by splitting it into two smaller rectangles and adding their areas together.

The Identity Property identifies $0$ as the Additive Identity and $1$ as the Multiplicative Identity. Adding $0$ to any whole number $a$ keeps it as $a$, and multiplying any whole number $a$ by $1$ keeps it as $a$. On a number line, adding $0$ means taking zero steps, staying at the same position.

Division by Zero is not defined for whole numbers. This is because division is repeated subtraction; if you try to subtract $0$ from a number repeatedly, you will never reach zero or change the original value, making the process infinite and undefined.

The Multiplication by Zero property states that any whole number multiplied by $0$ results in $0$. This can be visualized as having several groups, but each group contains zero items, resulting in a total of zero items.