====== Addition ======
{{anchor:addition}}
===== Properties of Addition =====

==== Commutative properties of addition ====

When adding [[en:math:algebra:number:integers|Integers]], the sum does not change when the places of the added numbers are changed. This property of addition in [[en:math:algebra:number:integers|Integers]] is called the **commutative property**.

In other words, for any numbers ''a'' and ''b'',

$$ a + b = b + a $$

==== Associative properties of addition ====

When adding three or more [[en:math:algebra:number:integers|integers]], which pair of numbers is added first has no effect on the result. This property of addition in [[en:math:algebra:number:integers|integers]] is called the associative property.

In other words, for any numbers ''a'' and ''b'',

$$ (a+b) + c = a + (b+c) $$

==== Inefficient (unit) element of addition ====

A number that does not change the result of the related operation is called **unit element**.

Adding a number [[en:math:algebra:number:zero|zero]] the result is the summed number. Therefore, the inefficient (unit) element of addition is [[en:math:algebra:number:zero|zero]].

$$ 5 + 0 = 5 $$

$$ 0 + (−98) = −98 $$

==== Inverse element of addition ====

A number whose sum with an [[en:math:algebra:number:integers|integer]] is equal to [[en:math:algebra:number:zero|zero]] is called the inverse of that [[en:math:algebra:number:integers|integer]] with respect to addition. In other words, two numbers whose sum is 0 are inverses of each other with respect to addition.

\begin{align}

\text{Inverso of } 0\text{ with respect to addition } \to 0 \\

\text{Inverso of } 98\text{ with respect to addition } \to −98 \\

\text{Inverso of } −32\text{ with respect to addition } \to +32 \\

\end{align}

====== Subtraction ======
{{anchor:subtraction}}

Subtraction is the opposite of addition.

===== Properties of Subtraction =====

<WRAP important round>
Subtraction does not have the property of commutative.
</WRAP>



====== Multiplication ======
{{anchor:multiplication}}

===== Properties of Multiplication =====

==== Commutative properties of multiplication ====

When the places of the factors are changed in multiplication, the result does not change. This property is called the **Commutative property** of multiplication.

In other words, for any numbers ''a'' and ''b'',

$$ a \cdot b = b \cdot a $$

==== Associative properties of multiplication ====

When multiplying by three or more [[en:math:algebra:number:integers|integers]], which pair of numbers is multiplied first has no effect on the result. This property of [[en:math:algebra:number:integers|integers]] multiplication is called the union property.

In other words, for any numbers ''a'' and ''b'',

$$ (a \cdot b) \cdot c = a \cdot (b \cdot c) $$

==== Inefficient (unit) element of multiplication ====

When we multiply a number by 1 in multiplication, the result is the multiplied number. Therefore, the ineffective (unit) element of multiplication is 1.

$$ 5 \cdot 1 = 5 $$

$$ 1 \cdot (−98) = −98 $$

==== Absorbing element of multiplication ====

A number that returns itself as a result of an operation with any number is called a **absorbing element (or annihilating element)**. 

In multiplication, a number [[en:math:algebra:number:zero|zero]] multiplied by [[en:math:algebra:number:zero|zero]] is equal to zero. Therefore, the element of the multiplication operation that absorbs is [[en:math:algebra:number:zero|0]].

==== The distributive property of multiplication ====

> Multiplication has the property of distributing over addition and subtraction.

For example, the operation $ -5 . (100 + 2) $ is solved with the dispersion property.

\begin{align}

−5 . ( 100 + 2 ) \\
= (−5 . 100) + (−5 . 2) \\
= (−500) + (−10)  \\
= −510 \\

\end{align}

====== Division ======
{{anchor:division}}

Division is the opposite of multiplication.

===== Properties of Division =====

<WRAP important round>
Division does not have the property of commutative.
</WRAP>