When adding Integers, the sum does not change when the places of the added numbers are changed. This property of addition in Integers is called the commutative property.

In other words, for any numbers a and b,

$$ a + b = b + a $$

When adding three or more integers, which pair of numbers is added first has no effect on the result. This property of addition in integers is called the associative property.

In other words, for any numbers a and b,

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

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

Adding a number zero the result is the summed number. Therefore, the inefficient (unit) element of addition is zero.

$$ 5 + 0 = 5 $$

$$ 0 + (−98) = −98 $$

A number whose sum with an integer is equal to zero is called the inverse of that 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 is the opposite of addition.

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,

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

When multiplying by three or more integers, which pair of numbers is multiplied first has no effect on the result. This property of integers multiplication is called the union property.

In other words, for any numbers a and b,

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

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

$$ 5 \cdot 1 = 5 $$

$$ 1 \cdot (−98) = −98 $$

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 zero multiplied by zero is equal to zero. Therefore, the element of the multiplication operation that absorbs is 0.

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 is the opposite of multiplication.

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