# Basic Properties of Algebra

The properties of math are like a set of rules that help us solve mathematical equations. The basic properties of algebra for real numbers involve various operations such as addition, subtraction, multiplication, and division.
Following are the four basic algebric properties of math: Let’s explore the properties using the real numbers a, b, and c:
• Associative Property
• Commutative Property
• Distributive Property
• Identity Property

### Associative Property

This property says that when adding three or more numbers, we can change the order by moving the parentheses, and the answer will remain the same.

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

Examples: consider a = 2, b = 3, and c = 4.

(a + b) + c = (2 + 3) + 4 = 9 Left side
a + (b + c) = 2 + (3 + 4) = 9 Right side

Both side expressions are the same, i.e., 9.

#### Associative Property of Multiplication:

This property says that when multiplying three or more numbers, we can change the order by moving the parentheses, and the answer will remain the same.

(a x b) x c = a x (b x c)

(a x b) x c = (2 x 3) x 4 = 24 Left side
a x (b x c) = 2 x (3 x 4) = 24 Right side
Both side expressions are the same, i.e., 24.

Remember: The associative properties only work with addition and multiplication; they do not apply to subtraction or division.

### Commutative Property

It says that when we add two or more numbers, the order in which we add the numbers does not affect the result.

a + b = b + a

Example: Consider a = 2, b = 3.
a + b = 2 + 3 = 5
b + a = 3 + 2 = 5

Both side expressions are the same, i.e., 5.

#### Commutative Property of Multiplication:

It says that when we multiply two or more numbers, the order in which we multiply the numbers does not affect the result.

a x b = b x a

a x b = 2 x 3 = 6
b x a = 3 x 2 = 6

Both side expressions are the same, i.e., 6.

Remember: The commutative properties only work with addition and multiplication; they do not apply to subtraction or division.

### Distributive Property

#### The Distributive Property of Multiplication Over Addition:

It states that multiplying a number by the sum of two other numbers produces the same result as multiplying the number by each other number separately.

It can be expressed as
a x (b + c) = (a x b) + (a x c)

Example: Consider a = 2, b = 3, and c = 4.

a x (b + c) = 2 x ( 3 + 4) = 2 x 7 = 14 Left side
(a x b) + (a x c) = (2 x 3) + (2 x 4) = 6 + 8 = 14 Right side

Both sides of the equation are equal, confirming that the distributive property holds true in this example.

#### The Distributive Property of Multiplication Over Subtraction:

It states that when you multiply a number by the difference of two other numbers, it is equal to the difference of the products of the distributed number.

It can be expressed as: a(b – c) = ab – ac

Example: Consider a = 3, b = 6, and c = 4.

a(b – c) = 3(6 – 4) = 3 x 2 = 6 Left side
ab – ac = 3 x 6 – 3 x 4 = 18 – 12 = 6 Right side

Both side expressions are the same, i.e., 6.

### Identity Property

It says that if we add zero to any number, the sum is equal to the number itself. Zero ( 0) is called additive identity.

a + 0 = a

8 + 0 = 8

#### Identity Property of Subtraction:

It says that if we multiply any number by 1, the product is equal to the original number.

a x 1 = a

8 + 1 = 8

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