# Basic Properties of Algebra

**Associative Property****Commutative Property****Distributive Property****Identity Property**

### Associative Property

**Associative Property of Addition:**

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

**Commutative Property of Addition:**

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

**Identity Property of Addition:**

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