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
Basic Properties of Algebra
