## Algebraic Properties

There are five basic properties in Algebra. They are commutative, associative, distributive, identity and inverse.

#### Commutative Property

If the result of an operation remains unchanged, with a change in the order of operands, then the operation is said to be commutative.

Let a and b be two operands.

Commutative law over Addition: a + b = b + a

Commutative law over Multiplication: a * b = b * a

#### Associative Property

If the order of performing multiple operations is not important, then the operation is said to be associative.

Associative law over Addition: (a + b) + c = a + (b + c)

Associative law over Multiplication: (a * b) * c = a * (b * c)

#### Distributive Property

Distributive property includes both the addition and multiplication of real numbers.

a * (b + c) = a * b + a * c.

Here addition is said to be distributive over multiplication.

#### Identity Property

If the value of the operand(s) remains unchanged even after performing an operation, then the operation is said to have identity property.

The additive identity is ‘0,’ i.e. a + 0 = 0 + a = a

The multiplicative identity is ‘1,’ i.e. a * 1 = 1 * a = a

#### Inverse Property

The value which gives the additive identity when added to the original number is called the additive inverse. The additive inverse is the negative of the value.

a + (- a) = 0

The value which gives the multiplicative identity when multiplied with the original number is called the multiplicative inverse. The multiplicative inverse is 1/a for a real number a.

a * 1/a=1

Example

The algebraic properties make calculations simple. Let us see the following example which proves it.

Evaluating 5 * 206 directly takes time, definitely, but applying distributive property here can lessen the calculation time and also the effort.

5 * (200 + 6) = 5 * 200 + 5 * 6

= 1000 + 30

= 1030

#### Try this problem

Describe the 5 basic properties of algebra.

The five basic properties of algebra are commutative, associative, distributive, identity and inverse.

1. If the result of an operation remains unchanged, with a change in the order of operands (such as a and b), then the operation is said to be commutative. Thus, a+b=b+a and a*b=b*a.
2. If the order of performing multiple operations is not important, then the operation is said to be associative. For example, (a+b)+c=a+(b+c) and (a*b)*c=a*(b*c).
3. The distributive property includes both the addition and multiplication of real numbers, such as a*(b+c)=a*b+a*c.
4. If the value of the operand(s) remains unchanged even after performing an operation, then the operation is said to have identity property. For example, if the additive identity is ‘0,’ then a+0=0+a=a. Also, if the multiplicative identity is ‘1,’ then a*1=1*a=a.

The value which gives the additive identity when added to the original number is called the additive inverse. The additive inverse is the negative of the value: for example a+(-a)=0.

The value which gives the multiplicative identity when multiplied with the original number is called the multiplicative inverse. The multiplicative inverse is 1/a for a real number a: for example a * 1/a=1.