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.