Identity Property

The identity property is a key concept in mathematics that applies to operations like addition and multiplication. It says that when a number (n) is combined with a specific identity element using an arithmetic operation, the result remains the same. 

The special numbers (0 and 1) are called identity elements because they keep the value of a number intact. The number stays the same after the operation as well.

The identity property shows how the numbers behave in arithmetic operations in different groups of numbers.

There are two kinds of identity properties:

Additive Identity: 0 is the identity for addition because, adding 0 to a number, keeps the number the same.
        n + 0 = n
    

Multiplicative Identity: 1 is the identity for multiplication because multiplying a number by 1 keeps it the same.                               
                       n × 1 = n
 

For the identity property to hold in a mathematical operation, the following conditions must be met:

• There must be a unique number, called the identity element that, when used in an operation with any number n, leaves n unchanged.

• The additive identity is 0, meaning a + 0 = 0 + a = a. This must be applied to all numbers in the given set.

• The multiplication identity is 1, meaning a × 1 = 1 ×  a = a. This must be true for all numbers in the set. 

• The mathematical operation should be properly defined within the number set (e.g., real numbers, integers) for the identity element to work correctly. 

• The identity element must be unique for each operation. There is only one additive identity (0) and one multiplicative identity (1) in standard number systems.

• Exponentiation does not have identity property. For example, a^e = e^a = a is only valid if both a and e are equal to 1, which means exponentiation does not satisfy the identity property for all real numbers.

Let’s understand the two identity property differences in a simple table format. 
 

The identity property isn’t just a math rule. It applies to real-life situations, too.

Whether dealing with money, shopping, or counting, this property helps us understand how numbers stay the same when adding zero or multiplying by one. Here are some everyday examples.

 

• Money Transactions (Additive Identity ): If you have $20 in your wallet. If you don’t add any money ($0), the total amount of money doesn't change. You still have $20. This follows the rule $20 + $0 = $20.

• Multiplying by One in Shopping (Multiplicative Identity): Let's say you are at a shop, and you buy a pack of gum for $2, the total price is $2 × 1 = $2. The number stays the same because multiplying by 1 doesn’t change the value. So, whether it's gum or fruits, if you buy only one item, the total cost is the price of that one item.