# Identity property

An identity element is a number that, when used in an operation with another number, results in the same number. The additive and multiplicative identities are two of the earliest identity elements people typically come across; the additive identity is 0 and the multiplicative identity is 1.

## Identity property of addition

The identity property of addition states that the sum of 0 and any other number is that number. This is because adding nothing to any number just leaves the same number, since 0 has no magnitude. It can be written as follows, where a is a variable that represents any number.

a + 0 = 0 + a = a

Any value can be substituted for a. It doesn't matter how large or small the number. Adding 0 (or subtracting 0 for that matter) will result in the same number.

Examples

1. a = 12:

12 + 0 = 12

2. a = 456,756,821:

0 + 456,756,821 = 456,756,821

One way to visualize the identity property of addition is to use objects to represent addition. Adding 0 objects to any number of objects just leaves the same number of objects. The figure below represents the addition problem 3 + 0.

## Identity property of multiplication

The identity property of multiplication states that the product of 1 and any number equals that number. It can be written as follows, where a is any number.

a × 1 = a

Examples

1. a = 8:

12 × 1 = 12

2. a = 3,452,789:

3,452,789 × 1 = 3,452,789

One way to visualize the identity property of multiplication is through the use of arrays. We can think of the identity property of multiplication as an a × 1 array, where a is any number. Regardless whether we have an an a × 1 array or a 1 × a array, there will be the same number of objects in the array since there is no way to organize a group of objects being multiplied by 1 such that the number of objects changes. The figure below depicts the multiplication problem 1 × 4 or 4 × 1.