Identity Property of Addition

Starting with your first math course, one of the first things that you will learn is how to add, subtract, multiply and divide numbers. These are the four basic mathematical operations that you need to understand properly. They are elements of what is called basic arithmetic or mathematics. There are many properties related to each of these mathematical operations, which need to be understood by you.

Adding Numbers

Before we define what is known as the identity property of addition, let me introduce what we mean by the math term - 'addition' of two or more numbers. The addition operation between two numbers, denoted by the '+' sign is summing two small numbers together to get a bigger number. Suppose you have 5 dollars in your pocket and your father gives you 10 dollars more. How may dollars are left with you?

To know the answer, you must 'sum up' or 'add' the two amounts. So you will have (5 + 10 = 15) dollars now. One way of visualizing addition of numbers is by looking at the number line as a series of steps in a staircase. Each step in the staircase is labeled with numbers starting from 1, 2, 3, 4, ... up till the maximum number possible in ascending order. When you add, you ascend steps and when you subtract, you descend steps downwards.

So if you are standing on step number 5, and you need to add 10, you ascend ten steps on the staircase, to reach 15! To make this number line and staircase analogy more precise, know that the staircase is never ending on both sides! There are positive and negative integers extending on both sides of it.

Definition

This property deals with the consequence of 'zero' to any positive or negative number. It states that, 'On adding zero to any number or any algebraic variable, the resultant sum is the same number again.' In the form of an equation, it can be stated as:

X + 0 = X

where X is any variable or a number. It is called the identity property, as when zero is added to a number, the resultant sum is the same as the number's identity! Here zero is known as the 'identity element'. Let me illustrate the concept further through examples in the next section.

Illustrative Examples

Here are some examples and illustrations which demonstrate the property succinctly.

• 1 + 0 = 1
• -1 + 0 = -1
• XY + 0 = XY
• 0 + 0 = 0
• X/Y + 0 = X/Y
• 0 + 88 = 88
• 99.99 + 0 = 99.99

This property of identity, in addition arises due to the peculiar nature of zero as a number, which denotes 'nothing.' In terms of our staircase analogy, adding zero is like taking neither a step up or step down on the staircase. It is just jumping on the same step, which leaves you right where you are!

It is one of the most important of properties when it comes to defining addition of numbers, though it looks trivial at face value. To put it in simplest of words, the property states that when you add 'nothing' (in the form of zero) to any number, all that you get is the same number back again! Remember this simple explanation and understanding the identity property is not that difficult.