The identity property is a fundamental property in arithmetic that applies to all numbers and algebraic expressions. This property is often used to prove more complex properties and theorems, so knowing this property is important as it opens a wide range of applications.
In this article, learn how the identity property is applied for the four core arithmetic operations: addition, subtraction, multiplication, and division. This article will also help you understand the roles of 0 and 1 in the four variations of the identity property. Begin by understanding the definition of identity property.
What Is Identity Property?
The identity property is one of the most fundamental properties that exist for numbers and arithmetic operations. The identity property focuses on the situations when a given number remains the same after being added, multiplied, subtracted, and divided by a particular constant. For the identity property of addition and subtraction, this constant is equal to 0. For multiplication and division, this constant is 1.
To understand the concept of identity property, imagine having a box of cookies. How many donuts should you add or take out from the box for you to have the same number of cookies? Similarly, suppose that you have a box of donuts, how can you group them so that each group has the same number of donuts?
The answers to these questions may come in easy but mathematicians want to establish these identities so that future properties have a strong foundation. This is why it’s important to study and understand the variations of the identity property – beginning with the identity property of addition and subtraction.
Understanding the Identity Property of Addition and Subtraction
The identity property of addition and subtraction establishes the fact that when 0 is added to or subtracted from a number, k, the result is equal to k. This is why 0 is called the additive identity and subtractive identity.
\begin{aligned}k + 0 &= k\\ k - 0 &= k \end{aligned}This means that no matter how large or small a number is when 0 is added or subtracted from it, the result will still be the same number. Go back to the representation of this identity using real-world objects and try to understand why this property is true.
- If there are three donuts and no other donuts were added, it makes sense that there will still be three donuts left.
- Similarly, if there are four cookies and none of the cookies were eaten, there will still be four cookies.
The identity property of addition (and subtraction) establishes this rule for all types of numbers. Meaning, this rule applies to integers, whole numbers, rational numbers, irrational numbers, and more. Here are some examples highlighting how the identity property is applied for different types of numbers:
|
Identity Property of Addition |
Fraction |
1/2 + 0 = 1/2 |
|---|---|---|
|
Decimal |
0.00185 + 0 = 0.00185 | |
|
Scientic Notation |
4 \times 10^{11} + 0 = 4 \times 10^{11} | |
|
Identity Property of Subtraction |
Irrational Number |
4\pi - 0 = 4\pi |
|
Percentage |
45\% - 0 = 45\% | |
|
Algebraic Expression |
4x^2 - 0 = 4x^2 |
Now, let’s move on to the identity property of multiplication and division. This time, the constant to focus on is 1. It’s time to think about what happens when a number is multiplied by 1 or divided by 1.
Understanding the Identity Property of Multiplication and Division
According to the identity property of multiplication (and division), when a number, k, is multiplied or divided by 1, the resulting product or quotient will still be the same number, k. Hence, 1 considered the multiplicative identity and divisive identity in arithmetic.
\begin{aligned}k \times0 &= k\\ k \div &= k \end{aligned}Once again, understand why this property is true by thinking real-world examples like the one provided in the earlier discussion. Think of multiplication as adding more boxes of donuts and division as grouping a box of cookies into an equal number of cookies.
- Suppose that a box has half a dozen donuts. To have exactly six donuts, only one box is needed, Showing that one is a multiplicative identity.
- Using a similar thought process, the only way to divide six cookies into exactly six cookies as well is by grouping them as one whole group of cookies. This is a reflection of the identity property of division.
Following through this discussion, the identity rule states that no matter how large or small a number is when it is multiplied or divided by 1, the result will still be the same number. This property extends for different types of numbers as the examples below will show you.
|
Identity Property of Multiplication |
Irrational Number |
\sqrt{5} \times 1 = \sqrt{5} |
|---|---|---|
|
Percentage |
120\% \times 1 =120\% | |
|
Algebraic Expression |
5x \cdot 1= 5x | |
|
Identity Property of Division |
Fraction |
4\frac{2}{3} \div 1= 4\frac{2}{3} |
|
Decimal |
12.56 \div 1 = 12.56 | |
|
Scientic Notation |
6 \times 10^{-6} \div 1 = 6 \times 10^{-6} |
These examples highlight how regardless of the number’s scale and type, the identity property remains absolute. This is why it’s important that properties like this are discussed and explored. Now, it’s time to learn how important these properties are by trying out different problems involving the identity property.
How to Apply the Identity Property?
When applying the identity property, remember the result when numbers are dealing with the two important constants: 0 and 1. Use the fact that when 0 is added or subtracted from a number, the result will still be the number. Similarly, when a number is multiplied or divided by 1, the result will still be the number.
|
Summary of the Identity Property |
|
|---|---|
|
Addition |
k + 0 = k |
|
Subtraction |
k - 0 = k |
|
Multiplication |
k \times 1 = k |
|
Division |
k \div = k |
Here’s a special note for the identity property of subtraction and division, the order of the numbers matters. Meaning the identity property won’t hold true when 1 is divided by k or when k is subtracted from 1.
0 – k ≠ k
1 ÷ k ≠ k
Just remember this while working on the problems shown below. Apply what you’ve just learned to solve the problems and by the end of these exercises, feel confident when working with 0 and 1 in different arithmetic operations!
Problem 1
Celine curates handcrafted soaps and offers them as a bundle in one box of four soaps. If her customer wants to receive four soaps, how many boxes does Celine need?
To find the total number of boxes that Celine needs to pack four soaps, use the fact that when a number is divided by 1, the result will still be the same number.
One box contains four soaps, so if the customer wants the same number of soaps, Celine would only need one box.
Problem 2
A meeting room currently has six chairs. If there are no more available chairs to be added, how much can be seated in the meeting room?
When a number is added by zero, the sum returns the same number. This is the core concept behind the identity property of addition. This means that when no more chairs are added, the same number of chairs will remain inside the meeting room.
Hence, there are a total of six people that can be seated inside the meeting room.
Problem 3
Fill in the blanks to make the following equations true.
a. 4 + \_\_\_ = 4
According to the identity property of addition, the only way for the sum of a number and constant will still be equal to the number, the constant has to be zero. This means that for the equation to be true, 4 + \boldsymbol{0} = 4 .
b. (5\pi -1) - 0 = \_\_\_
Similar to the previous problem, it has been established that zero is also a subtractive identity. Meaning that when zero is subtracted from any number, including irrational numbers, the difference returns the same number. Hence, (5\pi -1) – 0 = \boldsymbol{5\pi -1}.
c. \_\_\_ \times 1 = 5e
The only way for a product to return the same value as one of the two factors is when the other remaining factor is one. This means that for the product to be 5e with 1 as the remaining factor, then only possible equation will be \boldsymbol{5e} \times 1 = 5e .
d. \frac{2a + b}{1} = \_\_\_\_
Lastly, it has been established that one is also a divisive identity. This means that when an algebraic expression, such as 2a + b, will have the same expression when divided by 1. Hence, \frac{2a + b}{1} = \boldsymbol{2a +b }