Identity Property, also known as the Identity Element or Identity Law, is a fundamental concept in mathematics. It is used primarily in the study of groups, rings or fields in abstract algebra.
Identity Property ensures that there exists a special element within a set that leaves other elements unchanged when combined with them using a defined operation. In this article, we will discuss Identity Property in detail including its definition. We will also discuss the Identity Property of addition and multiplication as well.
What is the Identity Property?
Identity Property is a fundamental concept in mathematics that applies to arithmetic operations. It is defined as the property where if any arithmetic operations are used to combine an identity with a number (n), the end result will be n.
In simple words, when you add, subtract, multiply, or divide a number by a specific value, there’s a special number that won’t change the result. That special number is the Identity Element for the defined operation.
The identity property is applied to a group of numbers in the form of sets, and the identity of these numbers remains the same as 1 and 0 even when the numbers are added, subtracted, multiplied, and divided.
Identity Property Definition
For any number a and operation ” * “, identity property is defined as:
a * e = e * a = a
Where e is the identity element under operation ” * “.
Condition for Identity Property to Not Hold
Consider the set of real numbers. The operation we’re considering here is exponentiation, denoted by ^. According to the Identity Property of Exponentiation, for any real numbera, a^e = e^a = a.
As we know, for any two real number it only holds true if both a and e are 1, other than that this relation doesn’t hold true for any real number.
Thus, identity property doesn’t hold for real numbers under the operation of exponentiation i.e., a^e ≠ e^a.
Types of Identity Properties
There are two main types of Identity Properties:
- Identity Property of Addition
- Identity Property of Multiplication
Identity Property of Addition
For addition, the identity element is usually denoted as 0. The Identity Property of Addition states that for any element a in the set, a + 0 = 0 + a = a.
For example, 7 + 0 = 0 + 7 = 7 and −1 + 0 = 0 + (-1) = −1.
In both cases, adding 0 to a does not change the value of a, illustrating the Identity Property of Addition.
Note: 0 is the additive identity i.e., identity element for addition operation.
Identity Property of Multiplication
For multiplication, the identity element is typically denoted as 11. The Identity Property of Multiplication states that for any element a in the set, a × 1 = 1 × a = a.
For example, 5 × 1 = 1 × 5 = 5 and −2 × 1 = 1 × (-2) =−2.
In each case, multiplying a by 1 yields a, demonstrating the Identity Property of Multiplication.
Note: 1 is the multiplicative identity i.e., identity element for multiplication operation.
Additive Vs Multiplicative Identity
Let’s break down the concepts of additive and multiplicative identity:
| Property | Additive Identity | Multiplicative Identity |
|---|---|---|
|
Definition |
The additive identity is a number that, when added to any other number, leaves the number unchanged. |
The multiplicative identity is a number that, when multiplied by any other number, leaves the number unchanged. |
| Operation | Addition | Multiplication |
| Identity Element | 0 | 1 |
| Identity Property | a + 0 = 0 + a = a | a × 1 = 1 × a = a |
| Example | 5 + 0 = 5 | 7 × 1 = 7 |
| Example (Negative) | (−3) + 0 = −3 | (−2) × 1 = −2 |
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Practice Problems on Identity property
Problem 1: Use the multiplicative identity property to solve the following equations:
- 7 × 1 = ?
- -20 × 1 = ?
- 1 × 57 = ?
Problem 2: Solve the following problems using both the Additive and Multiplicative Identity Properties:
- 25 + 0 × 4
- 0 × (−6) + 7
- 3 × (1 + 9)
FAQs on the Identity Property
What is the Identity Property?
The Identity Property is a basic idea in mathematics. It talks about how some operations keep a certain number unchanged when you do them with that number. There are two types: the Additive Identity Property and the Multiplicative Identity Property.
What is the Additive Identity Property?
The Additive Identity Property states that when you add zero to any number, the result remains unchanged. Symbolically, for any real number a, a + 0 = a. Here, 0 acts as the identity element for addition.
Can the Identity Property be extended to other operations?
Yes, while the Identity Property is commonly associated with addition and multiplication, similar concepts exist for other operations as well.
What is the Multiplicative Identity Property?
The Multiplicative Identity Property asserts that when you multiply any number by one, the result stays the same. In mathematical terms, for any real number b, b × 1 = b. Here 1 ,serves as the identity element for multiplication.