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.
- When you add or subtract 0 from any number, the result is the number itself
- When multiply or divide 1 with any number, the result is the number itself.
The identity property is applied to a group of numbers in the form of sets, and the identity of these numbers remains the same.
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)