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VOOZH | about |
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VOOZH | about |
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.
The identity property is applied to a group of numbers in the form of sets, and the identity of these numbers remains the same.
For any number a and operation " * ", identity property is defined as:
a * e = e * a = a
Where e is the identity element under operation " * ".
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.
There are two main types of Identity Properties:
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.
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.
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 |
Realted Articles
Problem 1: Use the multiplicative identity property to solve the following equations:
Problem 2: Solve the following problems using both the Additive and Multiplicative Identity Properties:
