Multiplicative identity of a rational number is 1, because when any rational number is multiplied by 1, the result is the number itself. This means that when any rational number is multiplied by 1, it's value remains the same. The identity element is called "multiplicative" because it applies to multiplication.
For example, if you have the fraction 5/8 and multiply it by 1:
5/8 × 1 = 5/8
The fraction remains unchanged.
In this article, we will explore the definition of the multiplicative identity in detail.
Table of Content
Multiplicative Identity Definition
Multiplicative identity of a number is a value that when multiplied by the number leaves the number unchanged. For any number x:
x × 1 = x
Thus, 1 is the multiplicative identity.
Multiplicative Identity of Rational Numbers
For a rational number a/b where a and b are integers and b ≠ 0 multiplying by the 1 does not change its value:
\frac{b}{a} \times 1 = \frac{b}{a}
This shows that the multiplicative identity for the rational numbers is also 1.
Examples of Multiplicative Identity of Rational Numbers
Multiplicative identity of a rational number is 1 because multiplying any rational number by 1 results in the same number. Let's consider some rational number and their multiplication with multiplicative identity i.e., 1.
- Rational number: 5/7
Multiplication with Multiplicative Identity: 5/7 × 1 = 5/7
- Rational number: −3/4
Multiplication with Multiplicative Identity: −3/4 × 1 = −3/4
- Rational number: 8/5
Multiplication: 8/5 × 1 = 8/5
Multiplicative Identity vs. Additive Identity of Rational Number
| Property | Multiplicative Identity | Additive Identity |
|---|---|---|
| Definition | The number which, when multiplied by any rational number, leaves the number unchanged. | The number which, when added to any rational number, leaves the number unchanged. |
| Identity Element | 1 | 0 |
| Formula | a/b × 1 = a/b | a/b + 0 = a/b |
| Example | 3/7 × 1 = 3/7 | 3/7 + 0 = 3/7 |
Conclusion
The multiplicative identity of rational numbers is a fundamental concept that simplifies understanding the multiplication across the rational numbers. By multiplying by 1 the number remains unchanged in which has important applications in the algebra, calculus and everyday problem-solving in mathematics.
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