Multiplicative Identity of Rational Numbers
The multiplicative identity of rational numbers is 1. As per the property of multiplicative identity, a number when multiplied by the original number gives the product as 1. Thus, we can say that the multiplicative inverse of any number 'a' is a-1 or 1/a. In other words, the two numbers are said to be multiplicative inverses of each other if their product is 1. The multiplicative inverse of a number is also called the reciprocal of the number.
In this article, let's discuss the multiplicative identity of rational numbers with solved examples and practice questions.
What is Multiplicative Identity of Rational Numbers?
The multiplicative identity of a rational number is 1 as the product of a number and its multiplicative inverse is 1. When 1 is multiplied by any number, the product is the number itself. Therefore, 1 is the multiplicative identity for rational numbers, integers, and whole numbers. For any non-zero rational number a/b, there exists a rational number b/a, such that a/b x b/a = 1, where b/a is called the reciprocal or multiplicative inverse of a/b.
Multiplicative Identity Property
As per the multiplicative identity property, the product of a number and its multiplicative inverse is 1, which can be represented as x. x-1 = 1. For example, consider a rational number 2/3. The multiplicative inverse of 2/3 is 3/2. According to the property, 2/3. (3/2) = 1. Hence Proved.
Multiplicative Identity and Multiplicative Inverse of a Rational Number
In a given mathematical system, the multiplicative identity refers to an identity element that leaves unchanged any element by which it is multiplied, that is e·x = x·e = x. For the group of rational numbers, it is 1. The "Multiplicative Identity" is 1, because multiplying a number by 1 leaves it unchanged: a × 1 = 1 × a = a.
For additive inverse, the sum of a rational number and its inverse should be equal to zero. For multiplicative inverse, the multiplication of a rational number with its inverse should be equal to 1.
Important Notes
- The multiplicative inverse of a number is its reciprocal only.
- The product of a number and its multiplicative inverse is equal to 1.
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FAQs on Multiplicative Identity of Rational Numbers
What Is the Multiplicative Identity of a Rational Number?
The multiplicative identity of a rational number is 1 as the product of a number and its multiplicative inverse is 1. For example, the multiplicative inverse of the rational number 4/5 is 5/4, their product is (4/5).(5/4) = 1.
How To Find the Multiplicative Identity of a Rational Number?
Multiplicative identity refers to an identity element that leaves the number unchanged when multiplied by that number. In the case of rational numbers also, the multiplicative identity is 1 only. For any rational number x/y, e is the element element such that e·x/y = x/y·e = x/y.
Can the Multiplicative Identity of a Rational Number Be 0?
Multiplicative identity of any number leaves the number remains unchanged when multiplied by that number. Thus, it is always 1 and can't be 0.
Does the Multiplicative Identity of Rational Numbers the Same as Fractions?
Addition, subtraction, multiplication, and division of rational numbers are done in the same way as we do for fractions. Rational number 1 is the multiplicative identity for rational numbers as well as fractions.
What Is the Additive and Multiplicative Identity of Rational Numbers?
In algebra, each of the mathematical operations is associated with some major identities. Additive identity and multiplicative identity are the two basic algebraic identities. For rational numbers, natural numbers, whole numbers, and integers, the additive identity is 0 and the multiplicative identity is 1.