Fill in the blanks.
(i) Zero has $\qquad$ reciprocal.
(ii) The numbers $\qquad$ and $\qquad$ are their own reciprocals.
(iii) The reciprocal of -5 is $\qquad$ .
(iv) Reciprocal of $\frac{1}{x}$, where $x \neq 0$ is $\qquad$ .
(v) The product of two rational numbers is always a $\qquad$ .
(vi) The reciprocal of a positive rational number is $\qquad$ .
Answer (2)
Zero has no reciprocal. The numbers 1 and -1 are their own reciprocals, while the reciprocal of -5 is -1/5. The product of two rational numbers is always rational, and the reciprocal of a positive rational number is also positive.
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Zero has no reciprocal.
The numbers 1 and -1 are their own reciprocals.
The rational number that is equal to its negative is 0.
The reciprocal of -5 is − 5 1 .
The product of two rational numbers is always a rational number.
The reciprocal of a positive rational number is positive.
Explanation
Filling in the Blanks Let's analyze each statement and fill in the blanks with the correct answers.
(i) Zero has reciprocal. A reciprocal of a number 'a' is defined as a 1 . For zero, the reciprocal would be 0 1 , which is undefined. Therefore, zero has no reciprocal.
(ii) The numbers and are their own reciprocals. We need to find numbers 'x' such that x = x 1 . Multiplying both sides by 'x', we get x 2 = 1 . Taking the square root of both sides, we get x = ± 1 . So, the numbers are 1 and -1.
(iii) The rational number that is equal to its negative. We need to find a rational number 'x' such that x = − x . Adding 'x' to both sides, we get 2 x = 0 . Dividing both sides by 2, we get x = 0 . So, the rational number is 0.
(iv) The reciprocal of -5 is . The reciprocal of a number 'a' is a 1 . So, the reciprocal of -5 is − 5 1 or − 5 1 .
(v) The product of two rational numbers is always a . A rational number is a number that can be expressed in the form q p , where p and q are integers and q is not zero. When we multiply two rational numbers, say q 1 p 1 and q 2 p 2 , we get q 1 × q 2 p 1 × p 2 , which is also a rational number since the product of integers is an integer.
(vi) The reciprocal of a positive rational number is . Let the positive rational number be q p , where p and q are positive integers. The reciprocal of this number is p q , which is also a positive rational number since both q and p are positive.
Examples
Understanding reciprocals and rational numbers is crucial in various real-life scenarios. For instance, when calculating the combined work rate of two people working together, you often need to add their individual work rates, which are expressed as reciprocals of the time they take to complete a task. Similarly, in financial calculations, understanding rational numbers helps in dealing with fractions of money or interest rates. For example, if you invest half of your money ( 2 1 ) in stocks and one-third ( 3 1 ) in bonds, rational numbers help you track and manage your portfolio effectively. These concepts are also fundamental in physics, engineering, and computer science, making them essential for problem-solving in diverse fields.
