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The sum of a rational number and an irrational number is always:
- a)Rational
- b)Irrational
- c)Whole number
- d)Integer
Correct answer is option 'B'. Can you explain this answer?
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The sum of a rational number and an irrational number is always:a)Rati...
Understanding Rational and Irrational Numbers
Rational numbers are those that can be expressed as a fraction of two integers (e.g., 1/2, -4, 0.75). In contrast, irrational numbers cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions (e.g., √2, π).
Sum of Rational and Irrational Numbers
When you add a rational number to an irrational number, the result is always irrational. Here’s why:
Example for Clarity
Consider adding the rational number 1 (which can be written as 1/1) and the irrational number √2:
Conclusion
Thus, the result of adding a rational number and an irrational number is always an irrational number, confirming that option 'B' is indeed correct. This principle is fundamental in understanding the properties of numbers in mathematics.
Rational numbers are those that can be expressed as a fraction of two integers (e.g., 1/2, -4, 0.75). In contrast, irrational numbers cannot be expressed as simple fractions and have non-repeating, non-terminating decimal expansions (e.g., √2, π).
Sum of Rational and Irrational Numbers
When you add a rational number to an irrational number, the result is always irrational. Here’s why:
- Definition of Rational Numbers: A rational number can be represented as p/q where p and q are integers and q is not zero.
- Definition of Irrational Numbers: An irrational number cannot be expressed as p/q; its decimal form goes on forever without repeating.
- Contradiction in Results: If the sum (rational + irrational) were rational, it could be expressed as a fraction. This would imply that the irrational part could be isolated and expressed as a fraction, contradicting its nature.
Example for Clarity
Consider adding the rational number 1 (which can be written as 1/1) and the irrational number √2:
- 1 + √2 = x (assuming x is rational)
- This implies √2 = x - 1, which shows √2 could be expressed as a rational number, contradicting its definition.
Conclusion
Thus, the result of adding a rational number and an irrational number is always an irrational number, confirming that option 'B' is indeed correct. This principle is fundamental in understanding the properties of numbers in mathematics.
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The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer?.
The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of a rational number and an irrational number is always:a)Rationalb)Irrationalc)Whole numberd)IntegerCorrect answer is option 'B'. Can you explain this answer?.
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