Which option correctly describes the nature of \(\frac{1}{2}+\sqrt{2}\)?
Answer and explanation
Correct answer: Irrational number
\(\frac{1}{2}\) is rational and \(\sqrt{2}\) is irrational. If their sum were rational, then \(\sqrt{2}=(\frac{1}{2}+\sqrt{2})-\frac{1}{2}\) would be a difference of rationals and thus rational, contradicting that \(\sqrt{2}\) is irrational. Hence \(\frac{1}{2}+\sqrt{2}\) is irrational. The closest distractor "rational" is incorrect for the reason above; "integer" and "terminating decimal" are specific types of rationals so they are also incorrect. Exam tip: memorize that rational + irrational = irrational (contradiction proof is quick and useful in exams).
Frequently asked questions
What is the correct answer to this question?
Irrational number
Why is this the correct answer?
\(\frac{1}{2}\) is rational and \(\sqrt{2}\) is irrational. If their sum were rational, then \(\sqrt{2}=(\frac{1}{2}+\sqrt{2})-\frac{1}{2}\) would be a difference of rationals and thus rational, contradicting that \(\sqrt{2}\) is irrational. Hence \(\frac{1}{2}+\sqrt{2}\) is irrational. The closest distractor "rational" is incorrect for the reason above; "integer" and "terminating decimal" are specific types of rationals so they are also incorrect. Exam tip: memorize that rational + irrational = irrational (contradiction proof is quick and useful in exams).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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