Which option is always true?
Answer and explanation
Correct answer: Every irrational number is real
The real-number system is divided into two mutually exclusive classes: rational numbers and irrational numbers. Thus every irrational number belongs to the set of real numbers, so option A is always true. The converse is false because rational numbers such as 1/2, -3, and 0 are also real. Consequently, option B is false. Option C is false because a rational number cannot be irrational by definition, and option D is false because every integer can be written as a fraction, for example 3 = 3/1, so every integer is rational. The useful inclusion is irrational numbers ⊂ real numbers, while rational numbers ⊂ real numbers as well. Recognising these set relationships prevents the mistake of treating “real” and “irrational” as synonyms.
Frequently asked questions
What is the correct answer to this question?
Every irrational number is real
Why is this the correct answer?
The real-number system is divided into two mutually exclusive classes: rational numbers and irrational numbers. Thus every irrational number belongs to the set of real numbers, so option A is always true. The converse is false because rational numbers such as 1/2, -3, and 0 are also real. Consequently, option B is false. Option C is false because a rational number cannot be irrational by definition, and option D is false because every integer can be written as a fraction, for example 3 = 3/1, so every integer is rational. The useful inclusion is irrational numbers ⊂ real numbers, while rational numbers ⊂ real numbers as well. Recognising these set relationships prevents the mistake of treating “real” and “irrational” as synonyms.
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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