This article introduces the foundation of the real number system by distinguishing between rational and irrational numbers – a fundamental concept that appears regularly in DSE Paper 1 Section A(1). You will learn the definition of each type, the decimal expansion rule, how to classify numbers in DSE-style questions, and the key operations that preserve (or change) rationality. The article includes worked examples, exam techniques, practice questions, and a full set of homework with solutions. Mastery of this topic ensures you secure the easy 2–4 marks allocated to this concept in every DSE paper.
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Number Systems Explained: Rational and Irrational Numbers | HKDSE Mathematics
Question 1 MC
Which of the following numbers is not rational?
A. \( 0.\overline{5} \) B. \( \sqrt{144} \) C. \( \sqrt{12} \) D. \( \frac{22}{7} \)
Question 2 MC
Which of the following is a rational number?
A. \( \sqrt{18} \) B. \( \frac{\pi}{2} \) C. \( 0.121221222\ldots \) D. \( \sqrt[3]{-27} \)
Question 3 Short Answer
Classify \( \sqrt{72} \) as rational or irrational. Show your working.
Question 4 Short Answer
Express \( 0.\overline{36} \) as a fraction in its simplest form.
Question 5 MC
If \( m \) is a rational number and \( n \) is an irrational number, which of the following must be rational?
A. \( mn \) B. \( m + n \) C. \( \frac{m}{n} \) D. \( (m + n) + (m - n) \)
Question 6 Short Answer
Given that \( a = 2\sqrt{3} \) and \( b = \sqrt{12} \). Are \( a \) and \( b \) rational or irrational? Justify your answer.
Question 7 MC
Which of the following statements is always true?
A. The product of two rational numbers is irrational
B. The sum of two irrational numbers is irrational
C. The product of a rational number and an irrational number is rational
D. The sum of a rational number and an irrational number is irrational
Question 8 Short Answer
Is \( \frac{\sqrt{2}}{2} \) rational or irrational? Explain.
Question 9 Short Answer
Simplify \( \sqrt{98} \). Is the result rational or irrational?
Question 10 Short Answer
If \( x = 3 + \sqrt{7} \), find the value of \( x^2 - 6x \). Is the result rational or irrational?
Students often mistakenly classify \( \frac{\sqrt{2}}{2} \) as rational because it "looks like" a fraction. Remember: the numerator \( \sqrt{2} \) is irrational, so the entire quotient is irrational.