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.
By the end of this article, you should be able to:
Every year, DSE Maths Paper 1 Section A(1) tests your ability to classify numbers as rational or irrational. This seemingly simple topic is a guaranteed 2–4 marks – marks you cannot afford to lose.
Questions on number systems often appear in Section A(1) as short-answer classification questions or as part of algebraic manipulation problems where rationalising denominators is required.
A number is rational if it can be written as a ratio of two integers:
where:
| Property | Example |
|---|---|
| Terminating decimals | \( \frac{3}{4} = 0.75 \) |
| Recurring (repeating) decimals | \( \frac{10}{3} = 3.333\ldots \) or \( 3.\overline{3} \) |
| Integers are rational | \( 5 = \frac{5}{1} \) |
Rational = Ratio. If you can write it as a fraction \( \frac{p}{q} \), it's rational.
Common examples of rational numbers:
A number is irrational if it cannot be expressed as a ratio of two integers \( \frac{p}{q} \).
| Property | Example |
|---|---|
| Non-terminating, non-recurring decimal | \( 0.101101110\ldots \) |
| Square roots of non-perfect squares | \( \sqrt{2} \approx 1.41421356\ldots \) |
| Special constants | \( \pi \approx 3.14159265\ldots \) |
Common examples of irrational numbers:
Irrational = Infinite non-repeating decimal. Cannot be written as a simple fraction.
All rational and irrational numbers together form the real numbers.
| Type | Decimal Expansion | Example |
|---|---|---|
| Rational | Terminating or non-terminating recurring | \( 0.75 \), \( 0.\overline{3} \) |
| Irrational | Non-terminating and non-recurring | \( 1.41421356\ldots \) |
Some numbers look irrational but are actually rational:
| Expression | Actual Value | Classification |
|---|---|---|
| \( \sqrt{16} \) | \( 4 = \frac{4}{1} \) | Rational |
| \( \sqrt[3]{8} \) | \( 2 = \frac{2}{1} \) | Rational |
| \( \frac{\sqrt{9}}{\sqrt{4}} \) | \( \frac{3}{2} \) | Rational |
Questions often test whether the result of an operation is rational or irrational.
| Operation | Result | Example |
|---|---|---|
| Rational + Irrational | Irrational | \( 3 + \sqrt{2} \) is irrational |
| Rational × Irrational (non-zero) | Irrational | \( 5 \times \sqrt{3} \) is irrational |
| Rational ÷ Irrational | Irrational | \( \frac{4}{\sqrt{2}} \) is irrational |
| Irrational + Irrational | May be rational or irrational | \( \sqrt{2} + (-\sqrt{2}) = 0 \) (rational) |
| Irrational × Irrational | May be rational or irrational | \( \sqrt{2} \times \sqrt{2} = 2 \) (rational) |
To convert a recurring decimal to a fraction:
Example: Convert \( 0.\overline{27} \) to a fraction.
Let:
$$ x = 0.\overline{27} = 0.272727\ldots $$ $$ 100x = 27.272727\ldots $$ $$ 100x - x = 27.272727\ldots - 0.272727\ldots = 27 $$ $$ 99x = 27 \quad\Rightarrow\quad x = \frac{27}{99} = \frac{3}{11} $$Therefore, \( 0.\overline{27} = \frac{3}{11} \).
This conversion method appears regularly in DSE Paper 1 Section A(1). Memorise the shortcut:
\( 0.\overline{ab} = \frac{ab}{99} \)
Determine whether each number is rational or irrational:
(a) \( \sqrt{9} \) (b) \( \sqrt{7} \) (c) \( 3.\overline{4} \) (d) \( \pi \)
| Number | Classification | Reason |
|---|---|---|
| \( \sqrt{9} \) | Rational | \( \sqrt{9} = 3 = \frac{3}{1} \) |
| \( \sqrt{7} \) | Irrational | 7 is not a perfect square |
| \( 3.\overline{4} = 3.444\ldots \) | Rational | Non-terminating but recurring decimal |
| \( \pi \) | Irrational | Non-terminating, non-recurring decimal |
Question: Which of the following is an irrational number?
A. \( \sqrt{25} \) B. \( 0.\overline{12} \) C. \( \sqrt{2} \) D. \( \frac{22}{7} \)
C. \( \sqrt{2} \) is irrational because 2 is not a perfect square.
Question: If \( r \) is a rational number and \( s \) is an irrational number, which of the following must be irrational?
A. \( r + s \) B. \( r \times s \) (where \( r \neq 0 \)) C. \( r - s \) D. All of the above
D. All of the above
The sum, difference and product (with non-zero rational) of a rational and an irrational number are always irrational.
Question 1 MC
Which of the following numbers is irrational?
A. \( \sqrt{81} \) B. \( 0.\overline{7} \) C. \( \frac{3\pi}{2} \) D. \( \frac{\sqrt{25}}{\sqrt{9}} \)
Question 2 MC
Which of the following is a rational number?
A. \( \sqrt{2} \) B. \( \sqrt[3]{64} \) C. \( 0.101001000\ldots \) D. \( 2\pi \)
Question 3 Short Answer
Determine whether \( \sqrt{50} \) is rational or irrational. Explain your answer.
Question 4 Short Answer
The decimal expansion of a number is \( 0.142857142857\ldots \). Is this number rational or irrational? Explain.
Question 5 MC
If \( x = 4 + \sqrt{5} \) and \( y = 4 - \sqrt{5} \), which of the following statements is true?
A. \( x + y \) is irrational B. \( x \times y \) is irrational
C. \( x + y \) is rational D. \( x \div y \) is irrational
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An Exercise on Number Systems Explained: Rational and Irrational Numbers