Introduction
HSA Notes brings easy to understand , clear, exam focused notes designed to help Class 9 SSC Part One students score 90 plus marks in Punjab Board exams. This page covers Unit 1, Real Numbers, from Exercise 1.1, Exercise 1.2, Exercise 1.3 & Review Exercise 1 including identifying rational and irrational numbers and representing them on a number line.
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HSA Notes Features
- Very short, easy to remember definitions.
- Step by step solved answers for every question.
- Quick tricks and memory aids for faster revision.
- Exact alignment with the PCTB and PECTAA textbook.
- Number line constructions simplified into a clear method.
- Past paper pattern questions included throughout.
- A fast revision format built for last minute review.
- Trusted by thousands of Punjab students.
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Available Across All Punjab Boards and Top Schools
These notes are built for students across all Punjab Boards, including Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, and DG Khan, and are used at top schools including The City School, Beaconhouse School System, Roots International Schools, Lahore Grammar School, Karachi Grammar School, Froebel’s International School, The Educators, Army Public School, Divisional Public School, and Government Model School.
Definitions
Rational number: Any number that can be written in the form p over q, where p and q are integers and q is not equal to zero.
Irrational number: A number that cannot be written in the form p over q, with a decimal expansion that is both non terminating and non repeating.
Exercise 1.1, Identify Rational or Irrational
| No. | Number | Classification |
|---|---|---|
| i | 2.353535 repeating | Rational, since it is a repeating decimal |
| ii | 0.6 with the 6 repeating | Rational, since it is a repeating decimal |
| iii | 2.236067 continuing, approximately root 5 | Irrational, since the decimal is non repeating and non terminating |
| iv | Root 7 | Irrational, since 7 is not a perfect square |
| v | e | Irrational, a known mathematical constant |
| vi | Pi | Irrational, a known mathematical constant |
| vii | 5 plus root 11 | Irrational, since a rational number plus an irrational number is always irrational |
| viii | Root 3 plus root 13 | Irrational, since the sum of these two irrationals does not simplify to a rational number |
| ix | 15 over 4 | Rational, since it is already written as a simple fraction |
A quick correction on item (i): the number 2.353535, with the digits 35 repeating, is rational because it is a repeating decimal, not because it terminates. The decimal actually continues forever with the 35 pattern repeating, so the correct reasoning is repeating decimal rather than terminating decimal, even though the final classification, rational, is correct either way.
Exercise 1.1, Represent on a Number Line
The following numbers need to be marked on a number line: root 2, root 3, 4 and one third, negative 2 and one seventh, 5 and one eighth, and 2 and three quarters.
Quick method for irrational numbers, such as root 2 and root 3: Draw a right triangle with both legs measuring 1 unit. By the Pythagorean theorem, the hypotenuse of this triangle equals root 2. Using a compass, transfer this hypotenuse length from the number 0 along the number line to mark the position of root 2.
A similar construction using a leg of length root 2 and a second leg of 1 unit produces a hypotenuse of root 3, which can then be marked the same way.
Quick method for rational or mixed numbers, such as 4 and one third or negative 2 and one seventh: Convert the mixed number into an improper fraction or a decimal first, then divide the relevant unit segment on the number line into equal parts matching the denominator, and count the correct number of parts from the nearest whole number to mark the exact position.
Full step by step constructions and diagrams for each of these six numbers are included in the original PDF.
Why This Exercise Matters
Unit 1 sets the foundation for how numbers are classified and represented throughout the rest of the Class 9 syllabus, so being confident in distinguishing rational from irrational numbers, and in constructing accurate number line representations, pays off in later units that build on these same number properties. The number line construction method using a right triangle and a compass is a technique that reappears whenever an irrational square root needs to be placed precisely, so it is worth practicing this construction by hand rather than only memorizing the final positions.
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Frequently Asked Questions
What is the difference between a rational and an irrational number? A rational number can always be written as a fraction of two integers, while an irrational number cannot, and its decimal expansion never terminates or repeats.
Is 2.353535 with the 35 repeating a rational number? Yes, it is rational, since it is a repeating decimal. Repeating decimals, just like terminating decimals, can always be converted into a fraction of two integers.
How do you mark an irrational number like root 2 on a number line? Construct a right triangle with two legs of length 1 unit, which gives a hypotenuse of exactly root 2 by the Pythagorean theorem, then use a compass to transfer that length onto the number line from 0.
Are these notes aligned with the PCTB and PECTAA textbook? Yes, these notes are built to exactly match the PCTB and PECTAA New Book for Class 9 Mathematics.
Conclusion
With Exercise 1.1, Exercise 1.2, Exercise 1.3 & Review Exercise 1 fully solved, from classifying numbers as rational or irrational through constructing accurate number line representations, Class 9 students have a strong start on Unit 1, Real Numbers. Use these notes alongside regular practice with past papers to build the confidence needed for 90 plus marks in your Punjab Board exam.
Best of luck for your Class 9 Mathematics exam!