Class 9 Maths Chapter 1 Real Numbers Notes – Exercise 1.1 Solved (Free PDF Download)

Real Numbers is the first chapter of Class 9 Mathematics, and a clear understanding of this chapter makes the rest of the book much easier. Concepts from this chapter — rational numbers, irrational numbers, number lines and properties of real numbers — appear again and again in board exams.

These notes cover Exercise 1.1 completely, in handwritten style and in very simple words. Every question is solved step by step, and difficult parts carry a short, easy explanation before the solution, so slow learners can follow the method without getting stuck.

Read This First: Real Numbers Is Easier Than It Looks

Let us be honest with each other. When most students open Chapter 1 and see numbers like √2, π and 0.3737…, the first thought is, “Maths is not for me.” We hear this every year, and every year it turns out to be wrong. Exercise 1.1 only asks you to learn four small skills: telling rational and irrational numbers apart, placing a number on a number line, changing a repeating decimal into a fraction, and naming the property used in a line. That is all. There is nothing in this exercise that a slow and steady student cannot do.

These notes were written the way a good teacher explains at the board to the student sitting on the last bench — one idea at a time, in short sentences, with the reason spoken out loud before the answer is written. If you learn slowly, please do not call yourself weak. In our experience, the students who read slowly and practise one question at a time are the same students who stop losing silly marks in the board exam. Speed comes later. Understanding comes first, and these notes are built for understanding.

One small test will carry you through most of the exercise: if a decimal ends or repeats, the number is rational; if it never ends and never repeats, it is irrational. Keep this one line in your mind, and Question 1 becomes almost free marks.

Notes Details

Class	9
Subject	Mathematics
Chapter	1 – Real Numbers
Exercise	1.1 (complete)
Format	PDF (handwritten notes)
Pages	7
Language	Simple English
Price	Free

What Is Included in These Notes?

1. Rational and Irrational Numbers (Question 1)

A simple test decides whether a number is rational or irrational. All 10 parts of Question 1 are classified with a short reason for each answer, including special cases such as pi, e, and numbers containing square roots.

2. Numbers on a Number Line (Question 2)

Each number is first changed into a decimal, then marked at the correct place on the number line. Square root of 2, square root of 3, mixed fractions and a negative fraction are all shown with clear diagrams.

3. Repeating Decimals as Fractions (Question 3)

A repeating decimal is changed into a fraction p/q with a fixed 4-step method: call the number x, multiply by 10 or 100, subtract, and solve. All three parts are worked out in full, with an extra example for practice.

4. Properties of Real Numbers (Question 4)

Commutative, associative, identity, inverse and distributive properties are explained in plain words and grouped in an easy way, so the correct property name becomes simple to recognise in the exam.

5. Properties with Inequalities (Question 5)

Rules for less-than and greater-than signs are covered, including the additive, reciprocal, cancellation and trichotomy properties. One memory line explains the whole topic: positive keeps the sign, negative turns the sign.

6. Rational Numbers Between Two Numbers (Question 6)

Two rational numbers between the given numbers are found with the average method — add the two numbers and divide by 2, then repeat the same step once more. A simple method that works every time.

Exercise 1.1 Detailed Walkthrough: Every Question Explained

Before you download the PDF, read this honest walkthrough. It tells you what each question is asking, how the solution works, and where the marks are hiding — so when you open the handwritten notes, you already know the story of the exercise.

Question 1 — Rational or Irrational? (10 small parts)

This question gives you ten numbers and asks only one thing: rational or irrational? The whole game is the one-line test from page 1 of the notes. The decimal 2.353535 is rational because it ends. The number 0.666… is rational because the same digit repeats forever, and 2.236067… is irrational because it goes on without any pattern. The root numbers √7, 5 + √11 and √3 + √13 are all irrational, and so are the two famous guests, pi (π) and e. The fraction 15/4 is rational because it is already written as p/q. The last part, (2 − √2)(2 + √2), looks scary but quietly becomes 4 − 2 = 2, a whole number, so it is rational. Always give one short reason with each answer — that habit turns a guess into a full-mark answer.

Question 2 — Numbers on the Number Line (6 parts)

Many students fear this question because of the square roots, but the method never changes. Change the number into a decimal, then ask: between which two whole numbers does it live? √2 is about 1.4, so it sits between 1 and 2. √3 is about 1.7, a little further along. The mixed number 4⅓ becomes 13/3, about 4.33, between 4 and 5. The negative number −2 1/7 becomes −15/7, about −2.1, on the left side of zero between −3 and −2. The fraction 5/8 is 0.625, between 0 and 1, and 2¾ becomes 11/4 = 2.75, between 2 and 3. Every one of these is drawn on a clean number line in the notes, in a style you can copy directly into your exam copy.

Question 3 — Repeating Decimals as Fractions (3 parts)

This is the most “technical” question of the exercise, and also the most scoring, because the method is fixed. You call the number x. If one digit repeats, you multiply by 10; if two digits repeat, you multiply by 100. Then you subtract the original number, and the repeating tail cancels itself out like magic. In this way 0.444… becomes 4/9, 0.3737… becomes 37/99, and 0.2121… becomes 21/99. Three parts, one method, no surprises — practise it twice and it becomes free marks for life.

Questions 4 and 5 — Name the Property (14 small parts)

Here the examiner only wants the name of the rule used in each line, and students lose marks by memorising names without understanding the clue. These notes connect every name to a picture. If the order changes, it is commutative. If only the brackets move, it is associative. Adding 0 or multiplying by 1 is identity; a number meeting its opposite to make zero is the additive inverse; a number outside the bracket multiplying both terms inside is distributive. Question 5 applies the same thinking to inequalities: adding the same number to both sides is the additive property, reciprocals flip the direction, cutting the same factor from both sides is cancellation — and if that factor is negative, the sign must turn. Trichotomy, the last idea, says two numbers can only be greater, equal, or less.

Question 6 — Two Rational Numbers Between Two Numbers (3 parts)

The final question asks for two rational numbers between each given pair. Instead of guessing fractions and hoping, the notes use the average method: add the two numbers and divide by 2, and the result is guaranteed to stand between them. Repeat the same step with the new number and you have your second answer. Between 1/3 and 1/4, the answers are 7/24 and 15/48. Between 3 and 4, they are 7/2 and 13/4. Between 3/5 and 4/5, they are 7/10 and 13/20. A calm, certain finish to the exercise.

Why These Notes Help Slow Learners

• Simple words, short sentences and one idea at a time
• An easy explanation before the solved questions, where a topic is difficult
• Extra examples that are not in the exercise, for better understanding
• A one-page revision sheet at the end for quick preparation before a test
• Practice questions with answers for self-checking
• Handwritten style that feels like class notes and stays easy to read

How to Use These Notes

1. Read the easy explanation of a skill first.
2. Follow the solved question slowly, line by line.
3. Cover the answer and solve the same question again without help.
4. Attempt the practice questions at the end and check the answers.
5. Revise from the one-page revision sheet a day before the test.

Download Class 9 Maths Real Numbers Notes PDF

The complete Exercise 1.1 notes are available as a free PDF download. The PDF can be read on a mobile phone, saved for offline study, or printed for use in a notebook file. Every page carries the All Free Assets logo.

5 Common Mistakes Students Make in Exercise 1.1

These are real mistakes that cost students marks again and again. Read them once, and you will not repeat them.

1. “2.353535 is irrational because it looks long.” No. Look carefully — this decimal ends. A decimal that stops is always rational, no matter how long it looks. Do not add dots in your mind that the book did not print.
2. “Every square root is irrational.” Wrong. √4 = 2 and √9 = 3 are perfectly rational. Only the roots of numbers that are not perfect squares, like √2, √7 and √11, are irrational.
3. Multiplying by 10 when two digits repeat. For 0.3737…, the repeating part is “37” — two digits — so you must multiply by 100, not 10. Count the repeating digits first; that one decision decides the whole question.
4. Forgetting that the inequality sign turns with a negative number. In Question 5, when c is negative, < becomes >. Half the class loses this mark simply because they were in a hurry. Slow down for exactly this one step.
5. Guessing any two fractions in Question 6. Random fractions are often not between the given numbers at all. Use the average method given in these notes — add the two numbers, divide by 2 — and your answer is guaranteed to sit between them, every single time.

If you want more help like this, browse our complete Class 9 notes and the full collection of free maths notes — every chapter is explained in the same simple, step-by-step way.

Frequently Asked Questions

Are these Class 9 Maths notes free?

Yes. The complete notes are free to read and download. No payment and no account are needed.

Which exercise is covered in these notes?

These notes cover Chapter 1 Real Numbers, Exercise 1.1, Questions 1 to 6, fully solved.

Are these notes suitable for slow learners?

Yes. Difficult questions carry a simple explanation before the solution, extra examples are added for practice, and the language stays short and easy throughout the notes.

Which syllabus do these notes follow?

The notes follow the Class 9 Mathematics syllabus for PCTB (Punjab Curriculum and Textbook Board) and the concepts are the same for other boards teaching Real Numbers in Class 9.

I am very weak in maths. Where should I start?

Start at page 1 of the PDF and learn only the one-line test: end or repeat means rational. Then do Question 1 slowly. Leave Question 3 for the next day if you feel tired — the 4-step method will still be there, and it never changes. Small daily steps beat one long, painful sitting, especially for this chapter.

Can I print these notes or share them with my class fellows?

Yes, of course. Print the PDF for your notebook file, and share it freely with your friends and class fellows. The All Free Assets watermark on every page simply shows where the notes came from, so the next student who receives them can also download the rest of the chapters for free.

Conclusion

Real Numbers becomes an easy chapter once the basic ideas are clear. These handwritten notes explain Exercise 1.1 in simple words, solve every question step by step, and finish with a revision sheet and practice questions. Download the free PDF and start preparation today.

More free chapter-wise notes for Class 9, 10, 11, 12 and university level are available at AllFreeAssets.com.