Review Exercise 1 Class 9 Math Notes
Dear students we have covered all the basics of real numbers in pervious exercises of this unit such as Exercise 1.1, Exercise 1.2 and Exercise 1.3 . Review exercises in the end of each unit are for the revision of the concepts and to test your understanding of the topics discussed in that specific unit.
This article provides complete, solved notes for the Review Exercise 1 class 9 math. Designed specifically according to the new math book for class 9 under the Punjab Textbook Board (2025 edition). Here, you will find solutions to multiple-choice questions, short questions, simplification problems, and real-world applications of real numbers. Each section aims to reinforce your understanding of key concepts such as rational and irrational numbers, properties of real numbers, and algebraic identities. This resource will help you revise effectively, avoid common mistakes, and build a strong foundation for the chapters ahead.
PDF Notes
Key Concepts of Review Exercise 1 Class 9 Math
- Sets and types of numbers: Natural, Whole, Integers, Rational, Irrational, and Real Numbers.
- Properties of real numbers:
- Commutative Property
- Associative Property
- Distributive Property
- Number line representation.
- Laws of exponents and simplification of radicals.
- Rationalization of irrational denominators.
- Solving basic algebraic identities involving real numbers.
Common Mistakes to Avoid
Misapplying algebraic identities.
In my teaching experience i have seen many students confuse these two algebraic identities (a-b)²=a²- 2ab+b² and a²-b²=(a+b)(a-b). So you need to understand the difference between these identities and where to apply them.
Confusing irrational numbers with rational ones.
Sometimes rational number as disguised as irrational but solving the expression results in a rational number for example (√5+√3)(√5-√3) may seem irrational, but when we solve it result is a rational number (√5+√3)(√5-√3) =2. So make sure to solve the expression completely before choosing the answer.
Mistakes in Simplifying Radicals
1. Mixing Rational and Radical Terms
Students who lack the basic concepts may often mix radical and non-radical terms and add them e.g 4+4√a=8√a which is a big mistake and your solution will be marked as wrong. You can only add or subtract “like terms.” Just as you can’t add “apples” to “oranges” and get a single type of fruit (you have apples and oranges), you cannot directly add a rational number to an irrational number like this. They are fundamentally different kinds of real numbers.
2. Incorrectly Adding or Subtracting Radicals
Similarly students sometimes make the mistake of adding or subtracting radical terms incorrectly by combining them under a single square root sign for example, writing √3 + √3 = √6 or √5 − √3 = √2. This is a serious error. Square roots don’t combine that way. Think of √3 and √5 as different “types” of radicals — like trying to add square roots of apples and bananas. You can only add or subtract like radicals, just as you can only add like terms in algebra.
So, √3 + √3 = 2√3, not √6, and √5 − √3 stays just that it cannot be simplified any further.
3. Misapplying Square Root Rules
Many students wrongly simplify radical expressions by misapplying the rules of square roots. A common example is thinking that \(\sqrt{a + b} = \sqrt{a} + \sqrt{b} \quad \text{or} \quad \sqrt{a – b} = \sqrt{a} – \sqrt{b}\) This is not true. Square root operations don’t work like that. For instance, √(3 + 5) = √8 and never equal to √3 + √5. Always remember: you can’t split a square root over addition or subtraction inside the radical. Only multiplication and division work that way: √(a × b) = √a × √b, and √(a ÷ b) = √a ÷ √b — but not addition or subtraction.
To further clarify things here are some rules that might help you avoid the above mentioned mistakes.
Solved Problems
Q1: Simplify: √75 + √27
Both numbers can be factorized as 75= 3×5² and 27=3×3²
so the expression becomes
= √(3×5²) + √(3×3²) = 5√3 + 3√3 = 8√3
Q2: Rationalize the denominator: \( \frac{1}{\sqrt{2}}\)
\(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
Q3: Solve using identity: (a + b)² where a = 2, b = 3
= a² + 2ab + b² = (2)² + 2(2)(3) + (3)²
4 + 12 + 9 = 25
Key Definitions
- Rational Number: A number that can be expressed in the form p/q, where q is not equal to zero such as 4/5.
- Irrational Number: A non-repeating, non-terminating decimal that cannot be expressed as a fraction such as π.
- Real Number: A set that includes both rational and irrational numbers.
- Commutative Property: a + b = b + a, ab = ba
- Distributive Property: a(b + c) = ab + ac
Important Formulas
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- Rationalizing: \(\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{{a}}\)
🖊️ Short Answer Questions
Q: Is 0 a rational number? Explain.
Yes, 0 is a rational number because it can be written as \(\frac{0}{1}\).
Q: State the Trichotomy Property.
For any real number a, only one of the following is true:
a > 0 a = 0 a < 0
Q: Give an example of a non-terminating, non-repeating decimal.
√2 = 1.41421356… (continues forever without repeating)
Video Lecture
Dear students this sums up the Review Exercise Unit 1 class 9 math. With all these valuable resources i am confident that your concepts of real numbers will be clear. If you need more help regarding any concept or a specific question do leave a comment below and i will respond to it.