Number Systems: Chapter 1- NCERT Maths Class 9

The chapter Number Systems appears in NCERT mathematics textbook. It serves as a foundational chapter that introduces students to the classification and properties of different types of numbers, which will be essential for more advanced mathematical concepts in later chapters.

The class 9 maths chapter 1 systematically builds up the number system from natural numbers to real numbers, explaining how each set of numbers relates to the others. It also introduces important concepts like decimal expansions, operations with irrational numbers, and laws of exponents with rational powers.

This chapter provides a comprehensive introduction to number systems, focusing on the classification and properties of different types of numbers. Here’s a summary of the key topics:

Key Topics Covered in Chapter 1 Number Systems :

1. Rational Numbers:
   • Numbers that can be expressed as p/q where p and q are integers and q≠0
2. Irrational Numbers:
   • Numbers that cannot be expressed as p/q
3. Real Numbers:
   • The collection of all rational and irrational numbers.
   • Represented on the number line, where every point corresponds to a unique real number.
4. Operations on Real Numbers:
   • Sum, difference, product, and quotient of irrational numbers may be rational or irrational.
5. Decimal Expansions:
   • Terminating decimals (e.g., 0.50) and non-terminating recurring decimals (e.g., 0.3) are rational.
6. Square Roots and Other Roots:
   • Definition and properties of square roots and nth roots
   • Identities involving square roots
7. Laws of Exponents for Real Numbers:
   • Extended laws of exponents for rational exponents
   • Simplification of expressions with rational exponents
8. Representation of Numbers on the Number Line:
   • Techniques for locating irrational numbers like √2, √3, and √5 on the number line
   • Construction methods using geometric principles

Question and Answers on Number Systems

1. What is a number system? 

A number system is a way to represent and classify numbers. It includes different types of numbers such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

2. What are natural numbers? 

Natural numbers are counting numbers starting from 1: N = {1, 2, 3, 4, …}

3. What are whole numbers? 

Whole numbers include all natural numbers along with zero: W = {0, 1, 2, 3, …}

4. What are integers? 

Integers include all whole numbers and their negatives: Z = {…, -2, -1, 0, 1, 2, …}

5. What are rational numbers? 

A rational number can be written as p/q, where p and q are integers and q≠0.

• Examples: 1/2, -3, 0.75, 0.333… (repeating decimal)
• Decimal form: Either terminating (e.g., 0.5) or non-terminating recurring (e.g., 0.333…)

6. What are irrational numbers? 

Numbers that cannot be written as p/q (non-terminating, non-repeating decimals).

• Examples: √2, π, 0.101001000100001…

7. What are real numbers? 

The collection of all rational and irrational numbers. Every point on the number line represents a unique real number, and every real number corresponds to a unique point on the number line.

8. How do you identify if a number is rational or irrational?

• Rational: Terminating or repeating decimal expansion
• Irrational: Non-terminating, non-repeating decimal expansion

9. Is zero a rational number? 

Yes, because it can be written as 0/1.

10. Are all integers rational numbers? 

Yes, because any integer m can be written as m/1.

11. Are all square roots irrational? No, only square roots of non-perfect squares are irrational.

• Rational: √4 = 2, √9 = 3
• Irrational: √2 = 1.414213…, √3 = 1.732050…

12. How to find rational numbers between two numbers? Take the average or convert them to fractions with a common denominator.

• Example: Between 1 and 2, rational numbers include 3/2, 5/4, 7/4, etc.

13. What is the difference between terminating and non-terminating decimals?

• Terminating: Ends after a finite number of digits (e.g., 0.5)
• Non-terminating: Continues infinitely (can be recurring or non-recurring)

14. How to represent irrational numbers on a number line? 

Using geometric constructions. For example, √2 can be represented by drawing a right triangle with sides of length 1, and using the Pythagorean theorem to find the hypotenuse.

15. What are the laws of exponents for real numbers? 

For a > 0 and rational exponents p, q:

1. aᵖ · aᵍ = aᵖ⁺ᵍ
2. (aᵖ)ᵍ = aᵖᵍ
3. aᵖ/aᵍ = aᵖ⁻ᵍ
4. aᵖbᵖ = (ab)ᵖ

16. How to rationalize the denominator? 

Multiply numerator and denominator by an expression that eliminates the irrational part in the denominator.

• Example: 1/√2 = 1/√2 · √2/√2 = √2/2

17. Is π a rational number? 

No, π is irrational because its decimal expansion (3.141592…) never repeats or terminates.

18. Can the sum of two irrational numbers be rational? 

Yes. For example, (2 + √3) + (2 – √3) = 4, which is rational.

19. What is the difference between √4 and √5?

• √4 = 2 (rational)
• √5 = 2.236… (irrational)

20. Why is 1/0 not defined? 

Division by zero is undefined in mathematics because it leads to contradictions. If 1/0 were a number, then 0 × (1/0) = 1, which contradicts the fact that 0 multiplied by any number equals.

See more:

Click here for Heron’s Formula: NCERT Maths Class 9