What Is Number System Class 9: Complete Guide for Students

Understanding the Number System: Basic Definitions

The number system is a way to represent and classify numbers based on their properties. In Class 9 Mathematics, the NCERT syllabus covers several types of numbers:

• Natural Numbers ($ ext{N}$): Counting numbers starting from 1, 2, 3, ...
• Whole Numbers ($ ext{W}$): Natural numbers including zero (0, 1, 2, 3, ...)
• Integers ($ ext{Z}$): Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational Numbers ($ ext{Q}$): Numbers expressed as $rac{p}{q}$ where $p$ and $q$ are integers and $q

eq 0$

• Irrational Numbers: Numbers that cannot be expressed as a fraction, e.g., $ oot 2 ext{of } 2$, $ oot 3 ext{of } 5$, $ ext{pi } (.1415...)$

These sets form the foundation of the number system studied in Class 9 NCERT.

Classification of Numbers: From Natural to Real

Numbers can be classified into different sets based on their properties. Here's a simple classification:

Together, rational and irrational numbers form the Real Numbers ($ ext{R}$), which include all possible numbers on the number line.

Properties of Rational and Irrational Numbers

Understanding the difference between rational and irrational numbers is crucial:

• Rational Numbers:
• Can be written as $rac{p}{q}$ where $p, q$ are integers and $q

eq 0$

• Decimal expansion either terminates or repeats periodically
• Examples: $0.5 = rac{1}{2}$, $-1.25 = rac{-5}{4}$

• Irrational Numbers:
• Cannot be expressed as a fraction of two integers
• Decimal expansion is non-terminating and non-repeating
• Examples: $ oot 2 ext{of } 2 = 1.4142135...$, $ ext{pi} = 3.1415926...$

Worked Example:

Is $0.3333...$ (repeating) rational or irrational?

Since $0.3333... = rac{1}{3}$, it is a rational number.

Prime Factorisation and Its Role in Number Systems

Prime factorisation helps break down numbers into their basic building blocks — prime numbers.

• Every composite number can be expressed as a product of prime numbers.
• This is useful in finding the greatest common divisor (GCD) and least common multiple (LCM).

Example: Find the prime factorisation of 84.

$$84 = 2 imes 42 = 2 imes 2 imes 21 = 2^2 imes 3 imes 7$$

Prime factorisation helps in understanding the structure of numbers and is a key concept in the Class 9 NCERT Number Systems chapter.

Number Line and Representation of Real Numbers

The number line is a visual tool to represent numbers, especially real numbers.

• Natural, whole, and integers are represented as points on the number line.
• Rational numbers appear as points between integers.
• Irrational numbers fill the gaps between rational numbers, making the number line continuous.

Key Point:

Every real number corresponds to a unique point on the number line, and vice versa.

This helps in visualising concepts like ordering, density of rational numbers, and the completeness of real numbers.

Decimal Representation: Terminating and Non-Terminating Decimals

Decimals are another way to express rational and irrational numbers.

• Terminating decimals: Decimals that end after a finite number of digits (e.g., 0.75)
• Non-terminating repeating decimals: Decimals with a repeating pattern (e.g., 0.666...)
• Non-terminating non-repeating decimals: Decimals that neither end nor repeat (e.g., $ ext{pi}$)

Formula to convert a repeating decimal to fraction:

For example, convert $x = 0.ar{3}$

Multiply both sides by 10: $$10x = 3.333...$$ Subtract original equation: $$10x - x = 3.333... - 0.333...$$ $$9x = 3$$ $$x = rac{3}{9} = rac{1}{3}$$

This proves $0.ar{3}$ is rational.