CBSE Class 9 Maths Notes: Number System

📌 Representation of Numbers on the Number Line

This section explores how we can visually represent different types of numbers on a number line. This includes natural numbers, integers, and rational numbers.

🔍 Core Principles:

Number Line: A straight line with numbers marked at equal intervals.
Natural Numbers: Positive whole numbers (1, 2, 3, …). Represented on the number line starting from 1.
Integers: Whole numbers, including positive and negative numbers, and zero (…, -2, -1, 0, 1, 2, …).
Rational Numbers: Numbers that can be expressed as a fraction $p/q$, where $p$ and $q$ are integers and $q \ne 0$. They can be accurately located on the number line.

🔍 Plotting Terminating Decimals Using Magnification

Understand how to locate terminating decimals (e.g., 2.5, 3.75) precisely on a number line using magnification.

🔍 Example:

To plot 2.34, first locate it between 2 and 3. Then, divide the segment between 2 and 3 into 10 equal parts. Then, further magnify the segment to locate 2.3 and 2.4. Finally, magnify the required segment to pinpoint 2.34.

🔍 Displaying Recurring Decimals on the Number Line

Learn to represent recurring decimals (e.g., 0.333…, 1.666…) on the number line using an iterative approach.

🔍 Method:

Convert recurring decimals to fractions. Locate the fraction on the number line. If you are struggling with the fraction conversion, you can approximate by magnifying the sections that your decimal lays between, similar to terminating decimals.

🔍 Identifying Rational Numbers via Decimal Form

Determine whether a decimal represents a rational number based on its properties.

💬 Definitions:

Terminating Decimals: Decimals that end after a finite number of digits (e.g., 0.25, 1.5).
Recurring Decimals: Decimals that have a digit or a group of digits that repeats infinitely (e.g., 0.333…, 1.2727…).
Rational Number Condition: If a decimal is terminating or recurring, then it is a rational number.

📌 Irrational Numbers as Non-Repeating, Non-Terminating Decimals

Understand the nature of irrational numbers and their unique decimal representation.

💬 Definitions:

Irrational Numbers: Numbers that cannot be expressed as a fraction $p/q$, where $p$ and $q$ are integers.
Characteristics: Irrational numbers have non-repeating and non-terminating decimal expansions. (e.g., $\pi$, $\sqrt{2}$)

📌 Plotting Irrational Numbers (√2, √3, etc.)

Learn how to graphically represent irrational numbers like $\sqrt{2}$ and $\sqrt{3}$ on the number line using methods such as the Pythagorean theorem.

💡 Method:

$\sqrt{2}$: Construct a right-angled triangle with sides 1 and 1. The hypotenuse will be $\sqrt{2}$. Use a compass to transfer this length to the number line.
$\sqrt{3}$: Construct a right-angled triangle using the previously calculated value of $\sqrt{2}$ and a side of 1.

💬 Real Numbers and the Number Line

The core concept that every real number has a unique point on the number line, and every point on the number line corresponds to a real number.

💬 Core Principle:

There is a one-to-one correspondence between real numbers and points on the number line.

📌 Definition of the nth Root of a Real Number

Understand the definition and properties of the nth root.

💬 Definitions:

nth Root: The nth root of a number $a$ is a number $b$ such that $b^n = a$. Represented as $\sqrt[n]{a}$.

📌 Rationalization of Expressions

Learn how to remove radicals from the denominator of expressions.

💬 Examples & Formulaes:

Rationalizing $1/(a + b\sqrt{x})$: Multiply both the numerator and denominator by the conjugate $(a – b\sqrt{x})$.
$\frac{1}{a + b\sqrt{x}} \times \frac{a – b\sqrt{x}}{a – b\sqrt{x}} = \frac{a – b\sqrt{x}}{a^2 – b^2x}$
Rationalizing $1/(\sqrt{x} + \sqrt{y})$: Multiply both the numerator and denominator by the conjugate $(\sqrt{x} – \sqrt{y})$.
$\frac{1}{\sqrt{x} + \sqrt{y}} \times \frac{\sqrt{x} – \sqrt{y}}{\sqrt{x} – \sqrt{y}} = \frac{\sqrt{x} – \sqrt{y}}{x – y}$