Number System Class 9 ||Maths|| Chapter 1 Notes

Number System Class 9 ||Economics|| Chapter 1 Notes

1.1 Introduction

• In earlier classes, students learned about natural numbers, whole numbers, and integers.
• The chapter extends these concepts to understand rational and irrational numbers.

1.2 Irrational Numbers

• Rational Numbers: Numbers that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\mathrm{\ne }0$.
• Irrational Numbers: Numbers that cannot be expressed in the form $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating.

Examples of Irrational Numbers:

• $\sqrt{2},\sqrt{3},\pi$, etc.

Properties of Irrational Numbers:

• The sum or product of a rational number and an irrational number is always irrational.
• The product of two irrational numbers may be rational or irrational.

1.3 Real Numbers and their Decimal Expansions

• Real Numbers: All rational and irrational numbers together make up the real number system.

Decimal Expansions:

1. Terminating Decimal Expansion: Rational numbers like $\frac{1}{2}=0.5$, $\frac{3}{4}=0.75$, etc., terminate after a finite number of digits.
2. Non-Terminating, Repeating Decimal Expansion: Rational numbers like $\frac{1}{3}=0.333\dots$, $\frac{5}{6}=0.8333\dots$, etc., repeat after a fixed set of digits.

1.4 Representing Real Numbers on the Number Line

• Any real number can be represented on the number line.
• For irrational numbers like $\sqrt{2}$, you can approximate its location using the Pythagorean theorem.

1.5 Operations on Real Numbers

• Addition, Subtraction, Multiplication, Division can be performed on real numbers.
• Laws of Exponents hold for real numbers as well:
  • $a^m\times a^n=a^{m+n}$
  • ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$, etc.

1.6 Surds

• Surds: Irrational numbers of the form $\sqrt[n]{a}$, where $a$ is a positive rational number, and the root is not a perfect integer.

Examples:

• $\sqrt{2},\sqrt{3},\sqrt[3]{5}$, etc.

Simplification of Surds:

• $\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$
• $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, provided $b\mathrm{\ne }0$.

1.7 Rationalizing the Denominator

• Rationalization: The process of eliminating the surd from the denominator of a fraction.
• For example, to rationalize $\frac{1}{\sqrt{2}}$, multiply both numerator and denominator by $\sqrt{2}$, which gives $\frac{\sqrt{2}}{2}$.

1.8 Laws of Exponents for Real Numbers

• The chapter also revisits and extends the laws of exponents for real numbers:
  • $a^m\times a^n=a^{m+n}$
  • ${\left(a^m\right)}^n=a^{mn}$
  • $a^0=1$
  • $a^{-m}=\frac{1}{a^m}$, where $a\mathrm{\ne }0$.

Key Formulas

1. For any real numbers $a$ and $b$:

   • $a^m\times a^n=a^{m+n}$
   • $a^m÷a^n=a^{m-n}$
   • $(a^m{)}^n=a^{mn}$
   • $a^0=1$
   • $a^{-m}=\frac{1}{a^m}$
   • ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

2. Rationalization:

   • To rationalize $\frac{1}{\sqrt{a}}$, multiply numerator and denominator by $\sqrt{a}$.

Practice Problems

1. Simplify $\sqrt{72}$.
2. Rationalize $\frac{1}{\sqrt{5}}$.
3. Express $0.121212\dots$ as a fraction.
4. Represent $\sqrt{5}$ on the number line.

Tips for Study:

• Make sure you understand the different types of numbers and their properties.
• Practice representing numbers on the number line.
• Solve problems related to operations with rational numbers.
• Familiarize yourself with decimal expansions.