Class 9th Number Systems Notes/formulas

🧠 Important Concepts

• Natural Numbers (ℕ): Counting numbers starting from 1.
  Example: 1, 2, 3, …
• Whole Numbers (𝕎): Natural numbers + Zero.
  Example: 0, 1, 2, 3, …
• Integers (ℤ): Positive numbers, negative numbers, and zero.
  Example: …, -3, -2, -1, 0, 1, 2, …
• Rational Numbers (ℚ): Numbers that can be expressed in the form p/q, where q ≠ 0.
  Example: 1/2, -4/5, 3 (as 3/1)
• Irrational Numbers: Numbers that cannot be expressed in the form p/q. These are non-terminating and non-repeating decimals.
  Example: √2, π
• Real Numbers (ℝ): All rational and irrational numbers combined.

📌 Properties of Rational and Irrational Numbers

🔢 Decimal Expansion

• Terminating: Decimal comes to an end.
  Example: 1/4 = 0.25
• Non-Terminating Repeating: Repeats in a pattern.
  Example: 1/3 = 0.333…
• Non-Terminating Non-Repeating: No pattern, irrational number.
  Example: √2 = 1.414213…

✅ Laws of Exponents for Real Numbers

Let a > 0 and m, n be integers:

1. am×an=am+na^m × a^n = a^{m+n}
2. am÷an=am−na^m ÷ a^n = a^{m-n}
3. (am)n=amn(a^m)^n = a^{mn}
4. (ab)m=am×bm(ab)^m = a^m × b^m
5. (ab)m=ambm\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}

📘 Surds and Irrational Numbers

• Surd: An irrational root which cannot be simplified to remove the square root (or other roots).
  Example: √2, √3, √5
  Example: √50 = √(25 × 2) = 5√2

🔍 Representation on Number Line

To represent irrational numbers like √2 on the number line:

• Draw a right-angled triangle with both legs = 1 unit.
• Hypotenuse = √2.
• Use a compass to mark √2 on the number line starting from origin.

💡 Rationalization

Rationalization removes the irrational number from the denominator.

Example:
12=1×22×2=22\frac{1}{\sqrt{2}} = \frac{1 × \sqrt{2}}{\sqrt{2} × \sqrt{2}} = \frac{\sqrt{2}}{2}

📝 Formulas to Remember

• √a × √a = a
• √a × √b = √(ab)
• √(a/b) = √a / √b
• (√a + √b)(√a – √b) = a – b

🧪 Practice Examples

1. Is 0 a rational number?
   ✔ Yes, because 0 = 0/1
2. Convert 1.232323… into a fraction:
   Let x = 1.232323…
   100x = 123.232323…
   Subtracting:
   100x – x = 122
   ⇒ 99x = 122 ⇒ x = 122/99
3. Prove √2 is irrational:
   ✔ Use proof by contradiction: Assume √2 = p/q, where p and q are coprime integers. Then square both sides and show contradiction arises.

Surds & Radical Identities

Let a, b be positive real numbers:

• √(a × b) = √a × √b
• √(a/b) = √a ÷ √b
• (√a + √b)(√a − √b) = a − b
• (a + √b)(a − √b) = a² − b
• (√a + √b)² = a + b + 2√(ab)