NUMBER SYSTEM KEY CONCEPTS

Here’s a detailed explanation of key concepts from Chapter 1 of CBSE Class 9 Maths (Number Systems) with examples:

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1. Real Numbers

Real numbers include all rational and irrational numbers. Let’s explore them in detail:

Rational Numbers:

• Numbers that can be written in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.
• Examples: 34,−2,0,7.25\frac{3}{4}, -2, 0, 7.25 (decimal terminating or repeating).

Irrational Numbers:

• Numbers that cannot be expressed in the form pq\frac{p}{q}.
• Properties:
  • Decimal form is non-terminating and non-repeating.
  • Examples include 2,π,5\sqrt{2}, \pi, \sqrt{5}.
  • Sum or product of a rational and an irrational number is irrational. For example:
    • 2+32 + \sqrt{3} is irrational.
    • 5×2=525 \times \sqrt{2} = 5\sqrt{2} is irrational.

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2. Representation on the Number Line

• To represent real numbers, especially irrational numbers, on the number line:
  1. Example for 2\sqrt{2}:
     • Construct a right-angled triangle with base 11 unit and perpendicular 11 unit.
     • By Pythagoras theorem: Hypotenuse=12+12=2.Hypotenuse = \sqrt{1^2 + 1^2} = \sqrt{2}.
     • Use a compass to mark 2\sqrt{2} on the number line.
  2. Successive Magnification:
     • Locate a specific decimal value like 2.32.3 by zooming in between 2 and 3 on the number line.

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3. Laws of Exponents

These laws simplify calculations involving powers:

1. am⋅an=am+na^m \cdot a^n = a^{m+n}
   Example: 23⋅24=23+4=27=1282^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128.
2. aman=am−n\frac{a^m}{a^n} = a^{m-n}
   Example: 3532=35−2=33=27\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27.
3. (am)n=am⋅n(a^m)^n = a^{m \cdot n}
   Example: (23)2=23⋅2=26=64(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64.
4. (ab)m=am⋅bm(ab)^m = a^m \cdot b^m
   Example: (2⋅3)2=22⋅32=4⋅9=36(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \cdot 9 = 36.
5. ambm=(ab)m\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m
   Example: 4323=(42)3=23=8\frac{4^3}{2^3} = \left(\frac{4}{2}\right)^3 = 2^3 = 8.

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4. Rationalization

Rationalization involves removing an irrational number from the denominator.

Example 1:

Rationalize 12\frac{1}{\sqrt{2}}:

12×22=22.\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}.

Example 2:

Rationalize 13+2\frac{1}{\sqrt{3} + \sqrt{2}}:

13+2×3−23−2=3−23−2=3−2.\frac{1}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} – \sqrt{2}}{\sqrt{3} – \sqrt{2}} = \frac{\sqrt{3} – \sqrt{2}}{3 – 2} = \sqrt{3} – \sqrt{2}.

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5. Decimal Expansion

Decimal expansions are used to classify numbers:

1. Terminating: Ends after a finite number of decimals. Example: 14=0.25\frac{1}{4} = 0.25.
2. Non-Terminating, Repeating: Decimal repeats in a pattern. Example: 13=0.333…\frac{1}{3} = 0.333….
3. Non-Terminating, Non-Repeating: Indicates an irrational number. Example: π=3.14159265…\pi = 3.14159265….

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6. Properties of Irrational Numbers

• 2+3\sqrt{2} + \sqrt{3} is irrational.
• 2×3=6\sqrt{2} \times \sqrt{3} = \sqrt{6} is irrational.
• (2)2=2(\sqrt{2})^2 = 2 is rational.

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7. Example Problems

Problem 1: Show that 5\sqrt{5} is irrational.

• Assume 5=pq\sqrt{5} = \frac{p}{q}, where pp and qq are co-prime integers.
• Squaring both sides: 5=p2q25 = \frac{p^2}{q^2} or p2=5q2p^2 = 5q^2.
• This implies p2p^2 is divisible by 5, so pp is divisible by 5.
• Let p=5kp = 5k. Substituting, (5k)2=5q2(5k)^2 = 5q^2, which gives q2q^2 divisible by 5.
• Thus, pp and qq have a common factor, contradicting the assumption of co-primality. Hence, 5\sqrt{5} is irrational.

Problem 2: Simplify (2+3)(2−3)(2 + \sqrt{3})(2 – \sqrt{3}).

=22−(3)2=4−3=1.= 2^2 – (\sqrt{3})^2 = 4 – 3 = 1.

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