Mind map of chapter 1 : Number system

Number Systems

1. Types of Numbers

• Natural Numbers (N): 1, 2, 3, ...
• Used for counting.
• Smallest natural number is 1.
• Whole Numbers (W): 0, 1, 2, 3, ...
• Includes all natural numbers and zero.
• Integers (Z): ..., -2, -1, 0, 1, 2, ...
• Includes positive, negative, and zero.
• Rational Numbers (Q): Numbers of the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
• Examples: \( \frac{1}{2} \), \( \frac{-3}{4} \), 0.75, etc.
• Every integer is a rational number (e.g., \( 5 = \frac{5}{1} \)).
• Irrational Numbers: Numbers that cannot be expressed as \( \frac{p}{q} \).
• Examples: \( \sqrt{2} \), \( \pi \), \( e \), etc.
• Their decimal expansions are non-terminating and non-repeating.
• Real Numbers (R): All rational and irrational numbers combined.
• Includes all numbers on the number line.

2. Representation on Number Line

• Rational Numbers: Can be plotted using fractions or decimals.
• Example: \( \frac{1}{2} \) is plotted at 0.5.
• Irrational Numbers: Can be plotted using geometric constructions.
• Example: \( \sqrt{2} \) is plotted using the Pythagoras theorem.

3. Decimal Expansions

• Terminating Decimals: Decimals that end after a finite number of digits.
• Examples: 0.5, 0.75, 0.125, etc.
• Denominator of the fraction (in lowest terms) has only 2 and/or 5 as prime factors.
• Non-Terminating Repeating Decimals: Decimals that repeat endlessly.
• Examples: 0.\(\overline{3}\), 0.\(\overline{142857}\), etc.
• Denominator of the fraction (in lowest terms) has prime factors other than 2 and 5.
• Non-Terminating Non-Repeating Decimals: Irrational numbers.
• Examples: \( \sqrt{2} = 1.41421356... \), \( \pi = 3.14159265... \), etc.

4. Operations on Real Numbers

• Addition and Subtraction: Follow the rules of arithmetic.
• Example: \( 3 + 5 = 8 \), \( 7 - 2 = 5 \).
• Multiplication and Division: Follow the rules of arithmetic.
• Example: \( 4 \times 6 = 24 \), \( 10 \div 2 = 5 \).
• Rationalization: Process of eliminating radicals from the denominator.
• Example: \( \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \).

5. Laws of Exponents

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

6. Rational and Irrational Numbers

• Sum: Rational + Rational = Rational
• Example: \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \).
• Product: Rational × Rational = Rational
• Example: \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).
• Sum: Rational + Irrational = Irrational
• Example: \( 1 + \sqrt{2} \) is irrational.
• Product: Rational × Irrational = Irrational (if Rational ≠ 0)
• Example: \( 2 \times \sqrt{3} = 2\sqrt{3} \) is irrational.

7. Visualizing Real Numbers

• Number Line: Every real number corresponds to a unique point on the number line.
• Example: \( \sqrt{2} \) is located between 1.4 and 1.5 on the number line.
• Density: Between any two real numbers, there are infinitely many real numbers.
• Example: Between 1.4 and 1.5, there are numbers like 1.41, 1.42, etc.

8. Important Theorems

• Theorem 1: Every rational number has either a terminating or repeating decimal expansion.
• Example: \( \frac{1}{2} = 0.5 \) (terminating), \( \frac{1}{3} = 0.\overline{3} \) (repeating).
• Theorem 2: Every irrational number has a non-terminating and non-repeating decimal expansion.
• Example: \( \sqrt{2} = 1.41421356... \) (non-terminating and non-repeating).