In-Depth Notes on Number Systems, Fractions, Decimals, and Ratios

1. Number Systems

• Real Numbers vs Non-Real Numbers
  • Two main categories of numbers:
  • Real Numbers (R): Used in everyday life; includes integers, fractions, decimals, etc.
  • Non-Real Numbers (Complex Numbers, C): Numbers of the form $a + bi$ (where $a, b$ are real numbers and $i = \sqrt{-1}$).

1.1 The Set of Non-Real Numbers

• Complex numbers include:
  • Examples: $i$, $-3 + 4i$, $1 - 2i$, …

1.2 The Set of Real Numbers

• Composed of:
  • Rational Numbers (Q): Can be expressed as $rac{a}{b}$ where $a$, $b ext{ are integers and } b > 0$.
  • Irrational Numbers (Q′): Cannot be expressed as a simple fraction. Includes $\pi$, $\sqrt{2}$, etc.
• Integer Set (Z): ${0, 1, -1, 2, -2, 3, -3, …}$
• Non-Integer Set (Z′): ${a, b : a \notin Z, b \not= 0}$

2. Fractions

• Defined as $\frac{a}{b}$ where $a$ and $b$ have no common factors.
• They can be:
  • Proper Fractions: $a < b$ (e.g., $\frac{1}{7}$)
  • Improper Fractions: $a \geq b$ (e.g., $\frac{7}{2}$)
  • Mixed Fractions: A combination of an integer and a proper fraction (e.g., $1 \frac{1}{2}$).

2.1 Types of Fractions

• Proper Fractions: Example: $\frac{1}{2}, \frac{3}{20}$
• Improper Fractions: Example: $\frac{7}{2}, \frac{10}{3}$
• Mixed Fractions: Example: $1 \frac{1}{2}, 7 \frac{8}{9}$

2.2 Addition and Subtraction of Fractions

• With Common Denominators: Directly add numerators.
  • Example: $\frac{1}{2} + \frac{7}{2} = \frac{1 + 7}{2} = \frac{8}{2} = 4$
• With Unequal Denominators: Find the Lowest Common Denominator (LCD).
  • Definition: The smallest denominator that divides all denominators evenly.
  • Example of LCD computation:
  • Given ${2, 7}$, LCD = $2 \times 7 = 14$.
  • Add fractions via: $\frac{a}{b} + \frac{c}{d} = \frac{LCD \cdot a + LCD \cdot c}{LCD}$.

3. Multiplication and Division of Fractions

• Multiplication:
  • Formula: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
  • Example: $\frac{1}{2} \cdot \frac{3}{7} = \frac{1 \cdot 3}{2 \cdot 7} = \frac{3}{14}$.
• Division:
  • Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$.

3.1 Mixed Fractions Multiplication

• Convert mixed to improper first:
  • Example: $2 \frac{1}{5} \cdot \frac{3}{7}$
  • Improper: $\frac{11}{5} \cdot \frac{3}{7} = \frac{33}{35}$

4. Decimals

• Decimals can be terminating or repeating.
  • Example of terminating: $0.3$
  • Example of repeating: $0.333\ldots = 0.\bar{3}$
• Conversion methods:
  • Discontinuous: $0.3 = \frac{3}{10}$
  • Repetitive: Use geometric series for conversion; $0.\bar{3} = \frac{1}{3}$.

4.1 Scientific Notation

• Useful for large/small numbers: $245895531 = 2.45895531 \times 10^8$
• Conversion example: $0.26548 = 2.6548 \times 10^{-1}$

5. Ratios and Proportions

5.1 Ratios

• Define as $a:b$ where there is no common factor.
• Simplifying example: $100:50$ to $2:1$.

5.2 Proportions

• Proportion between different ratios; example of department expenditures E.g., Basic Education, Health, and Social Development.

5.3 Applications in Business and Finance

• Currency exchange, asset turnover, and liquidity ratios for measuring efficiency and solvency.