1. Basic Operations
- Addition: \( a + b \)
- Subtraction: \( a - b \)
- Multiplication: \( a \times b \) or \( ab \)
- Division: \( a \div b \) or \( \frac{a}{b} \)
2. Properties of Operations
- Commutative Property:
- Addition: \( a + b = b + a \)
- Multiplication: \( ab = ba \)
- Associative Property:
- Addition: \( (a + b) + c = a + (b + c) \)
- Multiplication: \( (ab)c = a(bc) \)
- Distributive Property:
- \( a(b + c) = ab + ac \)
3. Order of Operations (PEMDAS/BODMAS)
- Parentheses/Brackets
- Exponents/Orders (like squares and square roots)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
4. Fractions
- Addition/Subtraction:
- \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \)
- \( \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \)
- Multiplication:
- \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
- Division:
- \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \)
5. Percentages
- Percentage Formula:
- \( \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \)
- Finding the Whole:
- \( \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100 \)
6. Ratios and Proportions
- Ratio: \( a:b \) or \( \frac{a}{b} \)
- Proportion: If \( \frac{a}{b} = \frac{c}{d} \), then \( ad = bc \)
7. Exponents
- \( a^m \times a^n = a^{m+n} \)
- \( \frac{a^m}{a^n} = a^{m-n} \)
- \( (a^m)^n = a^{mn} \)
8. Square and Square Root
- Square of a Number: \( a^2 \)
- Square Root: \( \sqrt{a} \)
- Area of Rectangle: \( A = l \times w \)
- Area of Triangle: \( A = \frac{1}{2} \times b \times h \)
- Area of Circle: \( A = \pi r^2 \)
- Perimeter of Rectangle: \( P = 2(l + w) \)
- Circumference of Circle: \( C = 2\pi r \)
10. Statistics
- Mean (Average):
- \( \text{Mean} = \frac{\text{Sum of Values}}{\text{Number of Values}} \)
- Median: Middle value in a sorted list.
- Mode: Most frequently occurring value.