Basic Arithmetic: Numbers and Operations

Study Notes

Numbers and Types of Numbers:
- Natural Numbers: positive integers (1, 2, 3, …)
- Whole Numbers: natural numbers including zero (0, 1, 2, 3, …)
- Integers: whole numbers and their negatives (…, -3, -2, -1, 0, 1, 2, 3, …)
- Rational Numbers: numbers can be expressed as a fraction of two integers (e.g., 1/2, 3/4)
- Irrational Numbers: numbers cannot be expressed as a fraction of two integers (e.g., √2, π)
- Real Numbers: all rational and irrational numbers
Basic Operations:
- Addition (+): combining two numbers to get a sum
- Subtraction (−): finding the difference between two numbers
- Multiplication (×): repeated addition of a number
- Division (÷): splitting a number into equal parts
Properties of Operations:
- Commutative Property:
  - Addition: a + b = b + a
  - Multiplication: a × b = b × a
- Associative Property:
  - Addition: (a + b) + c = a + (b + c)
  - Multiplication: (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = (a × b) + (a × c)
- Identity Property:
  - Addition: a + 0 = a
  - Multiplication: a × 1 = a
- Inverse Property:
  - Addition: a + (−a) = 0
  - Multiplication: a × (1/a) = 1 (a ≠ 0)
Order of Operations (PEMDAS/BODMAS):
- Parentheses/Brackets: solve expressions inside parentheses first
- Exponents/Orders: calculate powers and roots
- Multiplication and Division: from left to right
- Addition and Subtraction: from left to right

Fractions:
- Numerator: the top part of a fraction
- Denominator: the bottom part of a fraction
- Equivalent Fractions: different fractions that represent the same value
- Simplifying Fractions: reducing fractions to their simplest form
Operations with Fractions:
- Addition/Subtraction: same denominator, add/subtract numerators
- Addition/Subtraction: different denominators, find a common denominator
- Multiplication: multiply numerators and denominators
- Division: multiply by the reciprocal

Decimals:
- Place Value: tenths, hundredths, thousandths, etc.
- Addition/Subtraction: align decimal points
- Multiplication: multiply as whole numbers, then place the decimal
- Division: move the decimal point to make the divisor a whole number

Percentages:
- Definition: a fraction of 100
- Conversion:
  - Fraction to percentage: multiply by 100
  - Decimal to percentage: multiply by 100
- Finding Percentages: e.g., 20% of 50 = 0.20 × 50 = 10

Ratios and Proportions:
- Ratio: a comparison of two quantities
- Proportion: an equation stating that two ratios are equal

Word Problems:
- Identify Keywords: sum, difference, product, quotient, etc.
- Set Up Equation: translate the problem into an equation
- Solve: use appropriate arithmetic operations to find the solution
- Check: verify the solution by substituting back into the original problem

Notes:
- π (Pi): approximately 3.14159
- Units: always ensure units are consistent
- Dimensions:
  - Area: measured in square units (e.g., cm², m²)
  - Volume: measured in cubic units (e.g., cm³, m³)

Angles:
- Definition: an angle is formed by two rays with a common endpoint called the vertex
- Types of Angles:
  - Acute Angle: less than 90°
  - Right Angle: exactly 90°
  - Obtuse Angle: more than 90° but less than 180°
  - Straight Angle: exactly 180°
  - Reflex Angle: more than 180°