Pre-Algebra: Comprehensive Mathematical Foundations

Mathematical Reasoning Components:

• Logical Thinking: Drawing valid conclusions from given information
• Pattern Recognition: Identifying regularities and structures
• Abstraction: Working with general concepts beyond specific numbers
• Problem-Solving: Applying strategies to unfamiliar situations
• Precision: Communicating with mathematical accuracy

Number System Hierarchy:

1. Natural Numbers (ℕ): {1, 2, 3, 4, ...} - Counting numbers
2. Whole Numbers (ℤ₀): {0, 1, 2, 3, ...} - Natural numbers + zero
3. Integers (ℤ): {..., -3, -2, -1, 0, 1, 2, 3, ...} - Whole numbers + negatives
4. Rational Numbers (ℚ): {a/b | a,b ∈ ℤ, b ≠ 0} - Fractions
5. Real Numbers (ℝ): ℚ + irrational numbers (π, √2, etc.)
6. Complex Numbers (ℂ): {a + bi | a,b ∈ ℝ, i = √-1}

Base-10 System Structure:

Our number system is positional and decimal (base-10):

• Each position represents a power of 10
• Digits 0-9 represent values in each place
• Position determines value (e.g., 5 in 500 vs 50 vs 5)
• Zero as placeholder enables positional notation

Key Number Theory Concepts:

• Prime Numbers: Natural numbers >1 with exactly two distinct positive divisors (1 and itself)
• Composite Numbers: Natural numbers >1 that are not prime (can be factored)
• Factors: Numbers that divide evenly into another number
• Multiples: Products of a number with integers
• Divisibility Rules: Patterns that determine if one number divides evenly into another

Decimal System Structure:

Decimals extend the base-10 system to fractional parts:

• Decimal point separates whole and fractional parts
• Each decimal place represents a negative power of 10
• Trailing zeros don't change value (0.5 = 0.50 = 0.500)
• Leading zeros don't change value (5 = 05 = 005)

Decimal Operation Rules:

1. Addition/Subtraction: Align decimal points, add/subtract as whole numbers
2. Multiplication: Multiply as whole numbers, count decimal places, place in product
3. Division: Make divisor whole number, move decimal in dividend same places
4. Rounding: Look at digit to right of target place, round up if ≥5

Fraction Fundamentals:

• Definition: a/b where a = numerator, b = denominator, b ≠ 0
• Proper: Numerator < denominator (1/2, 3/4)
• Improper: Numerator ≥ denominator (5/2, 7/4)
• Mixed: Whole number + proper fraction (2 1/2, 3 3/4)
• Equivalent: Fractions with same value (1/2 = 2/4 = 3/6)

Ratio Fundamentals:

• Ratio: Comparison of two quantities (a:b or a/b)
• Proportion: Equation stating two ratios are equal (a/b = c/d)
• Rate: Ratio comparing different units (miles/hour, $/hour)
• Unit Rate: Rate with denominator of 1 unit

Percentage Fundamentals:

• Definition: "Per hundred" - ratio with denominator 100
• Conversion: Percent ↔ Decimal ↔ Fraction
• Applications: Discounts, interest, statistics, probability
• Calculations: Part = Whole × Percentage

Integer Fundamentals:

• Definition: {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Absolute Value: Distance from zero (|x|)
• Opposites: Numbers same distance from zero, opposite signs
• Number Line: Visual representation of integers

Algebraic Concepts:

• Variables: Symbols representing unknown values (x, y, n)
• Expressions: Combinations of numbers, variables, and operations
• Equations: Mathematical statements with equality
• Inequalities: Relationships showing relative size

Equation Properties:

• Addition Property: If a = b, then a + c = b + c
• Subtraction Property: If a = b, then a - c = b - c
• Multiplication Property: If a = b, then a × c = b × c (c ≠ 0)
• Division Property: If a = b, then a ÷ c = b ÷ c (c ≠ 0)

Basic Geometric Concepts:

• Points, Lines, Planes: Fundamental geometric elements
• Angles: Acute, right, obtuse, straight
• Polygons: Triangles, quadrilaterals, regular polygons
• Circles: Radius, diameter, circumference, area
• Solids: 3D shapes and their properties

Basic Statistical Concepts:

• Data Types: Categorical vs numerical
• Measures of Center: Mean, median, mode
• Measures of Spread: Range, quartiles
• Data Representation: Tables, charts, graphs
• Basic Probability: Likelihood of events

Advanced Topics:

• Exponents: Repeated multiplication (aⁿ = a×a×...×a)
• Scientific Notation: Expressing large/small numbers (a × 10ⁿ)
• Square Roots: Numbers that when squared give original
• Pythagorean Theorem: a² + b² = c² for right triangles
• Coordinate Geometry: Plotting points on coordinate plane