Math
The Big Picture of Mathematics
Mathematics is a vast, interconnected landscape of ideas. Understanding how different areas relate to each other is crucial for self-directed learning. Here's the big picture:
The Mathematical Universe
The Foundation Layer
- Numbers and Operations: The building blocks - arithmetic, fractions, integers, rationals, reals
- Logic and Reasoning: How we think mathematically - proofs, sets, logical statements
- Algebra: The language of relationships - variables, equations, functions
- Geometry: Understanding space and shape - both abstract and applied
The Bridge Layer
- Functions: The central unifying concept connecting algebra, geometry, and calculus
- Trigonometry: The mathematics of periodic phenomena and triangles
- Precalculus: Integration of algebraic and geometric thinking
- Logic & Proofs: Mathematical reasoning and formal thinking
The Power Tools
- Calculus: The mathematics of change and accumulation
- Linear Algebra: The study of linear relationships and multidimensional spaces
- Discrete Mathematics: The mathematics of counting, graphs, and computer science
- Probability & Statistics: Understanding uncertainty and data
- Differential Equations: Modeling dynamic systems and change over time
How Mathematical Ideas Connect
- Algebra provides the symbolic manipulation skills needed for Calculus
- Geometry gives visual intuition that supports Linear Algebra and Calculus
- Functions from Precalculus become the objects studied in Calculus
- Linear Algebra provides the framework for Multivariable Calculus
- Logic & Proofs underpin all advanced mathematics
- Statistics uses Calculus for continuous distributions
- Discrete Math connects to Computer Science and Combinatorics
Your Self-Learning Journey
Learning Strategies
- Start with the big picture - understand where you're going before diving deep
- Build on solid foundations - don't skip prerequisites
- Make connections - always ask "how does this relate to what I know?"
- Practice regularly - mathematical skill requires consistent practice
- Learn through problems - work examples and solve problems, don't just read
- Use multiple resources - different explanations help solidify understanding
Prerequisites Map
Each section below includes explicit prerequisites. Follow this rough sequence: → Foundations → Intermediate → Advanced But within each level, some flexibility exists based on your interests.
Introduction
Mathematics is fundamental to understanding the world around us and plays a crucial role in personal and professional development. As a self-learner, you have the advantage of exploring mathematical concepts at your own pace and making connections that interest you most.
Critical Thinking and Problem-Solving
Math teaches systematic approaches to breaking down complex problems into manageable parts. These analytical skills transfer to every area of life, from financial planning to troubleshooting technical issues.
Foundation for STEM Fields
Mathematics is the language of science, technology, engineering, and medicine. Without a solid math foundation, advanced study in these fields becomes impossible.
Financial Literacy
Understanding percentages, interest rates, statistics, and basic algebra is essential for making informed financial decisions about loans, investments, insurance, and budgeting.
Career Opportunities
Many high-paying careers require mathematical competency, including:
- Software development and data science
- Engineering and architecture
- Finance and economics
- Research and academia
- Healthcare and pharmaceuticals
Logical Reasoning
Math develops the ability to think logically, identify patterns, and make valid deductions. These skills improve decision-making in all contexts.
Technology and Modern Life
As technology becomes more prevalent, mathematical concepts like algorithms, statistics, and computational thinking become increasingly important for digital literacy.
Universal Language
Mathematical principles are universal and transcend cultural and linguistic barriers, making it a powerful tool for communication and collaboration.
Foundations
The foundation layer provides the essential building blocks for all mathematics. These topics are interconnected and should be learned together.
Arithmetic & Number Systems
The foundation of all mathematical knowledge, arithmetic deals with basic number operations and properties.
Key Applications
- Personal finance and budgeting
- Unit conversions in science and cooking
- Basic data analysis and percentages
- The Four Basic Operations: Addition, Subtraction, Multiplication, Division
- Order of operations (PEMDAS/BODMAS)
- Properties: commutative, associative, distributive
- Number Types: Natural numbers, integers, rational numbers, irrational numbers, real numbers
- Understanding number line representation
- Absolute value and number comparison
- Fractions and Decimals: Converting between forms, simplifying fractions
- Mixed numbers and improper fractions
- Terminating vs. repeating decimals
- Ratios, Proportions, and Percentages: Real-world applications
- Cross-multiplication for solving proportions
- Percentage calculations and percent change
- The Four Basic Operations: Addition, Subtraction, Multiplication, Division
Basic Algebra
Introduces abstract thinking through the use of variables and symbolic manipulation. This is where mathematics becomes a language for expressing relationships.
Key Applications
- Modeling real-world relationships
- Problem-solving in business and science
- Foundation for programming and computer science
- Variables and Expressions: Using letters to represent unknown quantities
- Evaluating expressions by substitution
- Combining like terms and simplifying expressions
- Solving Linear Equations: Finding the value of unknowns
- One-step and multi-step equations
- Equations with variables on both sides
- Inequalities: Relationships between quantities
- Solving and graphing inequalities on number lines
- Compound inequalities (and/or)
- Introduction to Graphing: Coordinate plane and linear relationships
- Plotting points and understanding quadrants
- Graphing linear equations using slope-intercept form
- Variables and Expressions: Using letters to represent unknown quantities
Geometry
Explores shapes, sizes, and spatial relationships in two and three dimensions. Develops visual intuition and logical reasoning through proofs.
Key Applications
- Architecture and construction
- Computer graphics and game development
- Navigation and GPS systems
- Points, Lines, and Angles: Basic geometric elements
- Types of angles: acute, right, obtuse, straight
- Angle relationships: complementary, supplementary, vertical
- Parallel lines and transversals
- Polygons and Circles: Two-dimensional shapes
- Triangle classifications and properties
- Quadrilaterals: squares, rectangles, parallelograms, trapezoids
- Circle terminology: radius, diameter, chord, arc
- Area and Perimeter: Measuring two-dimensional space
- Formulas for common shapes
- Composite figures and problem-solving strategies
- Volume and Surface Area: Three-dimensional measurements
- Prisms, pyramids, cylinders, cones, spheres
- Real-world applications in construction and design
- Points, Lines, and Angles: Basic geometric elements
Logic & Mathematical Reasoning
The foundation of mathematical thinking: how to reason logically and construct valid arguments.
- Sets and Set Operations: Basic set theory, unions, intersections, complements
- Logic: Propositional logic, quantifiers (for all, there exists)
- Proof Techniques: Direct proof, proof by contradiction, proof by induction
- Mathematical Reasoning: How to read and write mathematical arguments
Key Applications
- Computer programming and algorithm design
- Legal reasoning and argumentation
- Scientific method and hypothesis testing
Intermediate
The bridge layer connects foundational concepts to powerful mathematical tools.
Precalculus (Functions & Advanced Algebra)
Functions are the heart of mathematics - understanding them deeply is crucial for everything that follows.
- Functions and their Graphs: Domain, range, transformations, compositions
- Polynomial and Rational Functions: Behavior, graphing, applications
- Exponential and Logarithmic Functions: Growth, decay, scaling
- Systems of Equations and Inequalities: Multiple variables, optimization
- Complex Numbers: Extending the number system
- Sequences and Series: Patterns and infinite sums
Key Applications
- Economic modeling (supply/demand, growth rates)
- Population dynamics and biological modeling
- Engineering and signal processing
Trigonometry
The mathematics of cycles, waves, and rotation - essential for understanding periodic phenomena.
- The Unit Circle: Defining trig functions geometrically
- Sine, Cosine, and Tangent: Properties and graphs
- Trigonometric Identities and Equations: Algebraic manipulation
- Law of Sines and Cosines: Solving any triangle
- Applications of Trigonometry: Waves, oscillations, navigation
Key Applications
- Sound and audio engineering
- Computer graphics and animation
- GPS and navigation systems
- Electrical engineering (AC circuits)
Calculus I (Single Variable)
The mathematics of change and accumulation - one of humanity's greatest intellectual achievements.
- Limits and Continuity: The foundation of calculus thinking
- Derivatives and their Applications: Rates of change, optimization
- Integration Techniques: Finding antiderivatives
- Fundamental Theorem of Calculus: Connecting derivatives and integrals
- Applications: Related rates, optimization, area and volume
Key Applications
- Physics (motion, forces, energy)
- Economics (marginal analysis, optimization)
- Biology (population models, drug concentration)
- Engineering (design optimization)
Advanced
The power tools of modern mathematics, science, and technology.
Discrete Mathematics
The mathematics of discrete structures - essential for computer science and modern technology.
- Combinatorics: Counting principles, permutations, combinations
- Graph Theory: Networks, trees, paths, connectivity
- Number Theory: Properties of integers, modular arithmetic
- Recursion and Induction: Mathematical induction, recurrence relations
- Boolean Algebra: Logic circuits, digital systems
Key Applications
- Algorithm design and analysis
- Cryptography and security
- Network analysis and social media
- Database design and optimization
Calculus II & III (Multivariable)
- Sequences and Series
- Parametric Equations and Polar Coordinates
- Vectors and the Geometry of Space
- Partial Derivatives and Multiple Integrals
Linear Algebra
The study of linear relationships - fundamental to modern applications in data science and machine learning.
- Vectors and Vector Spaces
- Matrices and Systems of Linear Equations
- Determinants and Eigenvalues
- Linear Transformations
Differential Equations
The mathematics of change and systems - how things evolve over time.
- First-Order Differential Equations: Separable, linear, applications
- Second-Order Linear Equations: Oscillations, systems
- Systems of Differential Equations: Multiple interacting variables
- Modeling Applications: Population dynamics, mechanical systems
Key Applications
- Population and epidemic modeling
- Mechanical and electrical systems
- Financial modeling and risk analysis
- Climate and weather prediction
Probability & Statistics
The mathematics of uncertainty and data - increasingly important in our data-driven world.
- Descriptive Statistics: Mean, median, mode, variance, correlation
- Probability Theory: Basic probability, conditional probability, Bayes' theorem
- Random Variables and Distributions: Binomial, normal, exponential
- Statistical Inference: Confidence intervals, hypothesis testing
- Regression Analysis: Linear models, correlation vs causation
- Experimental Design: Planning studies, avoiding bias
Key Applications
- Data science and machine learning
- Medical research and clinical trials
- Quality control in manufacturing
- Financial risk assessment
- A/B testing and product optimization