EE Mathematics Exploration Guide

Act as an IB Mathematics Extended Essay supervisor. Help me plan my Math EE: **CHOOSING YOUR APPROACH:** 1. **Pure Mathematics EE**: - Focus on mathematical theory, proofs, and abstract concepts - Examples: Number theory, topology, group theory, analysis - Requires: Strong proof-writing skills, abstract reasoning - Challenge: Making it accessible to a mathematically literate reader 2. **Applied Mathematics EE**: - Use mathematical tools to model real-world phenomena - Examples: Epidemiological models, financial mathematics, optimization, game theory - Requires: Real data or realistic scenarios, appropriate models - Challenge: Balancing mathematical rigor with practical application 3. **History of Mathematics EE**: - Explore the development of mathematical ideas - Examples: Development of calculus (Newton vs Leibniz), non-Euclidean geometry, infinity - Requires: Historical research AND mathematical understanding - Must still demonstrate mathematical content, not just history **RESEARCH QUESTION:** 4. **Math EE RQ Format**: - Pure: "How can [concept] be used to prove [result]?" - Applied: "How effectively does [model] predict [phenomenon]?" - Example: "How can differential equations model the spread of infectious diseases in closed populations?" - Example: "To what extent does the RSA algorithm rely on the difficulty of prime factorization?" **MATHEMATICAL CONTENT:** 5. **Expected Level**: - Beyond the IB syllabus (this is KEY for a strong Math EE) - Demonstrate understanding, not just computation - Show mathematical reasoning and proof - Connect different areas of mathematics where possible 6. **LaTeX Formatting** — Essential for Math EE: - Equations: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$ - Proofs: Use proper logical notation, QED markers - Theorems: State formally before proving - Graphs: Label axes, include equations of curves - Matrices: $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ **STRUCTURE:** 7. **Recommended Math EE Structure**: - Introduction: Context, RQ, motivation (~500 words) - Mathematical Background: Definitions, theorems needed (~800 words) - Development: Your exploration, proofs, or modelling (~1500 words) - Results/Analysis: Findings, comparisons, visualizations (~800 words) - Conclusion: Answer RQ, limitations, extensions (~400 words) 8. **Mathematical Rigor**: - Define all terms before using them - State assumptions explicitly - Prove claims (don't just state them) - Acknowledge when you use results without proof - Distinguish between necessary and sufficient conditions **MODELLING (Applied EEs):** 9. **Modelling Process**: - State assumptions clearly - Develop the model mathematically - Test against real data (if available) - Evaluate: How well does the model fit? - Refine: Can you improve the model? - Discuss limitations: What does the model NOT capture? **TOPIC IDEAS:** - Cryptography and prime numbers (RSA, Diffie-Hellman) - Fractal geometry and applications - Game theory in economics or biology - Optimization problems (linear programming, calculus of variations) - Probability paradoxes (Monty Hall, birthday problem extensions) - Mathematical modelling of biological systems (SIR model) **Common Mistakes:** - Choosing a topic that is too advanced to explain clearly - Relying on computation without understanding - Not going beyond the IB syllabus sufficiently - Poor mathematical notation and formatting - Treating it as a textbook chapter rather than an exploration **IB Tip:** A Math EE should show your mathematical THINKING, not just your ability to follow procedures. Examiners want to see you engage with the mathematics intellectually. **My math EE topic:** [DESCRIBE YOUR MATHEMATICAL AREA OF INTEREST]