Lecture Notes

Introduction

• Everything is Number: Importance of numerical tools for computer scientists.
  • Scale agreement: Rate your agreement from 1 to 10.

Instructor Introduction

• Sarah Santosh: Mathematician, background in arts and computing.
• Hobbies: Knitting, wrapping presents, dancing.
• Engage in discussion forums about personal interests.

Basics of Computing

• Basic Understanding: Knowledge of numerical foundations helps in effective problem solving.
  • All information in computing is handled as numbers.

Importance of Mathematics in Computing

• Patterns: Humans search for patterns; mathematics expresses these patterns.
• Connection to Computing: Mathematical tools are crucial for:
  • Graphics (transforming sound, designing VR, etc.)
  • Data analysis in business.

Tools and Topics Covered

1. Numbers: Binary, sequences, series.
2. Functions: Geometry, linear algebra, probability.
3. Numerical Problem Solving: Encouragement to engage in discussions.

Module Description

• Learning Approach: Collaborative journey with opportunities for personal exploration.
• Activities: Watch example videos, attempt quizzes early for feedback, engage in discussions, complete peer review tasks.

Deep Dive into Mathematics

• Machine Level Computing: Understanding how inputs are computed as binary (1s and 0s).
  • Light switch analogy: Basic operations involve switching.
  • Example calculations using binary.

Binary and Other Bases

• Understanding Binary: Everything is computed with binary as the base.
  • Operations with binary numbers.
• Other Bases: Learning how to convert and perform arithmetic operations in bases like octal, hexadecimal, etc.

Pattern Recognition and Mathematical Functions

• Mathematics as Pattern Recognition: Delve deeper into problem-solving using identifiable sequences.
  • Connecting mathematics with patterns, rhythms, and arts enhances understanding.

Concepts of Sequences

• Definition of Sequences: Ordered list of numbers with a pattern.
• Representing Sequences: Use subscript notation (a_n) to denote the nth term of a sequence.
• Types of Sequences:
  1. Arithmetic Progression: Common difference.
     • General term: a_n = a_1 + (n-1)d
  2. Geometric Progression: Common ratio.
     • General term: a_n = a_1 * r^(n-1)

Sums of Sequences

• Calculating Sums of a Series: Utilization of sigma (Σ) notation for concise representation.
  • Special formulas for arithmetic series: S_n = n/2 * (first_term + last_term)
  • Special formulas for geometric series: S_n = a * (1 - r^n) / (1 - r)

Practical Applications in Computing

• Sequences in Computer Science: Implementing numerical methods and algorithms.
  • How to use these sequences in computing scenarios for ratings, scores, or games.
• Graphs of Functions: Use graphs to illustrate sequences and functions.

Examining Functions through Graphs

• Coordinate System: Cartesian coordinates with (x, y).
  • Understanding positive and negative direction on axes and using intervals for defining boundaries of functions.
• Function Transformations: Translational movements (shifts), reflections, and scalings affect plots.

Final Thoughts

• Exploration of Sequences and Series: Applying knowledge to complex systems and observing patterns.
• Self-Assessment and Discussion: Engage with peers and enhance comprehension of numerical methods.

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Conclusion

• Encouragement to use these tools and insights in future numerical mathematics exploration and applications. Remember: Mathematics is not only about computation; it teaches us to think critically about problems.