Mathematics in the Modern World

Mathematics in our World

Definition of Mathematics

  • Mathematics: Study of relationships among numbers, quantities, and shapes.

    • Includes: arithmetic, algebra, trigonometry, geometry, statistics, calculus.

    • Helps organize patterns and regularities in the world.

Patterns in Nature and the World

  • Patterns: Regular, repeated forms in nature.

    • Examples include symmetries, spirals, waves, foams, and tessellations.

Symmetry

  • Ability to split an object into mirror images.

  • Types of Symmetry:

    1. Line or Bilateral Symmetry

    2. Rotational Symmetry

Spiral

  • A curve that moves away from a point as it revolves around it.

Wave

  • A propagating disturbance of one or more quantities in physics and engineering.

Foam

  • Material formed by trapping gas pockets in a liquid or solid.

Tessellation

  • Covering a surface with geometric shapes without overlaps or gaps.

Historical Context

  • Philosophers: Plato, Pythagoras studied patterns.

  • 19th Century: Joseph Plateau studied soap films; developed concept of minimal surfaces.

  • D'Arcy Thompson: Studied growth patterns in plants and animals.

The Fibonacci Sequence

  • A series of numbers where each number is the sum of the two preceding ones.

    • Starts at 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34…

  • Ratios of successive Fibonacci numbers approach the Golden Ratio, Φ ≈ 1.618.

  • Binet's Formula: $F_n = \frac{1}{\sqrt{5}} \left( \frac{1 + \sqrt{5}}{2} \right)^{n} - \frac{1}{\sqrt{5}} \left( \frac{1 - \sqrt{5}}{2} \right)^{n}$

Sequences in Mathematics

  • Types of Sequences:

    1. Arithmetic: Common difference between terms.

    2. Geometric: Common ratio between terms.

    3. Quadratic: Constant second difference between terms.

    4. Harmonic: Reciprocal of an arithmetic sequence.

    5. Mixed: Multiple operations in the rule.

    6. Square Number: Squares of integers.

    7. Cube Number: Cubes of integers.

Example Sequences

  • Identify rules and next terms in sequences, including:

    • 1, 10, 100, 1000

    • 2, 5, 9, 14, 20

    • 0, 1, 1, 2, 3, 5, 8

    • etc.

Applications of Mathematics

  • Organization: Helps organize patterns and regularities.

  • Prediction: Understanding natural behavior and phenomena.

  • Control: Enables human control of occurrences for advancement of civilization.