Basics

 

  • [Of the five Platonic solids]
    So their combinations with themselves and with each other give rise to endless complexities, which anyone who is to give a likely account of reality must survey. (Plato)
  • There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible. (Leibniz)
  • There is geometry in the humming of the strings, there is music in the spacing of the spheres. (Pythagoras)

You find here the Basics of DigiMat in short codes starting with (i) the construction of the natural numbers leading up to (ii) Calculus as the mathematics of change as the foundation of both the industrial and information society:

  1. Construction/representation of natural numbers in binary form by repetition of the basic operation of +1.
  2. Construction/representation of rational numbers in binary form.
  3. Computation with natural and rational numbers in binary form.
  4. Computation with rational numbers in decimal form.
  5. Coordinate systems (line, plane, screen).
  6. Position and motion of objects (along line, in plane, in 3d space, on screen) according to time stepping x = x + v*dt  or dx = v*dt solving dx/dt = v.
  7. Calculus as time stepping dx = f(x,t)*dt solving dx/dt = f(x,t).

Read Why Binary Numbers and recall DigiMat Pre.

Go though Basic Mathematics step by step supported by Basic Programming and use Model Workshop and Game Workshop to get perspective and inspiration.

The list of codes serves as your guide starting from very short simple codes leading you safely into a digital world of mathematics + programming under the motto: Follow the code!

When you have done that and understood what you have done, then you can tell your math teacher, or Mom and Dad and the World, that you now master the essence of school mathematics and introductory university mathematics. Thus get started!

(Sometimes restart is needed to activate mouse interaction).

Basic Mathematics

See code to read explanation if not given explicitly. Run code by clicking arrow.

You can skip 2 – 15 about (binary) numbers and get directly into describing the World starting in 19 – 32, and return later to constructing the numbers by repetition of  +1.

  1. Constructing the Natural Numbers  see Creation  compare Worldometer
  2. Binary Representation of Natural Numbers
  3. Binary Numbers: Illustration1  +  Illustration2
  4. Reading Binary Representation
  5. Binary Addition. Compare with 2048 Game.
  6. Binary Abacus1
  7. Binary Abacus2
  8. Abacus Adding Machine
  9. Binary Scale
  10. Binary Multiplication
  11. Binary Subtraction
  12. Natural Numbers: Base 3
  13. Natural Numbers: Any Base
  14. Binary Division
  15. Pocket Calculator
  16. Construction of Rational Numbers
  17. Primes
  18. Fibonacci Numbers
  19. Screen Geometry
  20. Motion-Change: x = x + v*dt
  21. Motion on Screen   Rabbit Motion
  22. Draw Line
  23. Newtonian Mechanics: Angry Birds Basic  (with image)
  24. Newtonian Mechanics
  25. Polynomials
  26. Exponential Function exp(t)
  27. Natural Logarithm log(t)
  28. Harmonic Series
  29. Trigonometric Functions cos(t) and sin(t) (Lissajous Figures) (Variant 2d) (Variant 3d)
  30. Polar Coordinates
  31. Complex Numbers
  32. Draw Circle
  33. Solving f(x)=0 by Bisection
  34. Solving f(x)=0 by Time Stepping x = x + f(x)*dt
  35. Solving x=g(x) by Fixed Point Iteration
  36. Solving f(x) = 0 by Newton’s Method
  37. Calculus: Compute x = integral f(t)dt by Time Stepping dx=f(t)*dt
  38. Time Stepping: Smart, Dumb and Midpoint Euler
  39. Integral: Midpoint Euler vs Forward/Endpoint Euler
  40. Compute Area of Circular Disk. Compute Pi. (check with Archimedes)
  41. Double Integral (3d plot) ()
  42. Level Curves in 2d
  43. L2 Projection on Continuous Piecewise Linears
  44. Piecewise Linear Approximation/Interpolation see Illustration
  45. Finite Element Method FEM 1d  (Compare with DigiMat Pro)
  46. FEM Error Estimation by Duality 1d
  47. FEM 2d -Laplace u = f = 1 (with f = delta function)
  48. FEM Error Estimation by Duality 2d
  49. Divergence Theorem Square
  50. Divergence Theorem Disc   (Newtonian Gravitation 2d)
  51. Stokes Theorem Disc
  52. Curve Integral 1
  53. Curve Integral 2
  54. rot(q)=0 iff q=grad(u), u scalar potential
  55. Weierstrass Function
  56. Taylor Polynomial Approximation

Basic Programming

We will use the p5.js JavaScript Library which comes with a very handy web editor. The p5.js-codes you will meet are short and simple and can easily be transferred to a programming language of your choice. Here you find an intro to JavaScript

You can also consult according to need and inspiration

To save your own p5.js codes, set up an account at p5.js JavaScript.

It is easy/fun to insert images of your choice.

Now’s The Time for DigiMat:

 

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