- [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:
- Construction/representation of natural numbers in binary form by repetition of the basic operation of +1.
- Construction/representation of rational numbers in binary form.
- Computation with natural and rational numbers in binary form.
- Computation with rational numbers in decimal form.
- Coordinate systems (line, plane, screen).
- 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.
- Calculus as time stepping dx = f(x,t)*dt solving dx/dt = f(x,t).
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).
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.
- Constructing the Natural Numbers see Creation compare Worldometer
- Binary Representation of Natural Numbers
- Binary Numbers: Illustration1 + Illustration2
- Reading Binary Representation
- Binary Addition. Compare with 2048 Game.
- Binary Abacus1
- Binary Abacus2
- Abacus Adding Machine
- Binary Scale
- Binary Multiplication
- Binary Subtraction
- Natural Numbers: Base 3
- Natural Numbers: Any Base
- Binary Division
- Pocket Calculator
- Construction of Rational Numbers
- Fibonacci Numbers
- Screen Geometry
- Motion-Change: x = x + v*dt
- Motion on Screen Rabbit Motion
- Draw Line
- Newtonian Mechanics: Angry Birds Basic (with image)
- Newtonian Mechanics
- Exponential Function exp(t)
- Natural Logarithm log(t)
- Harmonic Series
- Trigonometric Functions cos(t) and sin(t) (Lissajous Figures) (Variant 2d) (Variant 3d)
- Polar Coordinates
- Complex Numbers
- Draw Circle
- Solving f(x)=0 by Bisection
- Solving f(x)=0 by Time Stepping x = x + f(x)*dt
- Solving x=g(x) by Fixed Point Iteration
- Solving f(x) = 0 by Newton’s Method
- Calculus: Compute x = integral f(t)dt by Time Stepping dx=f(t)*dt
- Time Stepping: Smart, Dumb and Midpoint Euler
- Integral: Midpoint Euler vs Forward/Endpoint Euler
- Compute Area of Circular Disk. Compute Pi. (check with Archimedes)
- Double Integral (3d plot) ()
- Level Curves in 2d
- L2 Projection on Continuous Piecewise Linears
- Piecewise Linear Approximation/Interpolation see Illustration
- Finite Element Method FEM 1d (Compare with DigiMat Pro)
- FEM Error Estimation by Duality 1d
- FEM 2d -Laplace u = f = 1 (with f = delta function)
- FEM Error Estimation by Duality 2d
- Divergence Theorem Square
- Divergence Theorem Disc (Newtonian Gravitation 2d)
- Stokes Theorem Disc
- Curve Integral 1
- Curve Integral 2
- rot(q)=0 iff q=grad(u), u scalar potential
- Weierstrass Function
- Taylor Polynomial Approximation
You can also consult according to need and inspiration
Now’s The Time for DigiMat: