4. Functional MATRICES

Complementary to the conventional Number Line, I have used these shapes for visualising decimal numbers:

1. Decimal Pyramids that create
   • a left Diagonal of Squares and a right Diagonal of Squares – 1;
2. Digital Tables that separate
   • Leading from Last Digits
   • and create spaces for “Vertical Prime Pairs” of Primes and SemiPrimes;
3. Symmetrical Number Planes that reflect
   • the symmetries of complex numbers
   • and thus create the conceptual numerical link on 2D paper or screens to the physical metric world in 3D space;
4. Functional Matrices to produce:-
   • Diagonal Sums
     • creating diagonally symmetrical sums_ij and sums_ji by adding summands from row_i to summands in column_j;
   • Orthogonal Products
     • creating orthogonally symmetrical products_ij by multiplying factors from row_i with factors from column_j;
   • Products of SemiPrimes
     • creating orthogonally symmetrical matrices by multiplying Prime1 in row_i by Prime2 in column_j as SemiPrimes_ji and SemiPrimes_ij above and below the Top Left to Bottom Right Diagonal;
   • Factorial Tables of SemiPrimes
     • filling the gaps in the Digital Tables of Primes with SemiPrimes is achieved by following the series of results creating from multiplying Prime Factors_i with Prime Factors_j:

The matrices offer the full range of possible values from 1×1 to 109×109.

Next, we pick the SERIES of SemiPrimes, produced by the Prime Factors 7, 11 and 13:

And finally, altogether:

NEXT: Diagonal Sums, Orthogonal Products, Matrix of SemiPrimes and Tables of Prime Factors