Continued fraction/Arithmetic: Difference between revisions

==[[Continued fraction arithmetic/Continued fraction r2cf%28Rational N%29]]==

During these tasks I shal use the function described in this task to create continued fractions from rational numbers.

==Matrix NG==

Consider a matrix NG:

: <math>\begin{bmatrix}

a_12 & a_1 & a_2 & a \\

b_12 & b_1 & b_2 & b

\end{bmatrix}

</math>

and a function G(matrix NG, Number N₁, Number N₂)

which returns:

: <math>\frac{a_12*N_1*N_2 + a_1*N_1 + a_2*N_2 + a}{b_12*N_1*N_2 + b_1*N_1 + a_2*N_2 + b}</math>

: Convince yourself that NG = :

: <math>\begin{bmatrix}

0 & 1 & 1 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}</math> adds N₁ to N₂

:<math>\begin{bmatrix}

0 & 1 & -1 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}</math> subtracts N₂ from N₁

: <math>\begin{bmatrix}

1 & 0 & 0 & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}</math> multiplies N₁ by N₂

: <math>\begin{bmatrix}

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0

\end{bmatrix}</math> divides N₁ by N₂

: <math>\begin{bmatrix}

21 & -15 & 28 & -20 \\

0 & 0 & 0 & 1

\end{bmatrix}</math> calculates (3*N₁ + 4) * (7*N₂ - 5)

:Note that with N₁ = 22, N₂ = 7, and NG = :

: <math>\begin{bmatrix}

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0

\end{bmatrix}</math>

:I could define the solution to be N₁ = 1, N₂ = 1 and NG = :

: <math>\begin{bmatrix}

0 & 0 & 0 & 22 \\

0 & 0 & 0 & 7

\end{bmatrix}</math>

:So I can define arithmetic as operations on this matrix which make a₁₂, a₁, a₂, b₁₂, b₁, b₂ zero and read the answer from a and b. This is more interesting when N₁ and N₂ are continued fractions, which is the subject of the following tasks.