Continued fraction/Arithmetic

By popular demand, see Talk:Continued fraction#creating_a_continued_fraction and Talk:Continued fraction#Arithmetics.3F.3F, or be careful what you ask for.

This page is a placeholder for several subtasks which will eventually implement a function:

G(matrix NG, Continued Fraction N₁, Continued Fraction N₂)

which will perform basic mathmatical operations on continued fractions.

For these tasks continued fractions will be of the form:

$a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+\ddots }}}}}}$

so each may be described by the notation [$a_{0};a_{1},a_{2},...,a_{n}$]

Continued fraction arithmetic/Continued fraction r2cf(Rational N)

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

Matrix NG

Consider a matrix NG:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$

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

${\frac {a_{12}*N_{1}*N_{2}+a_{1}*N_{1}+a_{2}*N_{2}+a}{b_{12}*N_{1}*N_{2}+b_{1}*N_{1}+a_{2}*N_{2}+b}}$

Convince yourself that NG = :

${\begin{bmatrix}0&1&1&0\\0&0&0&1\end{bmatrix}}$ adds N₁ to N₂

${\begin{bmatrix}0&1&-1&0\\0&0&0&1\end{bmatrix}}$ subtracts N₂ from N₁

${\begin{bmatrix}1&0&0&0\\0&0&0&1\end{bmatrix}}$ multiplies N₁ by N₂

${\begin{bmatrix}0&1&0&0\\0&0&1&0\end{bmatrix}}$ divides N₁ by N₂

${\begin{bmatrix}21&-15&28&-20\\0&0&0&1\end{bmatrix}}$ calculates (3*N₁ + 4) * (7*N₂ - 5)

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

${\begin{bmatrix}0&1&0&0\\0&0&1&0\end{bmatrix}}$

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

${\begin{bmatrix}0&0&0&22\\0&0&0&7\end{bmatrix}}$

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.

• Investigate G(matrix NG, Contined Fraction N)
• The complete solution G(matrix NG, Continued Fraction N₁, Continued Fraction N₂)
• Compare two continued fractions