Continued fraction/Arithmetic: Difference between revisions
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so each may be described by the notation [<math>a_0 ; a_1, a_2, ..., a_n</math>] |
so each may be described by the notation [<math>a_0 ; a_1, a_2, ..., a_n</math>] |
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⚫ | |||
Subtasks: |
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During these tasks I shal use the function described in this task to create continued fractions from rational numbers. |
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⚫ | |||
==Matrix NG== |
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* Investigate matrix NG |
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Consider a matrix NG: |
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: <math>\begin{bmatrix} |
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a_12 & a_1 & a_2 & a \\ |
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b_12 & b_1 & b_2 & b |
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\end{bmatrix} |
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</math> |
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and a function G(matrix NG, Number N<sub>1</sub>, Number N<sub>2</sub>) |
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which returns: |
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: <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> |
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: Convince yourself that NG = : |
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: <math>\begin{bmatrix} |
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0 & 1 & 1 & 0 \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix}</math> adds N<sub>1</sub> to N<sub>2</sub> |
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:<math>\begin{bmatrix} |
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0 & 1 & -1 & 0 \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix}</math> subtracts N<sub>2</sub> from N<sub>1</sub> |
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: <math>\begin{bmatrix} |
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1 & 0 & 0 & 0 \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix}</math> multiplies N<sub>1</sub> by N<sub>2</sub> |
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: <math>\begin{bmatrix} |
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0 & 1 & 0 & 0 \\ |
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0 & 0 & 1 & 0 |
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\end{bmatrix}</math> divides N<sub>1</sub> by N<sub>2</sub> |
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: <math>\begin{bmatrix} |
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21 & -15 & 28 & -20 \\ |
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0 & 0 & 0 & 1 |
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\end{bmatrix}</math> calculates (3*N<sub>1</sub> + 4) * (7*N<sub>2</sub> - 5) |
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:Note that with N<sub>1</sub> = 22, N<sub>2</sub> = 7, and NG = : |
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: <math>\begin{bmatrix} |
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0 & 1 & 0 & 0 \\ |
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0 & 0 & 1 & 0 |
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\end{bmatrix}</math> |
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:I could define the solution to be N<sub>1</sub> = 1, N<sub>2</sub> = 1 and NG = : |
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: <math>\begin{bmatrix} |
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0 & 0 & 0 & 22 \\ |
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0 & 0 & 0 & 7 |
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\end{bmatrix}</math> |
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:So I can define arithmetic as operations on this matrix which make a<sub>12</sub>, a<sub>1</sub>, a<sub>2</sub>, b<sub>12</sub>, b<sub>1</sub>, b<sub>2</sub> zero and read the answer from a and b. This is more interesting when N<sub>1</sub> and N<sub>2</sub> are continued fractions, which is the subject of the following tasks. |
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* Investigate G(matrix NG, Contined Fraction N) |
* Investigate G(matrix NG, Contined Fraction N) |
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* The complete solution G(matrix NG, Continued Fraction N<sub>1</sub>, Continued Fraction N<sub>2</sub>) |
* The complete solution G(matrix NG, Continued Fraction N<sub>1</sub>, Continued Fraction N<sub>2</sub>) |
Revision as of 11:29, 6 February 2013
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:
so each may be described by the notation []
Continued fraction arithmetic/Continued fraction r2cf(Rational N)
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:
and a function G(matrix NG, Number N1, Number N2) which returns:
- Convince yourself that NG = :
- adds N1 to N2
- subtracts N2 from N1
- multiplies N1 by N2
- divides N1 by N2
- calculates (3*N1 + 4) * (7*N2 - 5)
- Note that with N1 = 22, N2 = 7, and NG = :
- I could define the solution to be N1 = 1, N2 = 1 and NG = :
- 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