Hello, beta12!
Now that LaTeX is back, I'll revise this post . . .
Let
Prove that: .
Simplify that awful equation: .
We have: .
Quadratic Formula: .
Since is positive: .
Let a = sqrt(3) -1. prove that
a = 1/(1 + 1/(2 + a) )
Use this to find the continued fraction expansion for a.
Deduce the continued fraction expansion for sqrt(3).
Check that your answer makes sense - that is , use the first 6 or 7 terms of the continued fraction expansion to give an approximation for sqrt(3) and make sure that this approximation is reasonable.
Can you teach me how to solve this question? Thank you very much.
Look at Soroban's Post
We know that,
a=[0;1,2,a]
Intuitively you can substitute "a" in the brackets for "a":
a=[0;1,2,1,2,a]
Again and again,
a=[0;1,2,1,2,1,2,1,2...]
Periodical expansion.
You can quickly evaluate the convergents of this continued fraction using the following recusion relations (I am sure you know them):
p_k=a_k*p_{k-1}+p_{k-2}
q_k=a_k*q_{k-1}+q_{k-2}
Where,
p_0=0
q_0=1
p_1=1
q_1=1
Thus, use those equations above to get,
p_2=(2)(1)+0=2
q_2=(2)(1)+1=3
p_3=(1)(2)+1=3
q_3=(1)(3)+1=4
p_4=(2)(3)+2=8
q_4=(2)(4)+3=11
p_5=(1)(8)+3=11
q_5=(1)(11)+4=15
p_6=(2)(11)+8=30
q_6=(2)(15)+11=41
Thus, the fraction,
30/41 Is the best possible approximation of that number.
The error is less than 1/(q_6)^2
Hi Perfecthacker,
Ok.
======================
Since,
a=[0:1,2,a]
Substitute the "a" into the "a" in the fraction,
a=[0,1,2,1,2,a]
Do that again,
a=[0,1,2,1,2,1,2,a]
See the pattern?
Yes. I got this part.
This should be belong to my question " use this to find the continued fraction expansion for a"
=====================
You can quickly evaluate the convergents of this continued fraction using the following recusion relations (I am sure you know them):
p_k=a_k*p_{k-1}+p_{k-2}
q_k=a_k*q_{k-1}+q_{k-2}
Where,
p_0=0
q_0=1
p_1=1
q_1=1
Thus, use those equations above to get,
p_2=(2)(1)+0=2
q_2=(2)(1)+1=3
p_3=(1)(2)+1=3
q_3=(1)(3)+1=4
p_4=(2)(3)+2=8
q_4=(2)(4)+3=11
p_5=(1)(8)+3=11
q_5=(1)(11)+4=15
p_6=(2)(11)+8=30
q_6=(2)(15)+11=41
Thus, the fraction,
30/41 Is the best possible approximation of that number.
Is this part answering my question " Deduce the continued fraction expansion for sqrt(3)?