convert a continued fraction into a normal fraction

[enoimreh7]

Hello!

Could you please help me converting the following continued fraction into a normal fraction, please? My calculator isn't goog enough, and I don't know how to solve it without.
I don't know how to write it like a fraction, so I write it like a term:

1+12+12+12+12+12+12+1:2))))))

I tried to do it "manual": 2+1:2=2.5 ; 2+1:2.5=2.4 ; 2+1:2.4=2.4166... ; and so on... But then, after a time, my calculator says it would be 0, but that's definitely wrong.
And then, I have to salve that term, too, but with my calculator, it's impossible:

(

-(1+12+12+12+12+12+12+1:2)))))))):

·100

Could you help me with this one, too, please?
Thank you very much!

Yours,
Enoimreh​

​

​

 

[Dr.Peterson]

Could you please help me converting the following continued fraction into a normal fraction, please? My calculator isn't good enough, and I don't know how to solve it without.

I don't know how to write it like a fraction, so I write it like a term:

1+1: (2+1: (2+1: (2+1: (2+1: (2+1: (2+1:2))))))

I tried to do it "manual": 2+1:2=2.5 ; 2+1:2.5=2.4 ; 2+1:2.4=2.4166... ; and so on... But then, after a time, my calculator says it would be 0, but that's definitely wrong.
And then, I have to salve that term, too, but with my calculator, it's impossible:

(

-(1+1: (2+1: (2+1: (2+1: (2+1: (2+1: (2+1:2)))))))):

·100

Could you help me with this one, too, please?​

​

​

Let's start with just the first.

I would do it manually, using fractions. Start, as you did, in the middle:

2 + 1/2 = 4/2 + 1/2 = 5/2

2 + 1/(5/2) = 2 + 2/5 = 10/5 + 2/5 = 12/5

and so on.

Can you finish? Note that the process amounts to alternately taking a reciprocal, then adding 2. You can do it easily on a calculator this way, especially if it has a reciprocal button.

 

[enoimreh7]

Hello Dr. Peterson!​

I am very surprised that my calculator worked this time. Before he has always said that there would have been an error. Now he brought out this:

2+12+12+12+12+1:2))))=169:70
2+1169:70)=408:169
1+1:(408:169)=577:408

(

-577:408):

x 100 =-1,50182509x10^-4

Is that all right?​

Thank you very much for enspireing me to try it again.

Yours,
Enoimreh​

​

​

 

[Dr.Peterson]

Hello Dr. Peterson!​

I am very surprised that my calculator worked this time. Before he has always said that there would have been an error. Now he brought out this:

2+1: (2+1: (2+1: (2+1: (2+1:2))))=169:70
2+1: (169:70)=408:169
1+1: (408:169)=577:408

(

-577:408):

x 100 =-1,50182509x10^-4

Is that all right?​

Thank you very much for enspireing me to try it again.
​

​

​

Yes, that is correct. I guessed that you made some little error in the long calculation, and either redoing it, or using a simpler approach, would get you past that.

The result shows, of course, that this continued fraction is a very good approximation to the square root of 2. The decimal value of 577/408 is 1.414215686..., while sqrt(2) is 1.414213562... .

Incidentally, I did my calculations even faster, using a spreadsheet to do each repetition for me.

 

[enoimreh7]

Thank you

Dear Dr. Peterson!

Thank you very much!

Yours,
Enoimreh