Math Q (1 Viewer)

axwe7 · Dec 22, 2015

Help needed with all 4 questions.

Thanks.

Edit 1: The first question is a bit hard to read, but it's 3+ 1/a1

leehuan · Dec 22, 2015

Most of this is calculator work I think.

SpiralFlex · Dec 22, 2015

$\\{\color{Red} 1.} \;$Notice that they kindly gave us$\; \pi \; $is approximately$\; 3 \;+ \;$(some fraction less than one). That is,$$

$\pi \approx 3 + \frac{1}{a_{1}}.$

$\\$To solve for$\; a_{1}\; $we simply subtract 3 from both sides and take the reciprocal.$$ .

$\pi - 3 = \frac{1}{a_{1}}$

$a_{1} = 7.0625....$ .

$\pi = 3 + \frac{1}{7.0625...} \;$

$\\$Well since$ \; 7.0625... \; $is$\; 7 \; + \; $(some fraction less than 1) we can write$\; a_{1} = 7.0625 = 7 + \frac{1}{a_{2}}$

$\\$Solve for$\; a_{2}\; $with the similar process as above, yields$$

$a_{2} = 15.99659....$

$\\$So$,\; 7.0625... \approx 7 + \frac{1}{15.99659...}$

$And, so,$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15.99659...}}\;$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15 + ...}}\;$

$\fbox{Can you now continue this process for say a few more steps and get another continued fraction approximation for $\pi$?}$

$\\$So putting this together and continuing the process we get,$$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15 +\frac{1}{1 + \frac{1}{292+...}}}}$

$\\$Which I will write using this notation,$$

$[3; 7, 15, 1, 292, ...]$

$\\{\color{Red} 2.} \;$It asks us to truncate the continued fraction at 7. The word truncate means we take the integer part of a number and discard the fractional part. So from below we chuck the [...] part away$$

$\pi \approx 3 + \frac{1}{7 + [....]} = 3 + \frac{1}{7}$ .

$\\$What we have done is taken the first convergent of the continued fraction.$$

$\\$We better give name to our previous notation, our continued fraction of$\; \pi \approx [3; 7, 15, 1, 292, ...],\; $we will call the numbers$$

$C_{1} = [3; 7] \; $the first convergent$$
$C_{2} = [3; 7, 15] \;$the second convergent$$
$C_{3} = [3; 7, 15, 1] \;$the third convergent and so on$$

$\\{\color{Red} 3.} \;$This question asks us for the 2nd convergent of the continued fraction$$

$\\{\color{Red} 4.} \;$This question asks us for the 3rd convergent of the continued fraction$$

$\\$Find those and you'll can see how much dp they differ by.$$

turntaker · Dec 22, 2015

Wow spiral.

Sent from my Nexus 4 using Tapatalk

axwe7 · Dec 22, 2015

SpiralFlex said:
$\\{\color{Red} 1.} \;$Notice that they kindly gave us$\; \pi \; $is approximately$\; 3 \;+ \;$(some fraction less than one). That is,$$

$\pi \approx 3 + \frac{1}{a_{1}}.$

$\\$To solve for$\; a_{1}\; $we simply subtract 3 from both sides and take the reciprocal.$$ .

$\pi - 3 = \frac{1}{a_{1}}$

$a_{1} = 7.0625....$ .

$\pi = 3 + \frac{1}{7.0625...} \;$

$\\$Well since$ \; 7.0625... \; $is$\; 7 \; + \; $(some fraction less than 1) we can write$\; a_{1} = 7.0625 = 7 + \frac{1}{a_{2}}$

$\\$Solve for$\; a_{2}\; $with the similar process as above, yields$$

$a_{2} = 15.99659....$

$\\$So$,\; 7.0625... \approx 7 + \frac{1}{15.99659...}$

$And, so,$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15.99659...}}\;$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15 + ...}}\;$

$\fbox{Can you now continue this process for say a few more steps and get another continued fraction approximation for $\pi$?}$

$\\$So putting this together and continuing the process we get,$$

$\pi \approx 3 + \frac{1}{7 + \frac{1}{15 +\frac{1}{1 + \frac{1}{292+...}}}}$

$\\$Which I will write using this notation,$$

$[3; 7, 15, 1, 292, ...]$

$\\{\color{Red} 2.} \;$It asks us to truncate the continued fraction at 7. The word truncate means we take the integer part of a number and discard the fractional part. So from below we chuck the [...] part away$$

$\pi \approx 3 + \frac{1}{7 + [....]} = 3 + \frac{1}{7}$ .

$\\$What we have done is taken the first convergent of the continued fraction.$$

$\\$We better give name to our previous notation, our continued fraction of$\; \pi \approx [3; 7, 15, 1, 292, ...],\; $we will call the numbers$$

$C_{1} = [3; 7] \; $the first convergent$$
$C_{2} = [3; 7, 15] \;$the second convergent$$
$C_{3} = [3; 7, 15, 1] \;$the third convergent and so on$$

$\\{\color{Red} 3.} \;$This question asks us for the 2nd convergent of the continued fraction$$

$\\{\color{Red} 4.} \;$This question asks us for the 3rd convergent of the continued fraction$$

$\\$Find those and you'll can see how much dp they differ by.$$

Wow, thanks so much!

Math Q (1 Viewer)

axwe7

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leehuan

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SpiralFlex

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turntaker

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axwe7

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