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anonymous
 one year ago
find four successive @rational approximations for root 3,each of them accurate to within 10^4 of the true value of the surd chosen. use the easiest starting point you can find.Use continued fractions, please explain every step and show all working.
Then use the method to find one single approximation to a larger surd such as root 91
anonymous
 one year ago
find four successive @rational approximations for root 3,each of them accurate to within 10^4 of the true value of the surd chosen. use the easiest starting point you can find.Use continued fractions, please explain every step and show all working. Then use the method to find one single approximation to a larger surd such as root 91

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jagr2713
 one year ago
Best ResponseYou've already chosen the best response.1Hey @Maretch hold up, we will find someone to help you ASAP :D

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.1Hes a great helper :D

TheSmartOne
 one year ago
Best ResponseYou've already chosen the best response.2@dan815 @nincompoop @jigglypuff314 @Michele_Laino @Hero @kirbykirby @iambatman A QH question for you QH's :P

TheSmartOne
 one year ago
Best ResponseYou've already chosen the best response.2Although none of the Qualified helpers are on, maybe @mathmate @freckles @Loser66 @ikram002p could help you :)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thanks, really appreciate you guys helping.

TheSmartOne
 one year ago
Best ResponseYou've already chosen the best response.2Unfortunately, today is a very quiet day on OpenStudy. Normally there would be a lot more people. Don't know what happened.

TheSmartOne
 one year ago
Best ResponseYou've already chosen the best response.2Aha, I spot a genius who just came online right now. @SithsAndGiggles could you help this user? (:

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7@Maretch Did you check the response of your other identical post?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah but it was through newtons method

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7@Maretch Have you learned how to do continued fractions for any number, or mainly for squareroots? I am not talking about the other answer. My question was: @Maretch Have you learned how to do continued fractions for any number, or mainly for squareroots?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0never learned newtons method and the title of the entire page is Continued fraction, so im assuming

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The thing is never learned, continued fractions

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7@Maretch I AM talking about continued fractions, NOT Newtons. I am not the "other" guy talking about Newtons.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i know absolutely nothing, i know how to convert continued fractions to a proper fractions, thats about it

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0this is why im having so much trouble with it, never even mentioned continued fractions in class

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7What course are you taking? Alg 2 or number theory?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Im australian, so its different

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0http://en.wikipedia.org/wiki/Mathematics_education_in_Australia

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and look at queensland

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Can you tell me what grade you're in, or the name of the course, so I can understand your teacher's expectations.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Im year 11, age 16, doing maths c assignment

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7So are you in form 5 or form 6?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Not sure, never heard about that

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7So form 5, which of the 8 courses?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0atm im doing vectors, matrices

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sorry, i dont understand most of that stuff, australia is alot different

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7lol, you sound like you're not from Australia, but studying there. But that's beside the point.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7So you need to find square root of 3 with continued fractions as an answer, right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeah, find approximations of root 3 with continued fractions, answer should be a fraction, that is accurate to 10^4 of the true value

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so like for root 2 an example of the answer would be like 17/12 or 41/29

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7And you have not learned HOW to find a continued fraction approximation, am I right, or you just don't remember how?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0even asked my classmates

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7The reason I am asking is there are different ways to approach the problem.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7If that's clear, we can use different approaches.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I will show you how to find sqrt(3) by continued fractions, but it involves a little work and concentration on your part. Are you ready for that?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Finding continued fraction approximations is a process called iteration, that means we get closer at each step, and probably never get the exact answer. We will stop when we have an accurate enough answer, or have found the rule.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ye, as the continued fraction grows, the answer gets more accurate and accurate

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Exactly, you get the idea.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Do you know how to find the first approximation?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7That is the integer part of the fraction.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i can convert the continued fraction to a proper or improper fraction, but i dont actually know how to setup the continued fraction

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0like i have no idea where they get the 1,1,2,1,2 etc

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Yes, that is understood. We are trying to solve a squareroot problem. Can you tell me the squareroot approximately equals what?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0square root equals 5.099, is that what you meant?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Very well. Now we have to introduce a concept of the "floor" function.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.726 because I'll use it to find the floor function of sqrt(26).

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7The floor function means the largest INTEGER that does not exceed a given number.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7For example, floor(sqrt(26)) = floor(5.099) = 5, the answer is always an integer.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Another example, floor(5) = 5, because 5 is an integer that does not exceed 5

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok, so like round to the nearest interger?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We'll find out! Can you tell me what is floor(3.3)?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so floor is rounding down and ceiling is rounding up?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Very good question, actually floor is ALWAYS rounding down.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok so floor of 3,3 is 3

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Exactly! how about floor 1.7

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7very good, a tough one here, floor(2.3)

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Very, very good! floor always round to a smaller number, not just dropping the decimal part!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7So 3 is smaller than 2.3, so floor(2.3)=3. All clear?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We're going to work on an algorithm. Do you know what it means?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what algorithym, like just a formula to get an answer, idk how to explain

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0formula to solve a problem

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Yes, a formula, but we have to use the formula many times to get the answer.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Most of the time, it is easier to work out the formula using a table to organize our calculations.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I'll start a table, and we will work on it together, ok?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7give me a minute to plan the table, please.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0btw is this question really difficult?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0or just really abstract, is that why not many people can help

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7It's not difficult, but 1. it belongs to number theory, and is not generally learned in elementary algebra courses. 2. it takes time to explain, especially if you have not done it before. 3. it's probably too early for the experts to come here, they usually work at night, night owls, you understand? lol

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7ok, I am going to draw a table, with lots of of blank spaces and notations. Don't be scared by it. We'll go through the steps.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Okay, this will eventually relate back to the question right, this is just things i need to know before solving right?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7dw:1433947126989:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7No, we will solve your problem together, sqrt(3), like an example, but it will be your problem.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so why am i learning this, if it doesnt relate.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7dw:1433947374403:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Because that is how you find the continued fraction for sqrt(3).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay, sorry, continue

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7dw:1433947464984:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7n represents the step number. Step 0 is already done for us! a0 means the a column on the first line (n=0) so m0=0 d0=1 a0=floor(sqrt(3))=floor(1.7320508...)=1 Are we good so far?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7dw:1433947582877:dw We will be using A0 later on.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so what does mn, dn and an stand for?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I don't really know, except A0, A1,A2... stand fordw:1433947729401:dw The others are just intermediate answers.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Now we need formulas to calculate M1,D1 and A1, which lie on the second line (where n=1), ok

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We will use the same formulas for every line, so instead of writing one set of formulas for each line, I will write it for line "n". So put n=1 for the second line, 2 for the third, etc, alright?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7ok, Mn+1 = DnAnMn Dn+1=(S(Mn+1)^2)/Dn An+1 = floor((A0+Mn+1)/Dn+1)

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We'll work them out one by one!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We'll first calculate Mn+1=M1. This means n=0, or the previous line, right?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7When n+1=1, n=0, is what I mean. So translating Mn+1=DnAnMN into M1=D0A0M0=1*10 Can you calculate M1 and put it in the table?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Yes, can you put it in the M1 cell of the table?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0trying to figure it out dw:1433948402569:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is that right, sorry i dont understand how to draw and stuff on this site.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Now we're going to work on D1, and recall that S = 3 the squareroot we of which we want. This translates to D1=(S  (M1)^2) / d0 = (31^2)/1 = ? Can you work that out,

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Right, can you now put it in the table?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433948679901:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Very well, we'll now calculate A1. If we substitute n=0, we have A1=floor( (A0+M1)/D1 ) can you fill select the numbers and calculate A1 for me?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7* can you substitute the numbers and calculate A1 for me?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433948843806:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Now we're going to calculate the second row, which is 2=n+1, so n=1, fair enough?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I mean third row, where n=2!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7See if you can post the values of M2, D2 and A2 before filling the table.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I don't have the same answers, let's check!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7for n+1=2, then n=1, so M2=D1A1M1 = 2*11 =1 right?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7You'll need to redo D2 and A2 because they depend on M2.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433949270224:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Yep, exactly what I've got too! Excellent!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Just for fun, we'll digress a little. What if we make up the continued fraction and check it's accuracy!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Can you do that using A0, A1 and A2?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433949392760:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ima horrible drawer.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7No problem, I am worse! Can you calculate the value?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Yes, except you cheated a little in assuming that A3 is 1, which it is!

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7So 7/4=1.75 is already quite close to 1.7320508...

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Isn't that encouraging?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Would you continue with the next row, 3=n+1, so use n=2?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We'll do at least two more rows, and we'll see why.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0sorry, havent we already dont n2

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7we're to do row n=3, but to fit in the formulas, n+1=3, so n=2. Yes All the M2,D2 and A2 are known, so we're calculating M3, but we call M3=Mn+1.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7In the formula, n+1 = new row, n=previous row.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433950128652:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7I don't have the same numbers for n=3.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Let's check: I have M3=1, so that's good. D3=(S(M3)^2)/D2=(31^2)/1=2

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Then A3=floor(A0+M3)/D3 = floor(1+1)/2=1

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Good, can we dw:1433950463947:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Can you do one more row and put it at the bottom of the table?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433950567103:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7ok, let's check, again I don't have the same numbers. M4=D3A3M3=2*11=1 Good D4=(S(M4)^2)/D3 = (31^2)/2 = 2/2 =1 (see if you agree)

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7A4=floor( (A0+M4)/(D4) ) = floor( (1+1)/1 ) =floor(2) = 2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1433951105030:dw

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Can you put in the A0, A1, A2....A4 and see what the continued fraction gives you so far?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7You probably got a numerical error somewhere. Depending on the last term, you should get either 45/26 or 26/15. Most probably 45/26 is what you would have got. Anyway, that is 1.730769... which is quite close to 1.7320508...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yeh i accidently multiplied the wrong thing

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7At this point, we don't have to do any more rows if you examine the table to look for patterns of _rows_.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7We have, from row 1: 1 2 1 1 1 2 1 2 1 1 1 2 So we should expect the next rows to have the same patterns because each row is derived from the previous.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7Sorry, OS was down the last while. We can now wrap up the problem.

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7@Maretch Since we see the pattern, what do you think the sequence of A would be? We have already calculated 1,1,2,1,2... How do you think it will continue?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.01,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, infinitely?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7There you go, so you can complete the problem by evaluating to the accuracy you want! Good job!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0how do i know which stage of the continued fraction will be within 10^4

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why do the rational approximations approach the surd in an alternating way, like over and under?

mathmate
 one year ago
Best ResponseYou've already chosen the best response.7@Maretch 1. Method 1 You would compare with the actual answer of 1.732050807568877 and calculate the absolute value of the difference. If the difference is less than 10^4, you have reached the answer. If not, you will have to continue with a longer chain of fractions. 2. Method 2 You can calculate the difference between the value of sqrt(3) using n terms and n+1 terms of An. If the difference is less than 10^4, there is a good chance that your answer will be within the given tolerance.
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