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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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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Hey @Maretch hold up, we will find someone to help you ASAP :D
Thanks
http://prntscr.com/7fc4ei

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Other answers:

Why? whats that do?
try it
there you go! :-)
Thanks
Hes a great helper :D
Although none of the Qualified helpers are on, maybe @mathmate @freckles @Loser66 @ikram002p could help you :)
Thanks, really appreciate you guys helping.
Unfortunately, today is a very quiet day on OpenStudy. Normally there would be a lot more people. Don't know what happened.
Aha, I spot a genius who just came online right now. @SithsAndGiggles could you help this user? (:
@Maretch Did you check the response of your other identical post?
yeah but it was through newtons method
@Maretch Have you learned how to do continued fractions for any number, or mainly for square-roots? 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 square-roots?
never learned newtons method and the title of the entire page is Continued fraction, so im assuming
The thing is never learned, continued fractions
@Maretch I AM talking about continued fractions, NOT Newtons. I am not the "other" guy talking about Newtons.
i know absolutely nothing, i know how to convert continued fractions to a proper fractions, thats about it
this is why im having so much trouble with it, never even mentioned continued fractions in class
What course are you taking? Alg 2 or number theory?
Im australian, so its different
http://en.wikipedia.org/wiki/Mathematics_education_in_Australia
and look at queensland
Can you tell me what grade you're in, or the name of the course, so I can understand your teacher's expectations.
Im year 11, age 16, doing maths c assignment
So are you in form 5 or form 6?
Not sure, never heard about that
So form 5, which of the 8 courses?
atm im doing vectors, matrices
ok, that's good.
sorry, i dont understand most of that stuff, australia is alot different
lol, you sound like you're not from Australia, but studying there. But that's beside the point.
So you need to find square root of 3 with continued fractions as an answer, right?
yeah, find approximations of root 3 with continued fractions, answer should be a fraction, that is accurate to 10^-4 of the true value
so like for root 2 an example of the answer would be like 17/12 or 41/29
And you have not learned HOW to find a continued fraction approximation, am I right, or you just don't remember how?
Have not learnt it
even asked my classmates
The reason I am asking is there are different ways to approach the problem.
If that's clear, we can use different approaches.
kk
I 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?
yeah im ready
Finding 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.
ok so far?
Ye, as the continued fraction grows, the answer gets more accurate and accurate
Exactly, you get the idea.
Do you know how to find the first approximation?
That is the integer part of the fraction.
i dont.
i can convert the continued fraction to a proper or improper fraction, but i dont actually know how to setup the continued fraction
like i have no idea where they get the 1,1,2,1,2 etc
Yes, that is understood. We are trying to solve a square-root problem. Can you tell me the square-root approximately equals what?
* square-root of 26
Huh?
square root equals 5.099, is that what you meant?
26*
yes, exactly!
Why 26?
Very well. Now we have to introduce a concept of the "floor" function.
26 because I'll use it to find the floor function of sqrt(26).
The floor function means the largest INTEGER that does not exceed a given number.
For example, floor(sqrt(26)) = floor(5.099) = 5, the answer is always an integer.
Another example, floor(5) = 5, because 5 is an integer that does not exceed 5
Ok, so like round to the nearest interger?
or am i wrong
We'll find out! Can you tell me what is floor(3.3)?
so floor is rounding down and ceiling is rounding up?
Very good question, actually floor is ALWAYS rounding down.
Ok so floor of 3,3 is 3
Exactly! how about floor 1.7
1
very good, a tough one here, floor(-2.3)
-3
Very, very good! floor always round to a smaller number, not just dropping the decimal part!
So -3 is smaller than -2.3, so floor(-2.3)=-3. All clear?
Yep, i understand
We're going to work on an algorithm. Do you know what it means?
what algorithym, like just a formula to get an answer, idk how to explain
formula to solve a problem
Yes, a formula, but we have to use the formula many times to get the answer.
Oh ok
Most of the time, it is easier to work out the formula using a table to organize our calculations.
Okay
I'll start a table, and we will work on it together, ok?
Ok
give me a minute to plan the table, please.
btw is this question really difficult?
or just really abstract, is that why not many people can help
It'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
Ahh okay
ok, 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.
Okay, this will eventually relate back to the question right, this is just things i need to know before solving right?
|dw:1433947126989:dw|
No, we will solve your problem together, sqrt(3), like an example, but it will be your problem.
Okay
so why am i learning this, if it doesnt relate.
|dw:1433947374403:dw|
Because that is how you find the continued fraction for sqrt(3).
okay, sorry, continue
|dw:1433947464984:dw|
n 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?
Yep
|dw:1433947582877:dw| We will be using A0 later on.
ok so far?
so what does mn, dn and an stand for?
I don't really know, except A0, A1,A2... stand for|dw:1433947729401:dw| The others are just intermediate answers.
kk
Now we need formulas to calculate M1,D1 and A1, which lie on the second line (where n=1), ok
kk
We 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?
Yeah
ok, Mn+1 = DnAn-Mn Dn+1=(S-(Mn+1)^2)/Dn An+1 = floor((A0+Mn+1)/Dn+1)
We'll work them out one by one!
Okay
We'll first calculate Mn+1=M1. This means n=0, or the previous line, right?
Yeah
When n+1=1, n=0, is what I mean. So translating Mn+1=DnAn-MN into M1=D0A0-M0=1*1-0 Can you calculate M1 and put it in the table?
its 1 right?
Yes, can you put it in the M1 cell of the table?
trying to figure it out |dw:1433948402569:dw|
Excellent!
is that right, sorry i dont understand how to draw and stuff on this site.
Now we're going to work on D1, and recall that S = 3 the square-root we of which we want. This translates to D1=(S - (M1)^2) / d0 = (3-1^2)/1 = ? Can you work that out,
2?
Right, can you now put it in the table?
|dw:1433948679901:dw|
Very 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?
* can you substitute the numbers and calculate A1 for me?
|dw:1433948843806:dw|
Perfect!!!!!
Now we're going to calculate the second row, which is 2=n+1, so n=1, fair enough?
I mean third row, where n=2!
Kk
See if you can post the values of M2, D2 and A2 before filling the table.
m2= 2 d2=2 a2=2?
I don't have the same answers, let's check!
for n+1=2, then n=1, so M2=D1A1-M1 = 2*1-1 =1 right?
yeah
You'll need to redo D2 and A2 because they depend on M2.
|dw:1433949270224:dw|
is that right?
Yep, exactly what I've got too! Excellent!
Just for fun, we'll digress a little. What if we make up the continued fraction and check it's accuracy!
Can you do that using A0, A1 and A2?
|dw:1433949392760:dw|
ima horrible drawer.
No problem, I am worse! Can you calculate the value?
is it 7/4
Yes, except you cheated a little in assuming that A3 is 1, which it is!
So 7/4=1.75 is already quite close to 1.7320508...
Isn't that encouraging?
yeah
Would you continue with the next row, 3=n+1, so use n=2?
We'll do at least two more rows, and we'll see why.
okay
sorry, havent we already dont n2
done*
we'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.
In the formula, n+1 = new row, n=previous row.
|dw:1433950128652:dw|
is that right?
I don't have the same numbers for n=3.
Let's check: I have M3=1, so that's good. D3=(S-(M3)^2)/D2=(3-1^2)/1=2
Then A3=floor(A0+M3)/D3 = floor(1+1)/2=1
Do you agree?
oh i see what i did
i substituted wrong
yeah i agree
Good, can we |dw:1433950463947:dw|
Can you do one more row and put it at the bottom of the table?
|dw:1433950567103:dw|
ok, let's check, again I don't have the same numbers. M4=D3A3-M3=2*1-1=1 Good D4=(S-(M4)^2)/D3 = (3-1^2)/2 = 2/2 =1 (see if you agree)
A4=floor( (A0+M4)/(D4) ) = floor( (1+1)/1 ) =floor(2) = 2
|dw:1433951105030:dw|
Great!
Can you put in the A0, A1, A2....A4 and see what the continued fraction gives you so far?
is it 35/26
You 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...
yeh i accidently multiplied the wrong thing
At this point, we don't have to do any more rows if you examine the table to look for patterns of _rows_.
We 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.
Do you agree?
Yeah
Sorry, OS was down the last while. We can now wrap up the problem.
@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?
1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, infinitely?
There you go, so you can complete the problem by evaluating to the accuracy you want! Good job!
how do i know which stage of the continued fraction will be within 10^-4
why do the rational approximations approach the surd in an alternating way, like over and under?
@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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