anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
jagr2713
  • jagr2713
Hey @Maretch hold up, we will find someone to help you ASAP :D
anonymous
  • anonymous
Thanks
Nnesha
  • Nnesha
http://prntscr.com/7fc4ei

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anonymous
  • anonymous
Why? whats that do?
Nnesha
  • Nnesha
try it
Nnesha
  • Nnesha
there you go! :-)
anonymous
  • anonymous
Thanks
jagr2713
  • jagr2713
Hes a great helper :D
jagr2713
  • jagr2713
@freckles
TheSmartOne
  • TheSmartOne
@dan815 @nincompoop @jigglypuff314 @Michele_Laino @Hero @kirbykirby @iambatman A QH question for you QH's :P
TheSmartOne
  • TheSmartOne
Although none of the Qualified helpers are on, maybe @mathmate @freckles @Loser66 @ikram002p could help you :)
anonymous
  • anonymous
Thanks, really appreciate you guys helping.
TheSmartOne
  • TheSmartOne
Unfortunately, today is a very quiet day on OpenStudy. Normally there would be a lot more people. Don't know what happened.
TheSmartOne
  • TheSmartOne
Aha, I spot a genius who just came online right now. @SithsAndGiggles could you help this user? (:
mathmate
  • mathmate
@Maretch Did you check the response of your other identical post?
anonymous
  • anonymous
yeah but it was through newtons method
mathmate
  • mathmate
@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?
anonymous
  • anonymous
never learned newtons method and the title of the entire page is Continued fraction, so im assuming
anonymous
  • anonymous
The thing is never learned, continued fractions
mathmate
  • mathmate
@Maretch I AM talking about continued fractions, NOT Newtons. I am not the "other" guy talking about Newtons.
anonymous
  • anonymous
i know absolutely nothing, i know how to convert continued fractions to a proper fractions, thats about it
anonymous
  • anonymous
this is why im having so much trouble with it, never even mentioned continued fractions in class
mathmate
  • mathmate
What course are you taking? Alg 2 or number theory?
anonymous
  • anonymous
Im australian, so its different
anonymous
  • anonymous
http://en.wikipedia.org/wiki/Mathematics_education_in_Australia
anonymous
  • anonymous
and look at queensland
mathmate
  • mathmate
Can you tell me what grade you're in, or the name of the course, so I can understand your teacher's expectations.
anonymous
  • anonymous
Im year 11, age 16, doing maths c assignment
mathmate
  • mathmate
So are you in form 5 or form 6?
anonymous
  • anonymous
Not sure, never heard about that
mathmate
  • mathmate
So form 5, which of the 8 courses?
anonymous
  • anonymous
atm im doing vectors, matrices
mathmate
  • mathmate
ok, that's good.
anonymous
  • anonymous
sorry, i dont understand most of that stuff, australia is alot different
mathmate
  • mathmate
lol, you sound like you're not from Australia, but studying there. But that's beside the point.
mathmate
  • mathmate
So you need to find square root of 3 with continued fractions as an answer, right?
anonymous
  • anonymous
yeah, find approximations of root 3 with continued fractions, answer should be a fraction, that is accurate to 10^-4 of the true value
anonymous
  • anonymous
so like for root 2 an example of the answer would be like 17/12 or 41/29
mathmate
  • mathmate
And you have not learned HOW to find a continued fraction approximation, am I right, or you just don't remember how?
anonymous
  • anonymous
Have not learnt it
anonymous
  • anonymous
even asked my classmates
mathmate
  • mathmate
The reason I am asking is there are different ways to approach the problem.
mathmate
  • mathmate
If that's clear, we can use different approaches.
anonymous
  • anonymous
kk
mathmate
  • mathmate
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?
anonymous
  • anonymous
yeah im ready
mathmate
  • mathmate
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.
mathmate
  • mathmate
ok so far?
anonymous
  • anonymous
Ye, as the continued fraction grows, the answer gets more accurate and accurate
mathmate
  • mathmate
Exactly, you get the idea.
mathmate
  • mathmate
Do you know how to find the first approximation?
mathmate
  • mathmate
That is the integer part of the fraction.
anonymous
  • anonymous
i dont.
anonymous
  • anonymous
i can convert the continued fraction to a proper or improper fraction, but i dont actually know how to setup the continued fraction
anonymous
  • anonymous
like i have no idea where they get the 1,1,2,1,2 etc
mathmate
  • mathmate
Yes, that is understood. We are trying to solve a square-root problem. Can you tell me the square-root approximately equals what?
mathmate
  • mathmate
* square-root of 26
anonymous
  • anonymous
Huh?
anonymous
  • anonymous
square root equals 5.099, is that what you meant?
anonymous
  • anonymous
26*
mathmate
  • mathmate
yes, exactly!
anonymous
  • anonymous
Why 26?
mathmate
  • mathmate
Very well. Now we have to introduce a concept of the "floor" function.
mathmate
  • mathmate
26 because I'll use it to find the floor function of sqrt(26).
mathmate
  • mathmate
The floor function means the largest INTEGER that does not exceed a given number.
mathmate
  • mathmate
For example, floor(sqrt(26)) = floor(5.099) = 5, the answer is always an integer.
mathmate
  • mathmate
Another example, floor(5) = 5, because 5 is an integer that does not exceed 5
anonymous
  • anonymous
Ok, so like round to the nearest interger?
anonymous
  • anonymous
or am i wrong
mathmate
  • mathmate
We'll find out! Can you tell me what is floor(3.3)?
anonymous
  • anonymous
so floor is rounding down and ceiling is rounding up?
mathmate
  • mathmate
Very good question, actually floor is ALWAYS rounding down.
anonymous
  • anonymous
Ok so floor of 3,3 is 3
mathmate
  • mathmate
Exactly! how about floor 1.7
anonymous
  • anonymous
1
mathmate
  • mathmate
very good, a tough one here, floor(-2.3)
anonymous
  • anonymous
-3
mathmate
  • mathmate
Very, very good! floor always round to a smaller number, not just dropping the decimal part!
mathmate
  • mathmate
So -3 is smaller than -2.3, so floor(-2.3)=-3. All clear?
anonymous
  • anonymous
Yep, i understand
mathmate
  • mathmate
We're going to work on an algorithm. Do you know what it means?
anonymous
  • anonymous
what algorithym, like just a formula to get an answer, idk how to explain
anonymous
  • anonymous
formula to solve a problem
mathmate
  • mathmate
Yes, a formula, but we have to use the formula many times to get the answer.
anonymous
  • anonymous
Oh ok
mathmate
  • mathmate
Most of the time, it is easier to work out the formula using a table to organize our calculations.
anonymous
  • anonymous
Okay
mathmate
  • mathmate
I'll start a table, and we will work on it together, ok?
anonymous
  • anonymous
Ok
mathmate
  • mathmate
give me a minute to plan the table, please.
anonymous
  • anonymous
btw is this question really difficult?
anonymous
  • anonymous
or just really abstract, is that why not many people can help
mathmate
  • mathmate
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
anonymous
  • anonymous
Ahh okay
mathmate
  • mathmate
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.
anonymous
  • anonymous
Okay, this will eventually relate back to the question right, this is just things i need to know before solving right?
mathmate
  • mathmate
|dw:1433947126989:dw|
mathmate
  • mathmate
No, we will solve your problem together, sqrt(3), like an example, but it will be your problem.
anonymous
  • anonymous
Okay
anonymous
  • anonymous
so why am i learning this, if it doesnt relate.
mathmate
  • mathmate
|dw:1433947374403:dw|
mathmate
  • mathmate
Because that is how you find the continued fraction for sqrt(3).
anonymous
  • anonymous
okay, sorry, continue
mathmate
  • mathmate
|dw:1433947464984:dw|
mathmate
  • mathmate
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?
anonymous
  • anonymous
Yep
mathmate
  • mathmate
|dw:1433947582877:dw| We will be using A0 later on.
mathmate
  • mathmate
ok so far?
anonymous
  • anonymous
so what does mn, dn and an stand for?
mathmate
  • mathmate
I don't really know, except A0, A1,A2... stand for|dw:1433947729401:dw| The others are just intermediate answers.
anonymous
  • anonymous
kk
mathmate
  • mathmate
Now we need formulas to calculate M1,D1 and A1, which lie on the second line (where n=1), ok
anonymous
  • anonymous
kk
mathmate
  • mathmate
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?
anonymous
  • anonymous
Yeah
mathmate
  • mathmate
ok, Mn+1 = DnAn-Mn Dn+1=(S-(Mn+1)^2)/Dn An+1 = floor((A0+Mn+1)/Dn+1)
mathmate
  • mathmate
We'll work them out one by one!
anonymous
  • anonymous
Okay
mathmate
  • mathmate
We'll first calculate Mn+1=M1. This means n=0, or the previous line, right?
anonymous
  • anonymous
Yeah
mathmate
  • mathmate
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?
anonymous
  • anonymous
its 1 right?
mathmate
  • mathmate
Yes, can you put it in the M1 cell of the table?
anonymous
  • anonymous
trying to figure it out |dw:1433948402569:dw|
mathmate
  • mathmate
Excellent!
anonymous
  • anonymous
is that right, sorry i dont understand how to draw and stuff on this site.
mathmate
  • mathmate
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,
anonymous
  • anonymous
2?
mathmate
  • mathmate
Right, can you now put it in the table?
anonymous
  • anonymous
|dw:1433948679901:dw|
mathmate
  • mathmate
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?
mathmate
  • mathmate
* can you substitute the numbers and calculate A1 for me?
anonymous
  • anonymous
|dw:1433948843806:dw|
mathmate
  • mathmate
Perfect!!!!!
mathmate
  • mathmate
Now we're going to calculate the second row, which is 2=n+1, so n=1, fair enough?
mathmate
  • mathmate
I mean third row, where n=2!
anonymous
  • anonymous
Kk
mathmate
  • mathmate
See if you can post the values of M2, D2 and A2 before filling the table.
anonymous
  • anonymous
m2= 2 d2=2 a2=2?
mathmate
  • mathmate
I don't have the same answers, let's check!
mathmate
  • mathmate
for n+1=2, then n=1, so M2=D1A1-M1 = 2*1-1 =1 right?
anonymous
  • anonymous
yeah
mathmate
  • mathmate
You'll need to redo D2 and A2 because they depend on M2.
anonymous
  • anonymous
|dw:1433949270224:dw|
anonymous
  • anonymous
is that right?
mathmate
  • mathmate
Yep, exactly what I've got too! Excellent!
mathmate
  • mathmate
Just for fun, we'll digress a little. What if we make up the continued fraction and check it's accuracy!
mathmate
  • mathmate
Can you do that using A0, A1 and A2?
anonymous
  • anonymous
|dw:1433949392760:dw|
anonymous
  • anonymous
ima horrible drawer.
mathmate
  • mathmate
No problem, I am worse! Can you calculate the value?
anonymous
  • anonymous
is it 7/4
mathmate
  • mathmate
Yes, except you cheated a little in assuming that A3 is 1, which it is!
mathmate
  • mathmate
So 7/4=1.75 is already quite close to 1.7320508...
mathmate
  • mathmate
Isn't that encouraging?
anonymous
  • anonymous
yeah
mathmate
  • mathmate
Would you continue with the next row, 3=n+1, so use n=2?
mathmate
  • mathmate
We'll do at least two more rows, and we'll see why.
anonymous
  • anonymous
okay
anonymous
  • anonymous
sorry, havent we already dont n2
anonymous
  • anonymous
done*
mathmate
  • mathmate
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.
mathmate
  • mathmate
In the formula, n+1 = new row, n=previous row.
anonymous
  • anonymous
|dw:1433950128652:dw|
anonymous
  • anonymous
is that right?
mathmate
  • mathmate
I don't have the same numbers for n=3.
mathmate
  • mathmate
Let's check: I have M3=1, so that's good. D3=(S-(M3)^2)/D2=(3-1^2)/1=2
mathmate
  • mathmate
Then A3=floor(A0+M3)/D3 = floor(1+1)/2=1
mathmate
  • mathmate
Do you agree?
anonymous
  • anonymous
oh i see what i did
anonymous
  • anonymous
i substituted wrong
anonymous
  • anonymous
yeah i agree
mathmate
  • mathmate
Good, can we |dw:1433950463947:dw|
mathmate
  • mathmate
Can you do one more row and put it at the bottom of the table?
anonymous
  • anonymous
|dw:1433950567103:dw|
mathmate
  • mathmate
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)
mathmate
  • mathmate
A4=floor( (A0+M4)/(D4) ) = floor( (1+1)/1 ) =floor(2) = 2
anonymous
  • anonymous
|dw:1433951105030:dw|
mathmate
  • mathmate
Great!
mathmate
  • mathmate
Can you put in the A0, A1, A2....A4 and see what the continued fraction gives you so far?
anonymous
  • anonymous
is it 35/26
mathmate
  • mathmate
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...
anonymous
  • anonymous
yeh i accidently multiplied the wrong thing
mathmate
  • mathmate
At this point, we don't have to do any more rows if you examine the table to look for patterns of _rows_.
mathmate
  • mathmate
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.
mathmate
  • mathmate
Do you agree?
anonymous
  • anonymous
Yeah
mathmate
  • mathmate
Sorry, OS was down the last while. We can now wrap up the problem.
mathmate
  • mathmate
@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
  • anonymous
1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, infinitely?
mathmate
  • mathmate
There you go, so you can complete the problem by evaluating to the accuracy you want! Good job!
anonymous
  • anonymous
how do i know which stage of the continued fraction will be within 10^-4
anonymous
  • anonymous
why do the rational approximations approach the surd in an alternating way, like over and under?
anonymous
  • anonymous
@Hero
anonymous
  • anonymous
@Michele_Laino
mathmate
  • mathmate
@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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