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Thanks

http://prntscr.com/7fc4ei

Why? whats that do?

try it

there you go! :-)

Thanks

Hes a great helper :D

Thanks, really appreciate you guys helping.

Aha, I spot a genius who just came online right now. @SithsAndGiggles could you help this user? (:

yeah but it was through newtons method

never learned newtons method and the title of the entire page is Continued fraction, so im assuming

The thing is never learned, continued fractions

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

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?

so like for root 2 an example of the answer would be like 17/12 or 41/29

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

yeah im ready

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.

like i have no idea where they get the 1,1,2,1,2 etc

* 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

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

Ahh okay

|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|

Yep

|dw:1433947582877:dw|
We will be using A0 later on.

ok so far?

so what does mn, dn and an stand for?

kk

Now we need formulas to calculate M1,D1 and A1, which lie on the second line (where n=1), ok

kk

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

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.

2?

Right, can you now put it in the table?

|dw:1433948679901:dw|

* 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!

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*

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|

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

yeh i accidently multiplied the wrong thing

Do you agree?

Yeah

Sorry, OS was down the last while.
We can now wrap up the problem.

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?