Question

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

Answer 1

Hey @Maretch hold up, we will find someone to help you ASAP :D

Answer 2

Thanks

Answer 3

http://prntscr.com/7fc4ei

Answer 4

Why? whats that do?

Answer 5

try it

Answer 6

there you go! :-)

Answer 7

Thanks

Answer 8

Hes a great helper :D

Answer 9

@freckles

Answer 10

@dan815 @nincompoop @jigglypuff314 @Michele_Laino @Hero @kirbykirby @iambatman A QH question for you QH's :P

Answer 11

Although none of the Qualified helpers are on, maybe @mathmate @freckles @Loser66 @ikram002p could help you :)

Answer 12

Thanks, really appreciate you guys helping.

Answer 13

Unfortunately, today is a very quiet day on OpenStudy. Normally there would be a lot more people. Don't know what happened.

Answer 14

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

Answer 15

@Maretch Did you check the response of your other identical post?

Answer 16

yeah but it was through newtons method

Answer 17

@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?

Answer 18

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

Answer 19

The thing is never learned, continued fractions

Answer 20

@Maretch
I AM talking about continued fractions, NOT Newtons.
I am not the "other" guy talking about Newtons.

Answer 21

i know absolutely nothing, i know how to convert continued fractions to a proper fractions, thats about it

Answer 22

this is why im having so much trouble with it, never even mentioned continued fractions in class

Answer 23

What course are you taking? Alg 2 or number theory?

Answer 24

Im australian, so its different

Answer 25

http://en.wikipedia.org/wiki/Mathematics_education_in_Australia

Answer 26

and look at queensland

Answer 27

Can you tell me what grade you're in, or the name of the course, so I can understand your teacher's expectations.

Answer 28

Im year 11, age 16, doing maths c assignment

Answer 29

So are you in form 5 or form 6?

Answer 30

Not sure, never heard about that

Answer 31

So form 5, which of the 8 courses?

Answer 32

atm im doing vectors, matrices

Answer 33

ok, that's good.

Answer 34

sorry, i dont understand most of that stuff, australia is alot different

Answer 35

lol, you sound like you're not from Australia, but studying there. But that's beside the point.

Answer 36

So you need to find square root of 3 with continued fractions as an answer, right?

Answer 37

yeah, find approximations of root 3 with continued fractions, answer should be a fraction, that is accurate to 10^-4 of the true value

Answer 38

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

Answer 39

And you have not learned HOW to find a continued fraction approximation, am I right, or you just don't remember how?

Answer 40

Have not learnt it

Answer 41

even asked my classmates

Answer 42

The reason I am asking is there are different ways to approach the problem.

Answer 43

If that's clear, we can use different approaches.

Answer 44

kk

Answer 45

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?

Answer 46

yeah im ready

Answer 47

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.

Answer 48

ok so far?

Answer 49

Ye, as the continued fraction grows, the answer gets more accurate and accurate

Answer 50

Exactly, you get the idea.

Answer 51

Do you know how to find the first approximation?

Answer 52

That is the integer part of the fraction.

Answer 53

i dont.

Answer 54

i can convert the continued fraction to a proper or improper fraction, but i dont actually know how to setup the continued fraction

Answer 55

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

Answer 56

Yes, that is understood.
We are trying to solve a square-root problem.
Can you tell me the square-root approximately equals what?

Answer 57

* square-root of 26

Answer 58

Huh?

Answer 59

square root equals 5.099, is that what you meant?

Answer 60

26*

Answer 61

yes, exactly!

Answer 62

Why 26?

Answer 63

Very well. Now we have to introduce a concept of the "floor" function.

Answer 64

26 because I'll use it to find the floor function of sqrt(26).

Answer 65

The floor function means the largest INTEGER that does not exceed a given number.

Answer 66

For example, floor(sqrt(26)) = floor(5.099) = 5, the answer is always an integer.

Answer 67

Another example, floor(5) = 5, because 5 is an integer that does not exceed 5

Answer 68

Ok, so like round to the nearest interger?

Answer 69

or am i wrong

Answer 70

We'll find out!
Can you tell me what is floor(3.3)?

Answer 71

so floor is rounding down and ceiling is rounding up?

Answer 72

Very good question, actually floor is ALWAYS rounding down.

Answer 73

Ok so floor of 3,3 is 3

Answer 74

Exactly! how about floor 1.7

Answer 75

1

Answer 76

very good, a tough one here, floor(-2.3)

Answer 77

-3

Answer 78

Very, very good! floor always round to a smaller number, not just dropping the decimal part!

Answer 79

So -3 is smaller than -2.3, so floor(-2.3)=-3.
All clear?

Answer 80

Yep, i understand

Answer 81

We're going to work on an algorithm. Do you know what it means?

Answer 82

what algorithym, like just a formula to get an answer, idk how to explain

Answer 83

formula to solve a problem

Answer 84

Yes, a formula, but we have to use the formula many times to get the answer.

Answer 85

Oh ok

Answer 86

Most of the time, it is easier to work out the formula using a table to organize our calculations.

Answer 87

Okay

Answer 88

I'll start a table, and we will work on it together, ok?

Answer 89

Ok

Answer 90

give me a minute to plan the table, please.

Answer 91

btw is this question really difficult?

Answer 92

or just really abstract, is that why not many people can help

Answer 93

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

Answer 94

Ahh okay

Answer 95

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.