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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

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  1. jagr2713
    • one year ago
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    Hey @Maretch hold up, we will find someone to help you ASAP :D

  2. anonymous
    • one year ago
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    Thanks

  3. Nnesha
    • one year ago
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    http://prntscr.com/7fc4ei

  4. anonymous
    • one year ago
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    Why? whats that do?

  5. Nnesha
    • one year ago
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    try it

  6. Nnesha
    • one year ago
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    there you go! :-)

  7. anonymous
    • one year ago
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    Thanks

  8. jagr2713
    • one year ago
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    Hes a great helper :D

  9. jagr2713
    • one year ago
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    @freckles

  10. TheSmartOne
    • one year ago
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    @dan815 @nincompoop @jigglypuff314 @Michele_Laino @Hero @kirbykirby @iambatman A QH question for you QH's :P

  11. TheSmartOne
    • one year ago
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    Although none of the Qualified helpers are on, maybe @mathmate @freckles @Loser66 @ikram002p could help you :)

  12. anonymous
    • one year ago
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    Thanks, really appreciate you guys helping.

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

  14. TheSmartOne
    • one year ago
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    Aha, I spot a genius who just came online right now. @SithsAndGiggles could you help this user? (:

  15. mathmate
    • one year ago
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    @Maretch Did you check the response of your other identical post?

  16. anonymous
    • one year ago
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    yeah but it was through newtons method

  17. mathmate
    • one year ago
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    @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?

  18. anonymous
    • one year ago
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    never learned newtons method and the title of the entire page is Continued fraction, so im assuming

  19. anonymous
    • one year ago
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    The thing is never learned, continued fractions

  20. mathmate
    • one year ago
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    @Maretch I AM talking about continued fractions, NOT Newtons. I am not the "other" guy talking about Newtons.

  21. anonymous
    • one year ago
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    i know absolutely nothing, i know how to convert continued fractions to a proper fractions, thats about it

  22. anonymous
    • one year ago
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    this is why im having so much trouble with it, never even mentioned continued fractions in class

  23. mathmate
    • one year ago
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    What course are you taking? Alg 2 or number theory?

  24. anonymous
    • one year ago
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    Im australian, so its different

  25. anonymous
    • one year ago
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    http://en.wikipedia.org/wiki/Mathematics_education_in_Australia

  26. anonymous
    • one year ago
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    and look at queensland

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

  28. anonymous
    • one year ago
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    Im year 11, age 16, doing maths c assignment

  29. mathmate
    • one year ago
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    So are you in form 5 or form 6?

  30. anonymous
    • one year ago
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    Not sure, never heard about that

  31. mathmate
    • one year ago
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    So form 5, which of the 8 courses?

  32. anonymous
    • one year ago
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    atm im doing vectors, matrices

  33. mathmate
    • one year ago
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    ok, that's good.

  34. anonymous
    • one year ago
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    sorry, i dont understand most of that stuff, australia is alot different

  35. mathmate
    • one year ago
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    lol, you sound like you're not from Australia, but studying there. But that's beside the point.

  36. mathmate
    • one year ago
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    So you need to find square root of 3 with continued fractions as an answer, right?

  37. anonymous
    • one year ago
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    yeah, find approximations of root 3 with continued fractions, answer should be a fraction, that is accurate to 10^-4 of the true value

  38. anonymous
    • one year ago
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    so like for root 2 an example of the answer would be like 17/12 or 41/29

  39. mathmate
    • one year ago
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    And you have not learned HOW to find a continued fraction approximation, am I right, or you just don't remember how?

  40. anonymous
    • one year ago
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    Have not learnt it

  41. anonymous
    • one year ago
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    even asked my classmates

  42. mathmate
    • one year ago
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    The reason I am asking is there are different ways to approach the problem.

  43. mathmate
    • one year ago
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    If that's clear, we can use different approaches.

  44. anonymous
    • one year ago
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    kk

  45. mathmate
    • one year ago
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    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?

  46. anonymous
    • one year ago
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    yeah im ready

  47. mathmate
    • one year ago
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    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.

  48. mathmate
    • one year ago
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    ok so far?

  49. anonymous
    • one year ago
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    Ye, as the continued fraction grows, the answer gets more accurate and accurate

  50. mathmate
    • one year ago
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    Exactly, you get the idea.

  51. mathmate
    • one year ago
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    Do you know how to find the first approximation?

  52. mathmate
    • one year ago
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    That is the integer part of the fraction.

  53. anonymous
    • one year ago
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    i dont.

  54. anonymous
    • one year ago
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    i can convert the continued fraction to a proper or improper fraction, but i dont actually know how to setup the continued fraction

  55. anonymous
    • one year ago
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    like i have no idea where they get the 1,1,2,1,2 etc

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

  57. mathmate
    • one year ago
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    * square-root of 26

  58. anonymous
    • one year ago
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    Huh?

  59. anonymous
    • one year ago
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    square root equals 5.099, is that what you meant?

  60. anonymous
    • one year ago
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    26*

  61. mathmate
    • one year ago
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    yes, exactly!

  62. anonymous
    • one year ago
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    Why 26?

  63. mathmate
    • one year ago
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    Very well. Now we have to introduce a concept of the "floor" function.

  64. mathmate
    • one year ago
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    26 because I'll use it to find the floor function of sqrt(26).

  65. mathmate
    • one year ago
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    The floor function means the largest INTEGER that does not exceed a given number.

  66. mathmate
    • one year ago
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    For example, floor(sqrt(26)) = floor(5.099) = 5, the answer is always an integer.

  67. mathmate
    • one year ago
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    Another example, floor(5) = 5, because 5 is an integer that does not exceed 5

  68. anonymous
    • one year ago
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    Ok, so like round to the nearest interger?

  69. anonymous
    • one year ago
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    or am i wrong

  70. mathmate
    • one year ago
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    We'll find out! Can you tell me what is floor(3.3)?

  71. anonymous
    • one year ago
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    so floor is rounding down and ceiling is rounding up?

  72. mathmate
    • one year ago
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    Very good question, actually floor is ALWAYS rounding down.

  73. anonymous
    • one year ago
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    Ok so floor of 3,3 is 3

  74. mathmate
    • one year ago
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    Exactly! how about floor 1.7

  75. anonymous
    • one year ago
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    1

  76. mathmate
    • one year ago
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    very good, a tough one here, floor(-2.3)

  77. anonymous
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    -3

  78. mathmate
    • one year ago
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    Very, very good! floor always round to a smaller number, not just dropping the decimal part!

  79. mathmate
    • one year ago
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    So -3 is smaller than -2.3, so floor(-2.3)=-3. All clear?

  80. anonymous
    • one year ago
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    Yep, i understand

  81. mathmate
    • one year ago
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    We're going to work on an algorithm. Do you know what it means?

  82. anonymous
    • one year ago
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    what algorithym, like just a formula to get an answer, idk how to explain

  83. anonymous
    • one year ago
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    formula to solve a problem

  84. mathmate
    • one year ago
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    Yes, a formula, but we have to use the formula many times to get the answer.

  85. anonymous
    • one year ago
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    Oh ok

  86. mathmate
    • one year ago
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    Most of the time, it is easier to work out the formula using a table to organize our calculations.

  87. anonymous
    • one year ago
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    Okay

  88. mathmate
    • one year ago
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    I'll start a table, and we will work on it together, ok?

  89. anonymous
    • one year ago
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    Ok

  90. mathmate
    • one year ago
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    give me a minute to plan the table, please.

  91. anonymous
    • one year ago
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    btw is this question really difficult?

  92. anonymous
    • one year ago
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    or just really abstract, is that why not many people can help

  93. mathmate
    • one year ago
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    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

  94. anonymous
    • one year ago
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    Ahh okay

  95. mathmate
    • one year ago
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    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.

  96. anonymous
    • one year ago
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    Okay, this will eventually relate back to the question right, this is just things i need to know before solving right?

  97. mathmate
    • one year ago
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    |dw:1433947126989:dw|

  98. mathmate
    • one year ago
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    No, we will solve your problem together, sqrt(3), like an example, but it will be your problem.

  99. anonymous
    • one year ago
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    Okay

  100. anonymous
    • one year ago
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    so why am i learning this, if it doesnt relate.

  101. mathmate
    • one year ago
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    |dw:1433947374403:dw|

  102. mathmate
    • one year ago
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    Because that is how you find the continued fraction for sqrt(3).

  103. anonymous
    • one year ago
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    okay, sorry, continue

  104. mathmate
    • one year ago
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    |dw:1433947464984:dw|

  105. mathmate
    • one year ago
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    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?

  106. anonymous
    • one year ago
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    Yep

  107. mathmate
    • one year ago
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    |dw:1433947582877:dw| We will be using A0 later on.

  108. mathmate
    • one year ago
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    ok so far?

  109. anonymous
    • one year ago
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    so what does mn, dn and an stand for?

  110. mathmate
    • one year ago
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    I don't really know, except A0, A1,A2... stand for|dw:1433947729401:dw| The others are just intermediate answers.

  111. anonymous
    • one year ago
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    kk

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

  113. anonymous
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    kk

  114. mathmate
    • one year ago
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    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?

  115. anonymous
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    Yeah

  116. mathmate
    • one year ago
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    ok, Mn+1 = DnAn-Mn Dn+1=(S-(Mn+1)^2)/Dn An+1 = floor((A0+Mn+1)/Dn+1)

  117. mathmate
    • one year ago
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    We'll work them out one by one!

  118. anonymous
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    Okay

  119. mathmate
    • one year ago
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    We'll first calculate Mn+1=M1. This means n=0, or the previous line, right?

  120. anonymous
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    Yeah

  121. mathmate
    • one year ago
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    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?

  122. anonymous
    • one year ago
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    its 1 right?

  123. mathmate
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    Yes, can you put it in the M1 cell of the table?

  124. anonymous
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    trying to figure it out |dw:1433948402569:dw|

  125. mathmate
    • one year ago
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    Excellent!

  126. anonymous
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    is that right, sorry i dont understand how to draw and stuff on this site.

  127. mathmate
    • one year ago
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    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,

  128. anonymous
    • one year ago
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    2?

  129. mathmate
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    Right, can you now put it in the table?

  130. anonymous
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    |dw:1433948679901:dw|

  131. mathmate
    • one year ago
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    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?

  132. mathmate
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    * can you substitute the numbers and calculate A1 for me?

  133. anonymous
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    |dw:1433948843806:dw|

  134. mathmate
    • one year ago
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    Perfect!!!!!

  135. mathmate
    • one year ago
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    Now we're going to calculate the second row, which is 2=n+1, so n=1, fair enough?

  136. mathmate
    • one year ago
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    I mean third row, where n=2!

  137. anonymous
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    Kk

  138. mathmate
    • one year ago
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    See if you can post the values of M2, D2 and A2 before filling the table.

  139. anonymous
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    m2= 2 d2=2 a2=2?

  140. mathmate
    • one year ago
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    I don't have the same answers, let's check!

  141. mathmate
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    for n+1=2, then n=1, so M2=D1A1-M1 = 2*1-1 =1 right?

  142. anonymous
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    yeah

  143. mathmate
    • one year ago
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    You'll need to redo D2 and A2 because they depend on M2.

  144. anonymous
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    |dw:1433949270224:dw|

  145. anonymous
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    is that right?

  146. mathmate
    • one year ago
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    Yep, exactly what I've got too! Excellent!

  147. mathmate
    • one year ago
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    Just for fun, we'll digress a little. What if we make up the continued fraction and check it's accuracy!

  148. mathmate
    • one year ago
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    Can you do that using A0, A1 and A2?

  149. anonymous
    • one year ago
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    |dw:1433949392760:dw|

  150. anonymous
    • one year ago
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    ima horrible drawer.

  151. mathmate
    • one year ago
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    No problem, I am worse! Can you calculate the value?

  152. anonymous
    • one year ago
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    is it 7/4

  153. mathmate
    • one year ago
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    Yes, except you cheated a little in assuming that A3 is 1, which it is!

  154. mathmate
    • one year ago
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    So 7/4=1.75 is already quite close to 1.7320508...

  155. mathmate
    • one year ago
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    Isn't that encouraging?

  156. anonymous
    • one year ago
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    yeah

  157. mathmate
    • one year ago
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    Would you continue with the next row, 3=n+1, so use n=2?

  158. mathmate
    • one year ago
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    We'll do at least two more rows, and we'll see why.

  159. anonymous
    • one year ago
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    okay

  160. anonymous
    • one year ago
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    sorry, havent we already dont n2

  161. anonymous
    • one year ago
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    done*

  162. mathmate
    • one year ago
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    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.

  163. mathmate
    • one year ago
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    In the formula, n+1 = new row, n=previous row.

  164. anonymous
    • one year ago
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    |dw:1433950128652:dw|

  165. anonymous
    • one year ago
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    is that right?

  166. mathmate
    • one year ago
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    I don't have the same numbers for n=3.

  167. mathmate
    • one year ago
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    Let's check: I have M3=1, so that's good. D3=(S-(M3)^2)/D2=(3-1^2)/1=2

  168. mathmate
    • one year ago
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    Then A3=floor(A0+M3)/D3 = floor(1+1)/2=1

  169. mathmate
    • one year ago
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    Do you agree?

  170. anonymous
    • one year ago
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    oh i see what i did

  171. anonymous
    • one year ago
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    i substituted wrong

  172. anonymous
    • one year ago
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    yeah i agree

  173. mathmate
    • one year ago
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    Good, can we |dw:1433950463947:dw|

  174. mathmate
    • one year ago
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    Can you do one more row and put it at the bottom of the table?

  175. anonymous
    • one year ago
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    |dw:1433950567103:dw|

  176. mathmate
    • one year ago
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    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)

  177. mathmate
    • one year ago
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    A4=floor( (A0+M4)/(D4) ) = floor( (1+1)/1 ) =floor(2) = 2

  178. anonymous
    • one year ago
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    |dw:1433951105030:dw|

  179. mathmate
    • one year ago
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    Great!

  180. mathmate
    • one year ago
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    Can you put in the A0, A1, A2....A4 and see what the continued fraction gives you so far?

  181. anonymous
    • one year ago
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    is it 35/26

  182. mathmate
    • one year ago
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    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...

  183. anonymous
    • one year ago
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    yeh i accidently multiplied the wrong thing

  184. mathmate
    • one year ago
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    At this point, we don't have to do any more rows if you examine the table to look for patterns of _rows_.

  185. mathmate
    • one year ago
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    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.

  186. mathmate
    • one year ago
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    Do you agree?

  187. anonymous
    • one year ago
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    Yeah

  188. mathmate
    • one year ago
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    Sorry, OS was down the last while. We can now wrap up the problem.

  189. mathmate
    • one year ago
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    @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?

  190. anonymous
    • one year ago
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    1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, infinitely?

  191. mathmate
    • one year ago
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    There you go, so you can complete the problem by evaluating to the accuracy you want! Good job!

  192. anonymous
    • one year ago
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    how do i know which stage of the continued fraction will be within 10^-4

  193. anonymous
    • one year ago
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    why do the rational approximations approach the surd in an alternating way, like over and under?

  194. anonymous
    • one year ago
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    @Hero

  195. anonymous
    • one year ago
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    @Michele_Laino

  196. mathmate
    • one year ago
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    @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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