## 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

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

2. anonymous

Thanks

3. Nnesha
4. anonymous

Why? whats that do?

5. Nnesha

try it

6. Nnesha

there you go! :-)

7. anonymous

Thanks

8. jagr2713

Hes a great helper :D

9. jagr2713

@freckles

10. TheSmartOne

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

11. TheSmartOne

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

12. anonymous

Thanks, really appreciate you guys helping.

13. TheSmartOne

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

14. TheSmartOne

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

15. mathmate

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

16. anonymous

yeah but it was through newtons method

17. 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?

18. anonymous

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

19. anonymous

The thing is never learned, continued fractions

20. mathmate

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

21. anonymous

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

22. anonymous

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

23. mathmate

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

24. anonymous

Im australian, so its different

25. anonymous
26. anonymous

and look at queensland

27. mathmate

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

Im year 11, age 16, doing maths c assignment

29. mathmate

So are you in form 5 or form 6?

30. anonymous

Not sure, never heard about that

31. mathmate

So form 5, which of the 8 courses?

32. anonymous

atm im doing vectors, matrices

33. mathmate

ok, that's good.

34. anonymous

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

35. mathmate

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

36. mathmate

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

37. 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

38. anonymous

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

39. mathmate

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

40. anonymous

Have not learnt it

41. anonymous

42. mathmate

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

43. mathmate

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

44. anonymous

kk

45. 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?

46. anonymous

47. 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.

48. mathmate

ok so far?

49. anonymous

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

50. mathmate

Exactly, you get the idea.

51. mathmate

Do you know how to find the first approximation?

52. mathmate

That is the integer part of the fraction.

53. anonymous

i dont.

54. anonymous

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

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

56. mathmate

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

* square-root of 26

58. anonymous

Huh?

59. anonymous

square root equals 5.099, is that what you meant?

60. anonymous

26*

61. mathmate

yes, exactly!

62. anonymous

Why 26?

63. mathmate

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

64. mathmate

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

65. mathmate

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

66. mathmate

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

67. mathmate

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

68. anonymous

Ok, so like round to the nearest interger?

69. anonymous

or am i wrong

70. mathmate

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

71. anonymous

so floor is rounding down and ceiling is rounding up?

72. mathmate

Very good question, actually floor is ALWAYS rounding down.

73. anonymous

Ok so floor of 3,3 is 3

74. mathmate

75. anonymous

1

76. mathmate

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

77. anonymous

-3

78. mathmate

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

79. mathmate

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

80. anonymous

Yep, i understand

81. mathmate

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

82. anonymous

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

83. anonymous

formula to solve a problem

84. mathmate

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

85. anonymous

Oh ok

86. mathmate

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

87. anonymous

Okay

88. mathmate

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

89. anonymous

Ok

90. mathmate

give me a minute to plan the table, please.

91. anonymous

btw is this question really difficult?

92. anonymous

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

93. 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

94. anonymous

Ahh okay

95. 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.

96. anonymous

Okay, this will eventually relate back to the question right, this is just things i need to know before solving right?

97. mathmate

|dw:1433947126989:dw|

98. mathmate

No, we will solve your problem together, sqrt(3), like an example, but it will be your problem.

99. anonymous

Okay

100. anonymous

so why am i learning this, if it doesnt relate.

101. mathmate

|dw:1433947374403:dw|

102. mathmate

Because that is how you find the continued fraction for sqrt(3).

103. anonymous

okay, sorry, continue

104. mathmate

|dw:1433947464984:dw|

105. 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?

106. anonymous

Yep

107. mathmate

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

108. mathmate

ok so far?

109. anonymous

so what does mn, dn and an stand for?

110. mathmate

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

111. anonymous

kk

112. mathmate

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

113. anonymous

kk

114. 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?

115. anonymous

Yeah

116. mathmate

ok, Mn+1 = DnAn-Mn Dn+1=(S-(Mn+1)^2)/Dn An+1 = floor((A0+Mn+1)/Dn+1)

117. mathmate

We'll work them out one by one!

118. anonymous

Okay

119. mathmate

We'll first calculate Mn+1=M1. This means n=0, or the previous line, right?

120. anonymous

Yeah

121. 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?

122. anonymous

its 1 right?

123. mathmate

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

124. anonymous

trying to figure it out |dw:1433948402569:dw|

125. mathmate

Excellent!

126. anonymous

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

127. 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,

128. anonymous

2?

129. mathmate

Right, can you now put it in the table?

130. anonymous

|dw:1433948679901:dw|

131. 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?

132. mathmate

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

133. anonymous

|dw:1433948843806:dw|

134. mathmate

Perfect!!!!!

135. mathmate

Now we're going to calculate the second row, which is 2=n+1, so n=1, fair enough?

136. mathmate

I mean third row, where n=2!

137. anonymous

Kk

138. mathmate

See if you can post the values of M2, D2 and A2 before filling the table.

139. anonymous

m2= 2 d2=2 a2=2?

140. mathmate

I don't have the same answers, let's check!

141. mathmate

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

142. anonymous

yeah

143. mathmate

You'll need to redo D2 and A2 because they depend on M2.

144. anonymous

|dw:1433949270224:dw|

145. anonymous

is that right?

146. mathmate

Yep, exactly what I've got too! Excellent!

147. mathmate

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

148. mathmate

Can you do that using A0, A1 and A2?

149. anonymous

|dw:1433949392760:dw|

150. anonymous

ima horrible drawer.

151. mathmate

No problem, I am worse! Can you calculate the value?

152. anonymous

is it 7/4

153. mathmate

Yes, except you cheated a little in assuming that A3 is 1, which it is!

154. mathmate

So 7/4=1.75 is already quite close to 1.7320508...

155. mathmate

Isn't that encouraging?

156. anonymous

yeah

157. mathmate

Would you continue with the next row, 3=n+1, so use n=2?

158. mathmate

We'll do at least two more rows, and we'll see why.

159. anonymous

okay

160. anonymous

sorry, havent we already dont n2

161. anonymous

done*

162. 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.

163. mathmate

In the formula, n+1 = new row, n=previous row.

164. anonymous

|dw:1433950128652:dw|

165. anonymous

is that right?

166. mathmate

I don't have the same numbers for n=3.

167. mathmate

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

168. mathmate

Then A3=floor(A0+M3)/D3 = floor(1+1)/2=1

169. mathmate

Do you agree?

170. anonymous

oh i see what i did

171. anonymous

i substituted wrong

172. anonymous

yeah i agree

173. mathmate

Good, can we |dw:1433950463947:dw|

174. mathmate

Can you do one more row and put it at the bottom of the table?

175. anonymous

|dw:1433950567103:dw|

176. 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)

177. mathmate

A4=floor( (A0+M4)/(D4) ) = floor( (1+1)/1 ) =floor(2) = 2

178. anonymous

|dw:1433951105030:dw|

179. mathmate

Great!

180. mathmate

Can you put in the A0, A1, A2....A4 and see what the continued fraction gives you so far?

181. anonymous

is it 35/26

182. 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...

183. anonymous

yeh i accidently multiplied the wrong thing

184. mathmate

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

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

Do you agree?

187. anonymous

Yeah

188. mathmate

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

189. 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?

190. anonymous

1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2, infinitely?

191. mathmate

There you go, so you can complete the problem by evaluating to the accuracy you want! Good job!

192. anonymous

how do i know which stage of the continued fraction will be within 10^-4

193. anonymous

why do the rational approximations approach the surd in an alternating way, like over and under?

194. anonymous

@Hero

195. anonymous

@Michele_Laino

196. 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.