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Infinite surds portfolio - As you can see in the first 10 terms of the infinite surd, they are all irrational numbers.

Extracts from this document...

Introduction

MATH PORTFOLIO

INFINTE SURDS

image00.png

image41.png

Submitted By Tim Kwok

Math 20 IB

Presented To Ms. Garrett

April 27, 2009

TABLE OF CONTENTS

Introduction to Surds and Infinite Surds        Page 2

Infinite Surd Example 1        Page 2

  • First Ten Terms of Sequence        Page 2-3
  • Formula for the Following Term        Page 3
  • Graph of First Ten Terms        Page 4
  • Relation Between Terms and Values in Infinite Surd        Page 4
  • Exact Value of the Infinite Surd        Page 5

Infinite Surd Example 2        Page 6

  • First Ten Terms of Sequence        Page 6-7
  • Formula for the Following Term        Page 7
  • Graph of First Ten Terms        Page 8
  • Relation Between Terms and Values in Infinite Surd        Page 8-9
  • Exact Value of the Infinite Surd        Page 9

Infinite Surd Example 3        Page 10

  • General Form of Infinite Surd Exact Value        Page 10        

Infinite Surd Example 4        Page 11

  • Values That Make an Infinite Surd an Integer        Page 11
  • General Statement for Values That Make an         Page 12      Infinite Surd an Integer        
  • Limitations to the General Statement        Page 13        

References        Page 14        

        Surds are used commonly in math, they just are not referred to as surds. A surd is any positive number that is in square root form.  Once you simplify the surd it must form a positive irrational number.  If a rational number is formed, it is not considered to be a surd.

image01.png

        Infinite surds are just surds forming a sequence that goes on forever.  The exact value of an infinite surd is expressed in the square root form.

...read more.

Middle

                                Since a negative number cannot be square                                                         rooted, the negative exact value is an                                                         extraneous root because it doesn't work.

x = image60.png                                Only the positive value is accepted.

image12.png

        We can further our understanding of infinite surds by looking at this next example:

image13.png

        Like the first infinite surd, the first 10 terms of this infinite surd's sequence are also all irrational numbers.

a1:        image62.png                                                        = 1.847759 ...

image47.pngimage14.png

a2:image63.png                                                = 1.961570 ...

a3:image64.png                                                = 1.990369 ...image15.png

a4:image65.png                                        = 1.997590 ...

image16.png

a5:image66.png                                = 1.999397 ...

image17.png

a6:image67.png                                = 1.999849 ...

image18.png

a7:image68.png                        = 1.999984 ...image19.png

a8:image69.png                = 1.999990 ...

image20.png

a9:image70.png        = 1.999997 ...

image22.png

a10:image71.jpgimage47.png= 1.999999 ...

image23.png

        From the first ten terms of the sequence you can see that the next sequence is image47.png

image72.pngthe previous term.  Turning that into a formula for an+1 in terms of an makes:image24.png

image25.png

image26.png

image16.png

image73.jpg

        The relationship shown from the plotted points of the infinite surd image74.png in graph 2 is that as n increases, the closer an gets to the value of about 2.000 but an will never pass that point.  The graph also shows how the rise of the slope is continually decreasing as n increase.  So as n gets larger, an- an+1 continues to decrease closer to 0.  Just like the infinite surd from example 1, it cannot be determined if an- an+1 ever equals 0 because the sequence also goes on forever.

image27.png

image28.png

        From graph 2, you know that as nimage51.pngimage52.png, an gets flatter and levels out at about 2.0. The value of the infinite surd which is about 2.0 can be considered as x. So, let x be:

x = image75.png

        (x)2 = image76.png2                Square both sides to create an equation                                                         to work with.

x2  = image78.png                        The infinite surd continues.

x2  = 2+x                                Substitute image75.png

...read more.

Conclusion

next term is the square root of the numerical value in the surd added with the previous term in the sequence.  From Example 3, we learned that the general form for the exact value of all infinite surds is image90.png and Example 4 showed that the exact value can be an integer if k = image105.png. In Example 4 we also learned that b = image107.png because if b were a rational number, the exact value of the infinite surd would not be integral.

References:

Table of mathematical symbols. Retrieved April 25, 2009, from

        Wikipedia Web site:

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

        (2008, August 25). Infinite surds - expression for which the exact value is

        an integer . Retrieved April 25, 2009, from Math Forum Help

        Web site: http://www.mathhelpforum.com/math-help/other-

        topics/46681-infinite-surds-expression-exact-value-integer.html

        How do i find the exact value for an infinite surd?. Retrieved April 25, 2009, from Yahoo Answers         Web site:         http://answers.yahoo.com/question/index;_ylt=Au.kibwg0umFF6o1yLGdENcjzKIX;_ylv=3?qid=        20090117014917AAnvdJz

        2009, March 12). Various questions about surds . Retrieved April 25,

        2009, from Math Help Forum Web site:

        http://www.mathhelpforum.com/math-help/other-topics/46681-

        infinite-surds-expression-exact-value-integer.html

        (2009, February 23). Infinite Surds . Retrieved April 16, 2009, from         http://thewonderfulworldofmath.blogspot.com/2009/02/infinite-surds.html

        M (2009, February 18). Mathematics Sl Portfolio Infinite Surds.

        Retrieved April 25, 2009, from Anti Essays Web site:

        http://www.antiessays.com/free-essays/34605.html

        tmgt5, "[Math SL] Infinite surds, All questions concerning this

        project ." IB Survival. 13 April 2009. 26 Apr 2009

        <http://www.ibsurvival.com/forum/index.php?

        showtopic=2347>.

        Suomalainen, Heikki . "INFINITE SURDS." 23 May 2008. 26

        Apr 2009 <www.lyseo.edu.ouka.fi/~hequel/portfolio.doc >.

Infinite Surds                Page

...read more.

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