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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.

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Introduction

MATH PORTFOLIO

INFINTE SURDS  Submitted By Tim Kwok

Math 20 IB

Presented To Ms. Garrett

April 27, 2009

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. 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.

Middle

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

x = Only the positive value is accepted. We can further our understanding of infinite surds by looking at this next example: Like the first infinite surd, the first 10 terms of this infinite surd's sequence are also all irrational numbers.

a1: = 1.847759 ...  a2: = 1.961570 ...

a3: = 1.990369 ... a4: = 1.997590 ... a5: = 1.999397 ... a6: = 1.999849 ... a7: = 1.999984 ... a8: = 1.999990 ... a9: = 1.999997 ... a10:  = 1.999999 ... From the first ten terms of the sequence you can see that the next sequence is  the previous term.  Turning that into a formula for an+1 in terms of an makes:     The relationship shown from the plotted points of the infinite surd 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.  From graph 2, you know that as n  , 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 = (x)2 = 2                Square both sides to create an equation                                                         to work with.

x2  = The infinite surd continues.

x2  = 2+x                                Substitute 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 and Example 4 showed that the exact value can be an integer if k = . In Example 4 we also learned that b = 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

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