- Level: International Baccalaureate
- Subject: Maths
- Word count: 1589
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
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.
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
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
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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