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Infinite Surds. The aim of this folio is to explore the nature of infinite surds.

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Introduction

Mathematics Folio: Infinite Surds        Michael Zhao

4/04/2012

The aim of this folio is to explore the nature of infinite surds. Infinite surds take the form
image00.pngimage00.png .

Part one: image38.pngimage38.png

Consider the surd where:

image54.png

image57.png

image63.png

Etc.

To begin, a recursive rule was determined from observing what changed from image12.pngimage12.png to image01.pngimage01.png and so on. The major change was that the previous surd was added to 1 and then square rooted to provide the next value. This observation provided the basis for the recursive rule below.

image06.png

The decimal values were calculated for image12.pngimage12.pngto image23.pngimage23.png and the results were then graphed.

n

a(n)

a(n) - a(n-1)

1

1.414214

-0.139560412

2

1.553774

-0.044279208

3

1.598053

-0.013794572

4

1.611848

-0.004273452

5

1.616121

-0.001321592

6

1.617443

-0.000408492

7

1.617851

-0.00012624

8

1.617978

-3.90113E-05

9

1.618017

-1.20552E-05

10

1.618029

-3.72528E-06

image33.png

It can be observed that the slope of the function gradually decreases asymptotes towards a value ofimage36.pngimage36.png. The values of image37.pngimage37.png suggest that as image39.pngimage39.png becomes larger, the function asymptotes towards a certain value. Finding the exact value of the surd requires being able to solve the surd. Solving the surd was done in the following method.

image40.png

image41.png

image42.png

image43.png

image44.png

image45.png

image46.png

image47.png

image48.png

image49.png

image50.pngimage50.png The exact value of the infinite surd image51.pngimage51.png

...read more.

Middle

image62.png

A table of values is composed for values of n in addition to a graph to identify possible asymptotes.

n

a(n)

a(n) - a(n-1)

1

1.847759065

-0.113811496

2

1.961570561

-0.028798893

3

1.990369453

-0.007221459

4

1.997590912

-0.001806725

5

1.999397637

-0.000451766

6

1.999849404

-0.000112947

7

1.999962351

-2.82371E-05

8

1.999990588

-7.05928E-06

9

1.999997647

-1.76482E-06

10

1.999999412

-4.41206E-07

image64.png

It can be observed that the values seem to approach 2 as image39.pngimage39.png becomes larger. The graph has a very similar shape to the graph in Part One but with a sharper turn and the values for image37.pngimage37.png also have a similar pattern in that they gradually decrease and become very small as image39.pngimage39.png becomes larger. However, the values are different in that they become more distinct. E.g. the big gaps are bigger, smaller gaps are smaller. To solve, a similar method to the method used in Part one will be used.

image65.png

image66.png

image67.png

image68.png

image69.png

image70.png
image71.png

image72.png

image73.png

image74.png
image75.png
image76.png

The exact value for the infinite surd is 2 whereimage77.pngimage77.png. These results correspond to the calculated results validating the observation that the infinite surd approached 2 with greater values ofimage78.pngimage78.png.

Part Three: The General Infinite Surd - image79.pngimage79.png

Consider the General Infinite Surd image80.pngimage80.png

...read more.

Conclusion

image18.png

image19.png

image20.png

image21.png

The results are tabulated to see if there is a correlation between the value for image05.pngimage05.png and the integer value of the surd.

Integer Valueimage22.pngimage22.png

image05.png

1

0

2

2

3

6

4

12

5

20

6

30

7

42

8

56

9

72

10

90

The table shows how there is a growth in the value of image05.pngimage05.png which correlates directly to the integer values of the infinite surd. The correlation can be represented by the formula image24.pngimage24.png orimage25.pngimage25.png or with the General expression for Infinite surds.

When put into an excel spread sheet, and tested, the values for image08.pngimage08.png all returned with the matching values forimage26.pngimage26.png. Those values of image05.pngimage05.png when substituted into the table of values for image27.pngimage27.png showed values which all approached integers which were the original image08.pngimage08.png values. Examples of these results are shown below.

image28.png

image29.png

image30.png

n

a(n)

1

20.9879465773076

2

20.9997130117844

3

20.9999931669461

4

20.9999998373082

5

20.9999999961264

6

20.9999999999078

7

20.9999999999978


image31.png

image29.png

image32.png

n

a(n)

1

13.9818002261237

2

13.9993499929862

3

13.9999767854445

4

13.9999991709087

5

13.9999999703896

6

13.9999999989425

7

13.9999999999622

image34.png

image29.png

image35.png




n

a(n)

1

25.99028852413850

2

25.99981324017810

3

25.99999640846470

4

25.99999993093200

5

25.99999999867180

6

25.99999999997450

7

25.99999999999950


Infinite Surds and their traits are all closely interconnected with each other. The value of the infinite surd and the value for
image05.pngimage05.png in the infinite surd have a unique correlation which means that they can easily be calculated at any time as long as one variable is at hand. In addition, the general expression of an Infinite Surd can be used to calculate the value of any surd effortlessly.

...read more.

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