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infintite surds

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Introduction

Alice Wang

Infinite Surds

IB Portfolio

An infinite surd is a number that can only be expressed exactly using a root sign. Surds are useful because they can represent irrational numbers that have an infinite number of non-recurring decimals.

This expression is known as an infinite surd:

image00.png

It can be considered as a sequence of terms an, where

image01.png

…..etc.

It can be inferred from the first ten terms of the sequence that the relation of two consecutive terms is image12.png.

Figure 1. Relationship between n and an

n

image20.png

1

1.414213562

2

1.553773974

3

1.598053182

4

1.611847754

5

1.616121207

6

1.617442799

7

1.617851291

8

1.617977531

9

1.618016542

10

1.618028597

From Figure 1, it can be inferred that the sequence of the two following terms as:

image27.png. The values of an, are not exact because the values of an, have an infinite number of decimal points since it is a surd. When the value of n

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Middle

n

bn

1

1.847759065

2

1.961570561

3

1.990369453

4

1.997590912

5

1.999397637

6

1.999849404

7

1.999962351

8

1.999990588

9

1.999997647

10

1.999999412

Table 2. Graph of infinite surd 2

image04.png

The value of bn approaches 2, but never reaches it (Refer to Figure 2 and Table 2 above).

Considering a generalization of an infinite surd, variables can be used to replace an integer.

image05.png

Let x be this value:

image06.png

image07.png

image08.png

image09.png


image10.png

image11.png

image13.png

Because the equation above is not factorable, the quadratic equation is used:

image14.png

image15.png

image16.png


image17.png

1+4k has to be a perfect square because the Pythagorean Theorem states thatimage18.png is an irrational if n is not a perfect square. Note that the solution of x should be positive (Refer to Figure 1 and 2). Thus, ignore the optional subtraction sign because it will make x negative. The value of x is an integer; the numerator should be even because it has to be divisible by 2. Because image19.png is added to 1, the value of image19.png

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Conclusion

Then, image28.png which thus is an integer.

We can test the validity of this equation by solving for k in the quadratic formula:

image17.png

image29.png

Table 3

k

x

0

1

2

2

6

3

12

4

20

5

30

6

42

7

56

8

72

9

90

10

Figure 3

image30.png

Figure 3 and Table 3 showfurther exemplify the equations image31.pngandimage32.png. There is direct relationship between k and x; when x increases, k increases. The limitations of the general equations are mainly based on the fact that x needs to be an integer. Thus 1+4k has to a perfect square and the square root should be odd in order for it to be divisible by two. The scope of k is infinite because the equation is based on a surd. The validity of the general statement of k only applies to even numbers that fit into the equation image31.png.  Figure 3 and Table 3 represent that the value of an infinite surd is not always irrational and can be integer.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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