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Math Surds

Extracts from this document...

Introduction

Infinite Surds

Mathematics Portfolio SL Type 1

    November 10, 2008

Although there is no significant application of infinite surds in our daily lives, infinite surds can be classified as a recursion which means that the function defined gets used in its own definition.  See below for an example.

image00.png

http://www.vdschot.nl/jurilog/images/metro.jpg

and this humorous definition of recursion:image01.png

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

A surd is defined as an irrational number which can only be expressed with the root symbol and has an infinite number of non-recurring decimals.

The following expression is an example of an infinite surd: image62.png

This surd can be considered as a sequence of terms image16.pngimage16.png where:

image79.png

image84.png

image88.png

image94.png

etc.

A pattern is evident where for every consecutive term in the sequence a new (+image02.pngimage02.png ) is added to the end of the previous expression under all the previous roots. The formula for image03.pngimage03.png in terms of image16.pngimage16.png is therefore:

The first eleven terms of the sequence to eleven decimal places including image34.pngimage34.png are as follows:

image48.png

image59.png

image61.png

image63.png

image64.png

image65.png

image66.png

image67.png

image68.png

image69.png

image70.png

The first term in the sequence is actually image71.pngimage71.png since for every consecutive term a new (+image02.pngimage02.png ) is added on. Individual terms of an infinite surd are not necessarily surds themselves as  image72.pngimage72.png  is equal to 1 which disqualifies it from being a surd.

Plotting the first eleven terms including image34.pngimage34.png on a graph reveals the behavior of the expression.

...read more.

Middle

This surd can once again be considered in terms of image16.pngimage16.png where:

image89.png

image90.png

image91.png

image92.png

etc.

A pattern is once again evident where for every consecutive term in the sequence a new (+image93.pngimage93.png ) is added to the end of the previous expression under all the previous roots. The formula for image03.pngimage03.png in terms of image16.pngimage16.png is therefore:

The first eleven terms of this sequence to eleven decimal places including image34.pngimage34.png are as follows:

image95.png

image96.png

image97.png

image98.png

image99.png

image100.png

image101.png

image102.png

image103.png

image104.png

image105.png

Plotting the terms on a graph reveals a similar behavior as seen in the previous surd. The following graph illustrates the relation between n andimage73.pngimage73.png.

image106.png

After the first five terms the data points increase by less and less suggesting an asymptote at 2. It is also clear that further points will not be much larger than image12.pngimage12.png as the points are already on a horizontal trend. Also, as the values get larger the values of image16.pngimage16.png and image03.pngimage03.png will be nearly identical with a difference that is too miniscule to be significant.

This realization allows us to actually make them the same variable (x) in the previously deduced formula:

image04.png

image05.pngimage05.pngGiving us a new formula:

We can further simplify it:

image06.png

By squaring the equation, the negative solution is removed because a negative multiplied by a negative gives a positive.

image07.png

image08.pngimage08.png

The variable can be isolated from the rest of the equation with the use of the quadratic formula:

image09.png

...read more.

Conclusion

The second major limitation is the inability of the formula to produce negative integers.

 For example:

image53.png

image54.png

image55.png

image56.png

image57.png

image58.png

The reason the formula does not work in producing negative integers is because the negative side of the equation at one point gets squared changing it to a positive.

The underlying reason for the inability of producing negatives is the fact that there cannot possibly be an infinite surd involving roots that produces negative integers. A root of a negative value gives an imaginary number which is part of the complex number system. This is because no two numbers of the same sign can be multiplied together to give a negative number.

The previously discussed pattern of increasing difference between k values that give integers in the answer can be easily justified.

For example: 0x1=0, 1x2=2, 2x3=6, 3x4=12, and 4x5=20.

It is clear that the difference in the products also increases by two as in the product of 0 and 1 is 0, and the product of 1 and 2 is 2. The difference between the two products is 2. The difference between the products of the next two consecutive integers then has a difference of 4.

The fact that this pattern exists, supports my claim that for an integer to be produced, k has to be the product of two consecutive integers in:

image27.png

This formula appears to work with unlimited numbers as long as k is a product of two consecutive integers:

image60.png

...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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