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

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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
 .

Part one:

Consider the surd where:



Etc.

To begin, a recursive rule was determined from observing what changed from  to  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.

The decimal values were calculated for to  and the results were then graphed.

It can be observed that the slope of the function gradually decreases asymptotes towards a value of. The values of  suggest that as  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.

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 The exact value of the infinite surd   is.

Only the positive is used as you cannot have a real negative surd. In addition,  1.61803398 which is very close to the value for  however the gaps between the values for  become smaller and the difference between  and  is approximately -3.72528E-06 which is a minute change compared to the gap between  and the value at which . These results support the answer which was provided algebraically due to the small margin between the calculated and algebraic results.

Part Two:

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