# Infinite surds portfolio - As you can see in the first 10 terms of the infinite surd, they are all irrational numbers.

MATH PORTFOLIO

INFINTE SURDS

Submitted By Tim Kwok

Math 20 IB

Presented To Ms. Garrett

April 27, 2009

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.  When the infinite surds in those sequences are simplified, they are allowed to be rational or irrational unlike a surd.

The following is an the first example of an infinite surd:

You may be wondering why this can be classified as a surd when  is not a surd.   simplified forms a natural number and cannot be classified as a surd.  As you can see in the first 10 terms of the infinite surd, they are all irrational numbers.

a1:                                                                = 1.414213 ...

a2:                                                                = 1.553773 ...

a3:                                                        = 1.598053 ...

a4:                                                = 1.611847 ...

a5:                                                = 1.616121 ...

a6:                                        = 1.617442 ...

a7:                                = 1.617851 ...

a8:                                = 1.617977 ...

a9:                        = 1.618016 ...

a10:                   = 1.618028 ...

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:

From the plotted points of the infinite surd  in graph 1, you can see that the greater n increases, the closer an gets to the value of about 1.61803. But an never touches ...