Infinite Surds

MATH PORTFOLIO

Infinite Surds

Introduction

Surds: a surd is a number that can only be expressed exactly using the root sign “” in other words it can be defined as a positive irrational number.

Thus, a number  is a surd if and only if:

1.   It is an irrational number

(b ) It is the root of a positive rational number.

The symbol  is called the radical sign. The index n is called the order of the surd and x is called the radicand.

For example:

√2 = (1.414.......) is a surd.

√3= (1.732......) is a surd.

√4= (2) is not a surd.

√5= (2.236....) is a surd.

Note: if n is a positive integer and a be a real number, then if a is irrational,  is not a surd. Again if  is rational, then also  is not a surd.

Surds have an infinite number of non-recurring decimal. Hence, surds are irrational numbers and are considered infinite surds.

Following expression is the example of an infinite surd:

Considering this surd as a sequence of terms  where:

=

=

= etc.

Q: to find a formula for  in terms of

=  = 1.4142...

=     or  = 1.5537....

=   or = 1.5980....

As it is observed that = ; therefore, it can be understood that  =  as the trend has been observed as this until now.

Hence,

=  = 1.6118...

=  = 1.6161...

=  = 1.6174...

=  = 1.6178...

=  = 1.6179...

=  = 1.6179...

=  = 1.6179...

[NOTE: VALUES OBTAINED USING THE GRAPHICAL DISPLAY CALCULATOR: CASIO fx – 9860G]

From this graph I understand that after the value keeps on increasing so does the value become similar i.e. 5 onwards the value is ...