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# Infinite Surds

Extracts from this document...

Introduction

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.

Middle

x=   OR x=

x =       (x cannot have a negative value.)

Q: Considering another infinite surd  where the first term is  , to repeat the entire process and to find the exact value for this surd.

=  = 1.8477...

=   OR   =  = 1.9615...

=    OR =  = 1.9903...

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

Hence,

→   =  = 1.9975...

=   = 1.9993...

=  = 1.9998...

=  = 1.9999...

=  = 1.9999...

=  = 1.9999...

=  = 1.9999...

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

 N 1 1.8477 2 1.9615 3 1.9903 4 1.9975 5 1.9993 6 1.9998 7 1.9999 8 1.9999 10 1.9999

From the above graph we realize as the value starts to increase after a certain number (1.999) it stabilizes and remains almost the same through out the graph. As we can see after 5 the value continues to remain the same till 10 and onwards. The general idea in both the question

Conclusion

- x – k=0 we get:

-x-30=0

-6x+5x -30 =0

→ x(x-6) +5(x-6) =0

→ (x+5) (x-6) =0

Thus, x=6 OR x=-5

Hence, proved.

1. Supposing  n =5

Then, on substituting 6 as the value of n in the general statement we get:-

= 6+6^2 = 42

k=42

Now substituting the value of k in the equation  - x – k=0 we get:

- x – 42=0

-7x+6x -42 =0

→ x(x-7) +6(x-7) =0

→ (x+6) (x-7) =0

Thus, x=7 OR x=-6

Hence, proved.

HENCE, IT CAN BE CONCLUDED THAT EVEN AFTER USING OTHER VALUES OF “k” THE ANSWER OBTAINED IS AN INTEGER WHICH PROVES THE OBTAINED GENERAL STATEMENT VALID.

The scope of the general statement is that the need to always obtain an integer value as the answer is always satisfied by this general statement. The limitation of the general statement is that the use of a negative integer is invalid.

CONCLUSION:

By this I conclude that them I learnt how to use the surds and the correct time to use them.

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

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2. ## Infinite Surds

--> --> --> We only use the positive values for "a" because we have observed this throughout the pattern. The general statement that represents the values of k for when the expression is an integer is. The reason for which one does not use the other answer to this formula

1. ## Infinite Surds. The aim of this folio is to explore the nature of ...

values of was that the original surd is added to a 2 and then square rooted much like the first infinite surd which was provided. A table of values is composed for values of n in addition to a graph to identify possible asymptotes.

2. ## Infinite Surds Investigation. This graph illustrates the same relationship as was demonstrated in the ...

the sequence are: n an 1 1.847759065 2 1.961570561 3 1.990369453 4 1.997590912 5 1.999397637 6 1.999849404 7 1.999962351 8 1.999997647 9 1.999999412 10 1.999999853 The graph for these values is: It is interesting that the curves of both graphs look very similar.

1. ## Math Portfolio type 1 infinite surd

19 1.618033989 20 1.618033989 21 1.618033989 22 1.618033989 23 1.618033989 24 1.618033989 25 1.618033989 Notice: The formula to calculate the second column: B2=SQRT(1+SQRT(1)) then following to the next values will be B3=SQRT(1+B2) etc... Plot the relation graph between n and an: (This graph I used the software Microsoft Excel 2008 to draw)

2. ## Infinite surds portfolio - As you can see in the first 10 terms of ...

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 ...

1. ## Infinite Surds Coursework

a2 = 1.9615705608 a3 = 1.9903694533 a4 = 1.9975909124 a5 = 1.9993976374 a6 = 1.9998494037 a7 = 1.9999623506 a8 = 1.9999905876 a9 = 1.9999976469 a10 = 1.9999994117 Using technology, plot the relation between n and an. Describe what you notice.

2. ## Infinite surds Maths Portifolio

bn can be shown: As can be seen from this graph is that the value of bn increases as n gets larger.

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