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Math Project Infinite Surds

Extracts from this document...

Introduction

Regina Nieuwenburg – Maths SL Type 1

2008

Project 1

Regina Nieuwenburg

Pages:  18

[REGINA NIEUWENBURG – MATHS SL TYPE 1]


Title Page

Infinitive Surds

Regina Nieuwenburg

Maths SL Type 1

Candidate Number:

I.B.

Contents Page

Title Page                                                                                             Page 2

Contents Page                                                                                              Page 3

Introduction                                                                                              Page 4

Terms, scopes and limitations                                                                              Page 5

Investigation:

  • The formula of an+1 in terms of an                                                                    Page 6
  •  General statement that represents all the values of K for which

 the expression is an integer                                                                     Page 14

Conclusion                                                                                            Page 17

Bibliography                                                                                                  Page 18

Introduction

In this project I will investigate the infinite surds. A surd is a number that cannot be changed into a fraction. They go on infinitely without any pattern. They are usually a square root of a number.

My project will include two formulas. One is the formula of an+1 in terms of an.

...read more.

Middle

=1,999990588

A10=√(2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+(√2+√2)))))))))

= 1,999997647

image07.png

Un =3

A1=√(3+√3)

= 1,732050808

A2=√(3+(√3+√3))

= 2,175327747

A3=√(3+(√3+(√3+√3)))

 = 2,274934669

A4=√(3+(√3+(√3+(√3+√3))))

 = 2,296722593

A5=√(3+(√3+(√3+(√3+(√3+√3)))))

 = 2,301460969

A6=√(3+(√3+(√3+(√3+(√3+(√3+√3))))))

 = 2,302490167

A7=√(3+(√3+(√3+(√3+(√3+(√3+(√3+√3)))))))

 = 2,302713653

A8=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3))))))))

= 2,302762179

A9=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3)))))))))

 =2,302772715

A10=√(3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+(√3+√3))))))))))

=2,302775003

image11.png

Un =4

A1=√(4+√4)

= 2,000000000

A2=√(4+(√4+√4))

 = 2,449489743

A3=√(4+(√4+(√4+√4)))

 = 2,539584561

A4=√(4+(√4+(√4+(√4+√4))))

 = 2,557261144

A5=√(4+(√4+(√4+(√4+(√4+√4)))))

 = 2,560714967

A6=√(4+(√4+(√4+(√4+(√4+(√4+√4))))))

 = 2,561389265

A7=√(4+(√4+(√4+(√4+(√4+(√4+(√4+√4)))))))

 = 2,561520889

A8=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4))))))))

= 2,561546581

A9=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4)))))))))

=2,561551596

A10=√(4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+(√4+√4))))))))))

 =2,561552575

image15.png

K=5

A1=√(5+√5)

= 2,236067977

A2=√(5+(√5+√5))

= 2,689994048

A3=√(5+(√5+(√5+√5)))

= 2,773083852

A4=√(5+(√5+(√5+(√5+√5))))

= 2,788025081

A5=√(5+(√5+(√5+(√5+)√5+√5)))))

= 2,790703331

A6=√(5+(√5+(√5+(√5+(√5+(√5+√5))))))

 = 2,791183142

A7=√(5+(√5+(√5+(√5+(√5+(√5+(√5+√5)))))))

 = 2,791269092

A8=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5))))))))

= 2,791284488

A9=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5)))))))))

=2,791287246

A10=√(5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+(√5+√5))))))))))

 =2,791287740

image17.png

Un =100

A1=√(100+√100)

= 10,000000000

A2=√(100+(√100+√100))

= 10,488088482

A3=√(100+(√100+(√100+√100)))

= 10,511331432

A4=√(100+(√100+(√100+(√100+√100))))

= 10,512436988

A5=√(100+(√100+(√100+(√100+(√100+√100)))))

= 10,512489571

A6=√(100+(√100+(√100+(√100+(√100+(√100+√100))))))

= 10,512492072

A7=√(100+(√100+(√100+(√100+(√100+(√100+(√100+√100)))))))

= 10,512492191

A8=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+√100))))))))

= 10,512492197

A9=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+√100)))))))))

=10,512492197

A10=√(100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100+(√100))))))))))

=10,512492197

image19.png

Uk= √(k+(√k – Uk+2))= √(k+(√k+√k))

 = (√(k+√k)) – (√(k+(√k+√k))) = Uk-1

= √2.

Ak – ak+2 = ak-1        

The general statement that represents all the values of K for which

The expression is an integer

U1

...read more.

Conclusion

- X-K=

image06.pngimage06.png1+√1)=√2

e.g.

image08.pngimage08.pngimage09.pngimage09.png4

=2

Working steps

image10.pngimage10.png(K+image10.pngimage10.pngK)=image10.pngimage10.png4

=2

image12.pngimage12.pngK = 4

K=(4image13.pngimage13.pngK)2

=(4image13.pngimage13.pngK) X (4image13.pngimage13.pngK)

=16image14.pngimage14.pngKimage13.pngimage13.png4K+K2

=16image13.pngimage13.png8K+K2

Given as a formula 16= K

image13.pngimage13.png8K = X

K2 = X2

image16.pngimage16.pngX2image13.pngimage13.pngXimage13.pngimage13.pngK=0

Plot the formula in the graphic calculator in math image16.pngimage16.pngsolver image18.pngimage18.png enter image18.pngimage18.png green alpha image18.pngimage18.png enter

=12.468871125

K=12.468871125

=image20.pngimage20.png12.468871125 +(image10.pngimage10.png12.468871125)) =image10.pngimage10.png4

Excel evidence

image21.png

Conclusion

The formula of an+1 in terms of an =

Un+1=√(K+Un)Which is explained in the first chapter.

The general statement that represents all the values of K for which the expression is an integer   =      

X2image13.pngimage13.pngXimage13.pngimage13.pngK=0 Which is explained in the second chapter.

Bibliography

-Mr. Lock (teacher)

-I.B. SL Study book – Mathematics for the international student – Third edition – Dave Hall, Rob Jones, Carlo Raffo, Ian Chambers and Dave Gray – British library Cataloguing Data

-Calculator –Texas TI 83

-http://en.wikipedia.org/wiki/Nth_root

-http://www.mathhelpforum.com/math-help/general-high-school-math-help/46681-infinite-surds-expression-exact-value-integer.html

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...read more.

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