• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Infinite Surds (IB Math SL portfolio)

Extracts from this document...


Infinite Surds

  • Introduction to Infinite Surds
  • Definition of a surd

- An irrational number whose exact value can only be expressed using the radical or root symbol is called a surd.

E.g.) image00.png is a surd, because the square root of two is irrational.

  • The origin of the word

          - In or around 825AD, Al-Khwarizmi who was an Arabic mathematician during the Islamic empire referred to the rational numbers as ‘audible’ and irrational as ‘inaudible’. Then, the European mathematician, Gherardo of Cremona, adopted the terminology of surds (surdus means ‘deaf’ or ‘mute’ in Latin) in 1150. In English language, the ‘surd’ appeared in the work of Robert Recorde’s The Pathway to Knowledge, published in 1551.

  • The symbol, use of the word radical

- The radical symbol image01.png depicts surds, with the upper line above the expression called the vinculum. Also, a cube root takes the formimage21.png, which corresponds to a1/3 when expressed using indices. So, all roots can remain in surd form.

  • A definition of infinite surds

    - An infinite surd is a never ending irrational number and its exact value would be left in square root form.

E.g.) The general infinite surd an = image31.png

 Therefore, a3 = image39.png

  • The following expression is an example of an infinite surd.


Consider this surd as a sequence of terms an

...read more.



Since a1 = image33.png and a2 =image34.png,

A2 can be written as image36.png

Therefore, an+1 = image37.png

  • The first ten terms of the sequence.


































  • Plot the relation between n and an


 I can see easily from the graph that the value of an approaches 2 but never reach it and the graph becomes less steep and the value of an does not increase as the value of n increases.

  • What does this suggest about the value of an+1 – an as n gets very large?





















As you can see from the table on the left, the differences between an+1 and an become smaller and rapidly approaching zero as n gets larger.

  • Find the exact value for this infinite surd.

 * Let an be X

      X = image51.png

      X2 = 2 + X

      X2 – X – 2 = 0

Use quadratic formula. image52.pngimage52.png

Therefore, X = image53.png

The value must be positive since the graph showed that there are no negative values, so disregard the negative sign.

Thus, X = image54.png = 2

  • Finding an expression for the exact value of following general infinite surd in terms k.

The general infinite surd image06.pngwhere the first term isimage07.png.

...read more.


  • Explain how you arrived at your general statement.

- First, I found a formula for the general infinite surd image06.pngwhere the first term isimage07.png. I considered this surd as a sequence of terms bn,so I could find that bn+1 =image08.png. Then, I could know from the calculation for the differences between an+1 and an that the differences between bn+1 and bn become smaller and rapidly approaching zero as n gets larger. After that, I substituted image08.pngfor bn+1 because the formula for bn+1 = image08.pngand I got image08.png= bn. I let bn be X, so I got X2 –X –K = 0 at the end. And then I used the quadratic formula to find the expression for the exact value of this general infinite surd which isimage10.png. After I got the expression for the exact value, I tried to find some values of k that make the expression an integer and I could realize that if X is an integer, the numerator has to be even because it is divided by 2. So, I found that image11.pnghas to be an odd perfect square to X be an integer. Since I found out this fact, I let m be any odd number thus, 1+4k = (m)2because the square of an odd number is also an odd number. Finally, I developed the equation and I arrived at my general statement which is k = image12.png.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    -?e91.��z�NO�<(tm)(�>���8l����#�H�"�3/4};�JmÚ´aK �ƶ^:��X=`��9�Î/��zȧ���2ryë­·n�é¦M2-z�''�HLe| "��1/4����� �8n7m��6zd���'��%X�\��~ڬ��dc��(r)]"��0�2I�m�|�(��X4�bB3/4E#...�8�b�XF��"�I��`�(c)�h�e��B QF�-V�����ZjU�ꧣK��Jl��VCjX<d���j�T:8��;��(tm)���?���[]�@2jA(�-��Jn�(��P'�E�(�<��|\}���N�\����LNw��g�� �_*�Q"���nE�.Cx��x#-� G"�# ��''["(c)9��;l\ 7����Ç��3/4Bv�>���Ou m3/4M�g�� "��W�B�m�o=��>M�U"V�`�0'O �����"��(tm)3� -K�...zj)��V�E�-;pf�Q,9>(w�F����)�-QK�7@�[�w��T�'6��3/4�n�'��j�O��#��(��..."U X...�Z@P.e#���.I\VV�.\�0����-t��8A�5tm\NX��a�e�ÅLe��v���G��4fÄ� �����3N<�D�1�c�-/_-<fA��F�*��]n��YQ t�d�<JxY���mB\�fÍ´iÓ´J�~����-�4- 3_�B�2�b�l?[��1/4�� �)�����zjV(c)#a{�JP�E� �8��_*�pã¤ï¿½ï¿½e�Ҧ�%K-�oK��٤"(c)�}��GF--(r)�J��C� ��"fo 4(r)�Un�Ȩ�H(tm)�7...��NÇ,���g�|(c)Rd"�#`�ܰOЪnm��(r)�!!2K�(tm)g����jßµ1/i��0��iÊ)�(c)�J�Ŀ�o&}sj(�0D�@�/���5��Ȩ���P(c)i�,D8h� Å�+E� Ȩ��p���TN�y�a���]ç¤ï¿½L@F-f����g*Ç-E�$Xa|�,T��5�V^\v�� �w(c)6��"��Z)>J�Ȩ��K`�(tm)2�O�z��!��c�[5�o��(tm)��'��_�Hl5�?��.�--�{Ý´"y}���2έ�+���Vrd�r(��`?G�"ih������[��5�Ȩ�I7ٲ�q�gEf�Vӱ�)e9��Zz�ÏZ��x-R ��Í7��S����Zz3/4/'Zz�S�%L"� �CX��J�:�"|�^�|"�T�"���f���6vIEw�Q��...�W�=S�J�S�l��nu!�d��i8|��3D�!G�Z�b"9�"�-QKrU�٭[�$j� ���(c)� �U�(����Z !S���+��j�@�[� �,?'QK��1j�_RÒ�`�@��p�b�4h� 5��N��"OO PO��D�-�4| Y릡3zj�zÏ=iÐ�)�Hq*, ^�>��Ln"��â[Ȩ��3��k�(r)ihI5m ��bÅj�P�L�QK��3��6�0v"m)w" ����4| uuu6l��G��s? ��"ZÛ¶m�l�'(c)'B J�PO� �1/2[&�2��� �Æ�W_�n3N@F����֭[;ßP `�`��...�^�K(c)% ��1/4jS�J �}��Z+(�'��dÔ�;�Õ��|3B5 s�ÎA�\�_'Qs^"�>�l���+�!�n�V+�(��\5j -$�i�k2g�P�...(�3/4��n���"�9���3F�+(r)>|��������oÌ1"&Mʸ�$6d-�40�&������Q[�lY�BS�N�3g��ٳ׭[�3/4S_"2u�(c)�CE5�9�}��lp��1ir�UWM�>�ݡ(i�3/4} 6����z���"lj�k56=��9<^hoy�g����v ���o~�'�|�'�Ê+�7o�]�l�2�FmÍ5� x�v1/2[�YPN6m7j�c��9�1/4a��ر�СC^- ...q�S��t�'-���lz}����N�1���Fm�ڵX(r)K/1/2��+�dC�1/2���1/2{wL���m۶����"ƨ'�ÉFY���'�OOѬ�$#JW3+pơ�=-?����_�$ -Z�8p�ßW����װn3/4��ݲ(tm)�W�^/��r$Ms�믿��|�Ν'pP!(r)h�b�L(tm)2t�P�R�I)�h.�(��tÓ²s�gD�yù�C�dӣ�pƨ��'O+C� a!�/o 4#�*l&�(tm):l��@....o� ��,_�>�VJ7�7θE� I}��)...(S�L"��d�T��XW����1/4ͲקO�x��Y��$���Pww�;��Èd�"��F�#�44h4k�lÓ¦M5�� �_9�% ����(c)

  2. Math IB SL BMI Portfolio

    When the functions are allowed to extrapolate beyond the data points provided, one can see the great difference between the two: As expected, the sinusoidal function curves down, increasing at the rate it is decreasing from its max point at Age = 20.

  1. Infinite Surds

    To help analyze this, one must think about what 4k means. According to 1+ 4k, 4k will always give us a positive number alone (4�2 = 8, 4�5=20, 4�8 = 32), however if we add one to this then we will always get a negative number (8+1 = 9, 20 + 1 = 21, 32 + 1 = 33).

  2. Math Portfolio type 1 infinite surd

    Plot the relation graph between n and an : (This graph I used the software Microsoft Excel 2008 to draw) Evaluation: (I repeat the entire process above) As the graph shown, as the n gets very large, the values of an still be the same.

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

    Finally, the value for where the value of the infinite surd is equal to 4 can be worked out in order to see if there is a clear pattern. The results are tabulated to see if there is a correlation between the value for and the integer value of the surd.

  2. Infinite Summation - In this portfolio, I will determine the general sequence tn with ...

    to check the validity of general statement written above. So let's take a = 1.5 and x = 8 a = 1.5 and x = 9 a = 1.5 and x = 10 a = 1.5 and x = 11 a = 1.5 and x = 12 Values of a and x are positive, x is increasing.

  1. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    The quartic equation however gives a much higher value of 330 centimeters; 3 metres and 30 centimeters high for the gold medal. Again, it seems based on past trends very unlikely that humans are able to break a world record of 330 centimeters, let alone 273 centimeters by 2016.

  2. Math SL Fish Production IA

    ________________ The quartic model that was developed earlier does not fit the new data; it does not connect any of the points. The data points form a curve and the total mass seem to increase at a constant rate from the 1st year onwards.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work