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Infinite Surds (IB Math SL portfolio)

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Introduction

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.

image50.png

Consider this surd as a sequence of terms an

...read more.

Middle

n.

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.

Term

Surd

Decimal

a1

image38.png

1.847759065

a2

image40.png

1.961570561

a3

image41.png

1.990369453

a4

image42.png

1.997590912

a5

image43.png

1.999397637

a6

image44.png

1.9998494404

a7

image45.png

1.9999849404

a8

image46.png

1.999990588

a9

image47.png

1.999997647

a10

image48.png

1.999999412

  • Plot the relation between n and an

image49.png

 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?

Terms

Difference

a2-a1

0.113811496

a3-a2

0.028798892

a4-a3

0.007221459

a5-a4

0.001806725

a6-a5

0.0004518034

a7-a6

0.0001355

a8-a7

0.0000056476

a9-a8

0.000007059

a10-a9

0.000001765

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.

Conclusion

  • 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.

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