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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.) 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 depicts surds, with the upper line above the expression called the vinculum. Also, a cube root takes the form , 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 = Therefore, a3 = • The following expression is an example of an infinite surd. Consider this surd as a sequence of terms an

Middle

n.

Since a1 = and a2 = ,

A2 can be written as Therefore, an+1 = • The first ten terms of the sequence.
 Term Surd Decimal a1 1.847759065 a2 1.961570561 a3 1.990369453 a4 1.997590912 a5 1.999397637 a6 1.9998494404 a7 1.9999849404 a8 1.999990588 a9 1.999997647 a10 1.999999412
• 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?
 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 = X2 = 2 + X

X2 – X – 2 = 0

Use quadratic formula.  Therefore, X = The value must be positive since the graph showed that there are no negative values, so disregard the negative sign.

Thus, X = = 2

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

The general infinite surd where the first term is .

Conclusion

• Explain how you arrived at your general statement.

- First, I found a formula for the general infinite surd where the first term is . I considered this surd as a sequence of terms bn,so I could find that bn+1 = . 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 for bn+1 because the formula for bn+1 = and I got = 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 is . 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 has 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 = .

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

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