Infinite Surds

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Math SL

February 8, 2008

Part 1: The Infinite Surd

1.1 Patterns with

In this section, we are looking for the exact value of the infinite surd . In

order to do this, we are considering a sequence of surds of the form:

etc.

Investigating these terms reveals a pattern. Since and , it

follows that .

Using a GDC, I found the following values for the first ten terms of the sequence.

1 1.414213562

2 1.553773974

3 1.598053182

4 1.611847754

5 1.616121207

6 1.617442799

7 1.617851291

8 1.617977531

9 1.618016542

10 1.618028597

I used Corel Quattro Pro to produce the following graph of the relation between n and .

n

The term values seem to be approaching approximately 1.61803 (also known as the golden

number or ). The difference between a term and its predecessor becomes smaller and smaller

as n gets larger. This suggests that as n gets very large the difference will get smaller

and smaller until it is effectively zero.

In other words, as n approaches infinity , and therefore .

Since , we can use our formula for to find the value of the

infinite surd.

1.2 Finding the exact value of the infinite surd

Let with a very large n value. We could also say that , since as n

approaches infinity.

We know that , so as n approaches infinity:

We can use the quadratic formula to solve for x (which is the value of the infinite surd).

Since the value of the surd cannot be negative, we reject . Therefore, the value of the

infinite surd is . Since , this agrees with our

previous observations about the sequence and seems reasonable.

Part 2: The Infinite Surd

2.1 Patterns in the sequence

In this section, we are looking for the exact value of the infinite surd .

In order to do this, we are considering a sequence of surds of the form:

etc.

As in the first investigation, there is a pattern.. Since and , it

follows that .

Using a GDC, I found the following values for the first ten terms of the sequence.

1 1.847759065

2 1.961570561

3 1.990369453

4 1.997590912

5 1.999397637

6 1.999849404

7 1.999962351

8 1.999990588

9 1.999997647

10 1.999999412

Once again, I used Corel Quattro Pro to produce the following graph of the relation between n

and .

n

The term values seem to be approaching 2. Again, the difference between a term and its

predecessor becomes smaller and smaller as n gets larger.

2.2 Finding the exact value of the infinite surd

As with the previous example, let with a very large n value. We could also say that

, since as n approaches infinity.

We know that , so as n approaches infinity:

Factoring, we get which yields solutions of and . Again,

since the value of the surd cannot be negative, we reject -1. Therefore, the value of the infinite

surd is 2. This agrees with our previous observations about the

sequence in the both the graph and table in section 2.1.

Part 3: The Infinite Surd

As in the previous two examples, we will consider a sequence of surds. This time they will take

the form:

etc.

It is important to note that this form only applies to if we restrict ourselves to real number

solutions. Also, if , then the value of the infinite surd is 0. Therefore we will only

consider k values that are greater than 0.

This time the pattern is . Since the difference between and will get

very small as n gets large, it follows that as n approaches infinity.

Let for some very large n. We know that , so as n approaches

infinity:

Using the quadratic formula, we get the solution . Since the negative solution is not

possible, the value of the infinite surd is for all (an

explanation for not including 0 will follow in section 5).

Part 4: Infinite surds that are integers

We know that is an integer for . The infinite surd will have an

integer value for any k which yields an integer in the expression .

4.1 k values whose infinite surds are integers

We can begin by finding the k value whose infinite surd is equal to 0. It is immediately obvious

that yields a result of 0. Therefore, is the

integer 0 when .

Next, we will find the k value whose infinite surd is equal to 3. Since the value of the infinite

surd is given by , we can find the k that results in an infinite surd with a value of 3 by

solving the following equation.

Therefore, the infinite surd .

Lets find the k value whose infinite surd is equal to 4.

Therefore, the infinite surd .

Lets find the k value whose infinite surd is equal to 5.

Therefore, the infinite surd .

4.2 The general statement for all k whose infinite surds are integers

So far, we have found that the k values 0, 2, 6, 12, 20 yield integers in the expression

. These numbers follow a pattern that can be outlined in the

general statement for (or where n is a positive integer).

4.3 Testing the general statement

The previous general statement was obtained from a pattern of only 5 numbers. Lets check to

see if it holds true for other n values. We will check the next two numbers in the pattern, and

, then a larger number, .

4.3a Testing

Since , we will check it in the expression . We want to make sure that the

expression yields an integer. Lets check by substituting 30 for k.

Since the result is an integer, and the general statement holds

true.

4.3b Testing

Since , we will check it in the expression . Again, we want to make sure that

the expression yields an integer.

Since result is an integer, and the general statement holds true.

4.3c Testing

This time so we will check to see if the expression yields an integer.

The result is an integer. Therefore and again the general

statement works.

4.3d Testing

If we test , then where n is a positive integer. We will simplify the expression

where .

The expression simplifies to n. This means that if k is obtained from the general term, the

corresponding infinite surd will yield an integer. Apparently, our general statement holds true for

all n that are positive integers.

The general term yields infinite surds that are integers for all positive integer

values of n.

5 Scope and limitations of the general statement

The general statement gives us all k values where is an

integer. However, these k values do not all work in the expression .

5. 1 Investigating

The infinite surd is obviously equal to 0. Although, the k value of 0 is

included in my general term , its infinite surd is not accurately described by the

expression . If we substitute 0 for k we get the following result:

Since is not equal to 1, the expression does not describe the

value of this infinite surd.

It is interesting to note that the (initially rejected) expression does represent the value of

this infinite surd. However, this is the only surd for which is appropriate.

Again, since the value of is immediately obvious, it seems unnecessary

to modify the expression for the sake of this surd. Instead, we should say that

is an integer when , where and that

the surd:

i) has a value of for all positive values of k

ii) has a value of 0 if

iii) is undefined if

5.2 Simplifying when it yields an integer

A pattern seems to emerge for the value of the infinite surd obtained from the general statement.

If we use (or 30) for k, then the corresponding infinite surd has a value of 6. If we use (or

12) for k, then the corresponding infinite surd has a value of 4. If we use (or 420) for k, then

the corresponding infinite surd has a value of 21. It seems that when the value

of the infinite surd will be . Again, an exception occurs for (the k value of zero)

We have shown that this is true for several specific n values in section 4.3a, 4.3b, and 4.3c. In

4.3d, we showed that the expression simplifies to n as long as k takes the form

. In other words, section 4.3d shows that as long as k is obtained from the general

term, the infinite surd . We must stipulate that n needs to be an

integer greater than 1 because, as we outlined in section 5.1, there is an exception for (the k

value of 0).

5.3 Arriving at the general statement

In section 4.2 I found that the k values of 0, 2, 6, 12, 20 gave integer values for their respective

infinite surds. Initially, I used a guess and test method to find a pattern and hence a general

term. I noticed that , , , , and .

Generally, the pattern was . Testing this general statement in section 4 showed

that it seemed to work for all positive integer values of n. However, since I obtained my general

statement from just a few examples, I did investigate another method for finding the general

term.

If we want to find the k values that give us infinite surds that are integers we could set up an

equation , where n is a postive integer. Now we can solve for k.

This means that for an infinite surd to have an integer value, its k must be of the form

where n is a positive integer, hence the general statement , for

.

5.4 Rational infinite surds

Just as some integer values of k yield integer results for , we expect that

some rational values of k will yield rational results for the infinite surd.

Lets consider this sequence:

If we continue the sequence for the first ten terms, then we get the following decimal

approximations using Microsoft Excel.

n

1 0.666666667

2 1.054092553

3 1.224147458

4 1.291739874

5 1.317643472

6 1.327436596

7 1.33112022

8 1.332503157

9 1.333021981

10 1.333216571

It appears that our values are approaching or the rational number . We can use the

expression to find the exact value of the infinite surd .

If we substitute for k we get . This

confirms our suspicion that the infinite surd .

Now we can apply the same method we used in section 5.3 to determine a formula for rational

infinite surds. A rational number takes the form where a and b are both integers. We

will only consider positive rational numbers since negative numbers do not have a square root

and we have already dealt with 0. If we want our infinite surd to have a rational value, then

must be rational.

Let be a positive rational number. We can solve the following equation to determine which

types of k values will yield infinite surds that are rational.

According to our work above, any positive k value of the form where a and b are positive

integers should yield a rational value for . In order for to be

positive, a must be greater than b.

We can test this by trying some values for a and b. Lets try and . Since

, this gives us a k value of .

The following values were obtained from Microsoft Excel for the corresponding sequence

.

n

1 0.748331477

2 1.143823184

3 1.305305782

4 1.365761979

5 1.387718263

6 1.395606772

7 1.39843011

8 1.399439213

9 1.399799704

10 1.399928464

According to the formula , the infinite surd . Since the

sequence is approaching (or 1.4), our formula for finding an infinite surd that is a rationalnumber seems to work.

To sum up, an infinite surd is a rational number if ,

where and . The set of integer solutions is contained in this statement sinceinteger is simply a rational number with a denominator (or b value) of 1.

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