The Golden ratio, Fibonacci series and Continued fractions

8/9/2019 The Golden ratio, Fibonacci series and Continued fractions

1/23

The Golden Ratio and Continued

Fractions

Srinivasan Nenmeli Ph D

Introduction

The aim of this article is to introduce Continued

Fractions to high school students. Fibonacci series and

the Golden ratio [phi = 1.618......] are convenient

starting points for this exploration of continued

fractions.

Unfortunately 'Continued Fraction' [CF for short ] is

not taught in schools and do not find a place in text

books of high school math or even undergraduate texts.

This subject is of fundamental importance in number

theory and finds many fascinating applications

,especially in computer algorithms for approximations

and electrical networks.

Many noted mathematicians like Euler, Gauss and

Lagrange have contributed to this topic ; the formal


2/23

representation is due to John Wallis. John Wallis

[1616-1703] coined the term "Continued Fractions".

Legend has it that the Indian mathematician and

astronomer Aryabhata [5th century AD] first used CF for

solving linear Diaphantine equation.

We shall examine many interesting approximations to

pi , 'e' [the base of the natural logarithm] and finding

square roots and some applications in this article.

Fibonacci Series and the Golden Ratio

You may be familiar with the Fibonacci series, due to

Fibonacci [Leonardo of Pisa] :

0,1,1,2,3,5,8,13, 21, 34, 55,89, 144 .......

The pattern is 'obvious'; we add the previous two

numbers ,starting with 0 and 1.

If F(n) is the nth Fibo number, then

F(n) = F (n- 1) + F(n-2) -----------(1)

[Fibonacci developed the series by counting the number

of rabbits that would be born in an idealized breeding

pattern!]


3/23

Let us take the ratio of any two adjacent numbers:

F(n+1)/F(n) -->

8/5 = 1.6

34/21= 1.619

144/89 = 1.6179775

We can round off this ratio to 1.618 for all practical

computations with a calculator.

You will see that this ratio tends towards the Goldenratio phi = 1.618 or the limit of F(n)/F(n-1) goes to

phi.

The Golden ratio was the favorite ratio used by Greek

architects for aesthetic appeal; for instance , the

Parthenon in Athens had a height /width ratio close to

1.62. Several flower petals follow a spiral governed by

the Golden Ratio. Seurat painted a picture with golden

ratio in various blocks inside the picture.

Note that phi is an irrational number, like pi,'e' and

√2 and other square roots. Irrational numbers cannot be

expressed as ratio p/q where p and q are two integers.

Of course, some approximations can be found; for


4/23

instance we often replace pi = 22/7 =3.14285 as being

close to pi = 3.14159-- an approximation you use in

middle school. Some ancient astronomers employed

square root of 10 ,√10 = 3.1623 as an approximation for

pi--- an error of 0.02 .

We shall handle several irrational numbers using

'continued fractions' or CF in this article.

Quadratic Equations and Continued Fraction

The Golden ratio of phi is actually the positive root

of this deceptively simple quadratic equation:

x 2 - x - 1 =0 ---------(2)

Using the quadratic formula,

we get x= 1+ √(5)/ 2 = 1.618......

Note that sqrt(5) is an irrational number and so you can

write that number to several decimal places.

Let us rewrite equation (2) as follows:

x 2 = 1 + x

Divide this equation by x on both sides:

x = 1 + 1/ x -------------(3)


5/23

Equation (3) is the clue to Continued Fraction method!

Replace the 'x' in the denominator on the right side by

the same expression!

x = 1 + 1

--------

1+ 1/x

Well, you can continue doing this ,forming a " Continued

Fraction" for phi, which appears like a stair case:

x = 1 + 1

------

1 + 1

-------

1+ 1

------

1+ 1/x --------(4)

-----and so on.

This infinite expansion is called 'Continued Fraction

Expansion' or CFE.

This simple continued fraction for phi can be written as

follows:


6/23

phi = {1; 1,1,1,1,1,....} or simply

phi = { 1; ] --------(5)

As you can appreciate, this CF is the simplest of

continued fractions out there: with just '1' s only.

The numbers 1 repeating here are called 'partial

quotients.

Approximations to 'phi':

Now let us 'fold' or contract the CF by working

backwards from the last step in the stair case:

From equation (4) we get:

phi = 1 + 1

--------

1+ 1

------

1 + 1

------

1

phi = 1 +1


7/23

-------

1 +1

-----

2/1

phi= 1+1

-----

3/2

phi= 1+ 2/3 = 5/3 = 1.666

This, of course, is a poor approximation to phi [1.618]

but then I took only four steps down the stair case to

get this number.

Exercise : Repeat the process to find the approximation

to phi by taking 8 steps in the Continued fraction.

Answer: The answer is simple: you will get:

phi ~ 34/21 = 1.619, the two numbers from the Fibonacci

sequence down the series!

The numbers in Fibonacci series give the approximate

ratio of p/q for phi as found from the continued

fraction expansion or CFE. There is no surprise here.

You can see how the golden ratio is tied to a


8/23

Quadratic equation which is related to Fibonacci series

and how the Golden ratio can be expanded as a continued

fraction.

This CF with '1's as partial quotients is helpful and is

the simplest there is.

Can we find similar continued fractions for pi, e

,sqrt(2) and so on.

Well, if you find a new Continued fraction for a number

,your name would be etched into the hall of fame of

mathematics.!

The general pattern for CF

In general, a CF is written as follows:

x = a(0) + b(1)

-------------

a(1) + b(2)

---------

a(3) + b(3)

----------

a(4) + b(4)

---------


9/23

and so on.

The numbers b(1),b(2) ... can be 1, in which case we get

a "simple Continued Fraction" , as is the CF for phi.

The numbers a(1) a(2),a(3) ... are called partial

quotients.

We can express a CF as follows:

x = {a(0); a(1),a(2),....,b(1),b(2)......}

If the 'a' s repeat , write simply a with a bar over

that: .

Exercise

1 Solve the equation x 2 -5x-1 =0 and find the

Continued Fraction expansion for the positive root.

[Ans: x = {5; 5,5,5,5,5...} = 5.0495097....

2 Find the approximation using three partial quotients.

A few mixed ratios and their continued fractions

To get our feet wet for CF, let us examine a few

simple mixed ratios and expanding them into CF's.

Example 1 : x= 29/19

29/19 = 1 + 10/19 = 1+ 2 (5/19)


10/23

= 1 + 2

-------

3+ 1

------

1 + 4/4

We are done! x={1;2,3,1}

These are called "finite CF" because the steps terminate

after k number of steps.

Example 2 : x = 415/93

Find the continued fraction expansion.

415/93 = 4 + 43/93

1/93/43 = 1/ 2+7/43

7/43 = 1/ (43/7) = 1/ 6+ (1/7)

We are done: x = {4;2,6,7]

These are CF s with finite terms or quotients and

rather easy to develop.

Example 3: Expand pi = 3.1416 into CF

pi = 31416/10000 = 3927/1250

= 3 + 177/1250


11/23

1250/ 177 = 7 + 11/177

177/11 = 16 + 1/11

Therefore pi ~ 3.1416 = { 3; 7,16,11}

Taking the first-order approximation, pi = 3 + 7/16=

55/16=

3.4375----not accurate enough !

Pi and its continued fractions

pi is the ratio of the circumference of a circle to its

diameter.

π = circumference of a circle/ diameter.

pi is a irrational number and finding its fairly

accurate value to a few decimal places have intrigued

many minds.

One method is to inscribe a regular polygon of n-sides

in a circle and find its perimeter. The perimeter is a

close approximation to the circumference.

You can inscribe a square or octagon and keep going for

higher number of sides. Archimedes, the intrepid

mathematician, went up to 96 sided polygon or 96-gon and


12/23

found pi is approximately 22/7 . You may still use this

ratio in the middle school problems.

Remember that Archimedes had to find square roots in

every step and did not have the decimal system and hadno pocket calculator either!

Ptolemy did the same with 360 sided polygon and got pi=

3.1416.

A Chinese mathematician [430-501] used the ratio 355/113

= 3.14159292.

The Indian mathematician Bhaskara I [1114-1185] found

the ratio 3927/1250 = 3.1416.

[See the previous section in this article; most probably

both the Chinese and the Indian mathematicians used

Continued fraction expansion ,as we would show shortly.]

The modern version of CF for pi is as follows:

π = [3; 7,15,1,292,1]

This requires some computation to get the value of pi.

Let us 'fold' this expansion up to the third step in

this staircase:

pi= 3+ 1/ 7 + 1


13/23

-------

15 + 1/1

= 3 + 1/7 + 1/16/15

= 3 + 1

---------

7 + 15/16

= 3 + 1/ (127/16) = 3 + 16/127 = 3.12598

= 397/127.

This is not a good approximation. [The error =

3.1416-3.1259 = 0.02 or 0.02 x100/ 314= 0.006% .

That means that the CF expansion does not converge fast;

we have to include more quotients in our folding

process.

There are other better CF's for pi. One elegant CF was

given by

L J Lange in 1999 .

His expansion is as follows:

4/ pi = 1 + 1

------

3 + 4


14/23

-------

5 + 9

------

7 + 16

--------

9 + 25

-----

and so on.

We had given earlier pi ~ 355/113 = 3.14159292 as a

useful one. The error is 0.000000266

The next approximation is : pi~ 103993/33102 ;

Check this ratio with your pocket calculator:

pi~ 3.141592653

My CASIO pocket calculator gives '4' in the last decimal

place! { most probably a pocket calculator uses such a

ratio in its software!]

Srinivasa Ramanujan [1887-1920] , a self-taughtmathematician from India, gave many results without

formal proofs or derivations. He simply wrote them down!


15/23

Among his result, the following relation was found:

π 4 = 2143/22

This relationship can be expanded into CF:

2143/22 = [97; 2,2,3,1,16539,1.....]

He was a master of continued fractions and perhaps used

this method. This CF yields : pi = 3.141592653.

Euler and Continued Fraction for 'e'

Leonhard Euler [1707-1783] , the prolific mathematician ,

developed many theorems and formulas for e- the base of

natural logarithm .

Again 'e' is an irrational number.

He developed the CF for e as follows:

e = {2; 1,2,1,1,4,1,1,6 ,1,1,8,1,1....}

Here the partial quotients are repeating with '1's and

also multiples of 2.

This CF converges rapidly and taking the first few steps

could give a good approximation:

e ~ {2;1,2,1,1,4] = 2.7272727

The Casio calculator gives: e = 2.718281828


16/23

Euler also gave the CF for sqrt(e):

sqrt(e) = {2;1,1,1,5,1,1,9,1,1...]

Exercise:

1 Find the approximate value of √ e taking the first

six partial quotients in Euler's expansion. and check

the result with your calculator and find the error %.

---------------------------------------

Square roots and continued fractions

square roots which are irrational can be expanded into

continued fractions.

[Pythagoras was intrigued by the irrational nature of

square root of 2---which did not fit his view of the

Universe with whole numbers or fractions!

Heron of Alexandria found a simple method to find

square roots. His iterative formula is as follows:

x(i+1) = [1/2]( x(i) + N /x(i) ] for square root of N

where x(i) is an initial guess for the square root.

This formula can be derived from the well-known

Newton-Raphson method which involves Calculus.]

The CF's for some square roots are given here. we will


17/23

find some interesting patterns.

√2 = {1; 2,2,2,2,2 ...} = { 1; }

√3 = {1;1,2,1,2,1,2,1,2 } = {1; }

√5 = {2;4,4,4,4......} = {2; }

and so on.

Note that these CF's have repeated quotients, with one

or more numbers repeating in the sequence.

The CF for phi was also repeating with 1's.

To 'derive' these CF's we need an iterative formula;Let us shall derive for sqrt(2):

Write √2 = 1+ 1/x or

x = 1/ (√2-1) = (√2 +1)/2-1

x = √2 +1 = 1 + 1/x +1 = 2 + 1/x

So ,repeating this expression in the denominator 'x' in

the right side, we get the CF:

√2 = 1 + 1/x = 1 + 1

-------

2 + 1

-------

2 + 1/x


18/23

and so on.

Therefore √2 = { 1; 2,2,2,2....}

Let us see how we can approximate √2 by using CF:

Truncating the CF to 5 quotients:

√2 ≈ {1;2,2,2,2,2} we get 99/70

99/70 = 1.414285714

The value given by a Casio calculator is:

√2 = 1.414213562

The error in this approximation is: err= 0.00007 only.

Therefore, if you come across √2 in your problems ,

replace that by 99/70.

Exercise:

1 Find the CF expansion for √13 and √18.

[When you find the convergent folding the CF expansion,

with each step ,you will be getting positive and

negative errors alternating; that is the approximation

is either below or above the actual value. This is

characteristic of CF's.]

Log function and CF's

For log 2 , we have the CF : = {0; 3,3, 9}


19/23

Taking the two quotients 3,3 , we get log 2 =3/10 = 0.3

---a good approximation for 0.3010.

Another example: log 2 (5) = {2;3,9,--]= 65/28

Trig functions and CF's

It is easy to relate trig functions with the square

roots we obtained:

sin 45 = cos 45 = 1/√2

sin 60 = cos 30 = √3/2

There are many other relationships for hyperbolic

functions that can be expanded into CF's.

Applications

1 In the early days of computing using digital

computers, the available memory {RAM} was limited and

the speed of operation was also slow. Therefore computer

scientists looked for simplified approximations for

pi,e,square root and trig and other functions. The

approximations such as given earlier were very useful in

computer algorithms for faster execution with limited


20/23

memory.

2 Polynomials and rational functions were approximated

by CF or by using Pade' approximation or similar schemes

which could be derived from CF.

3 Many complex functions which are usually stored in

tabular form could be written using approximations

derived from CF; These are called "look-up tables".

Examples are log tables and tables for error functionintegral or Gaussian distribution .

4 Continued fractions are regularly used in the analysis

of electrical networks.

5 We may employ infinite series expansions or Taylor

series for approximating functions [such as sin or

tangent functions.] But continued fractions often give

better or more accurate approximations with fewer

computations.

Linking phi,'e' and pi:

We are familiar with Euler relation:

e iπ +1 = 0

This relates e,pi, i , 1 and 0.


21/23

Srinivasa Ramanujan [1887-1920] gave a remarkable

Continued Fraction relating phi,pi as follows:

y = [√(2 + φ) - φ ] e 2π/5

y = 1 + x / (1+ x 2) / (1 + x 3 ) ------

where x = e -2π and φ is the Golden ratio.

Recent Applications

In recent decades, besides communication theory, CF's

have been used in deterministic chaos,fractals and

Mendelbrot sets and so on.

It appears that irrational numbers, mixed with rational

numbers, can create chaos as well as beautiful patterns

in the real world .

References

1 M R Schroeder --Number theory in Science and

Communication [Springer 1999]

2 Donald Knuth-- "Art of computer programming " vol 2

{seminumerical algorithms}[ Pearson educational,]


22/23

3 William Dunham -Journey through genius ,[John Wiley

/Penguin 1990]

4 M Abramowitz and I Stegun --Handbook of Mathematical

Functions [Dover,1970]

5 G H Hardy and E M Wright --An introduction to the

theory of numbers --{Oxford U P ,1984]

6 John D Barrow ---Chaos in numberland ---[+plus

magazine online]

7 L J Lange --Amer Mathematical Monthly vol 106 [1999]

[Note: The author received his doctoral degree in

engineering from Columbia University, New York .]

---------------- XXXXXXXXXXXX-----------------


23/23

The Golden ratio, Fibonacci series and Continued fractions

Documents

Transcript of The Golden ratio, Fibonacci series and Continued fractions