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Continued Fractions: Invisible Patterns Ramana Andra Thomas Jackson FTLOMACS OCT 14, 2017

Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

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Page 1: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Continued Fractions: Invisible Patterns

Ramana Andra

Thomas Jackson

FTLOMACS

OCT 14, 2017

Page 2: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

• Representations on Numbers

• Continued Fraction Notation

• Finite Ratio Example

• Convergents

• Radicals

• Algebraic Numbers

• π

• Java Programs

• Invisible Connections

AGENDA

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Page 3: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Infinite Series, Periodic and Nonperiodic Decimal Expansions,

Integrals

𝜋 = 1 +1

2 + 1

3+1

4−1

5+1

6+1

7+1

8+1

9+ −

1

10+1

11+1

12−1

13…

1 − 𝑥2𝑑𝑥 =𝜋

2

−∞

𝜋

4= 1 −

1

3+1

5−1

7+1

9− ⋯ =

(−1)𝑘

2𝑘 + 1

𝑘=0

3.1415926535897932384626433832795…….

TRADITIONAL

REPRESENTATIONS OF NUMBERS

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Page 4: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

𝜑 =1 + 5

2= 1.6180339887 …

𝜑 = 1 + 1 + 1 + ⋯

e = 1

𝑛!∞𝑛=0 = 1 +

1

1+1

1•2+1

1•2•3+

1

1•2•3•4+⋯

e = lim𝑛→∞(1 +

1

𝑛)𝑛 = 2.718281828459…

2 = 1.41421356237…

TRADITIONAL

REPRESENTATIONS OF NUMBERS

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Page 5: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

4

𝜋 = 1 +

12

2 + 32

2 + 52

2 + 72

2 + 92

2 + 112

2 + …

EXAMPLES OF CF’S

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Page 6: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

e = 2 +1

1 + 1

2 + 1

1 + 1

1 + 1

4 + 1

1 + 1

1 + 16 + …

Φ = 1 +1

1 + 1

1 + 1

1 + 1

1 + 1

1 + 11 + …

EXAMPLES OF CF’S

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Page 7: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

𝜋

2 = 1 -

1

3 − 2•3

1− 1•2

3 − 4•5

1 − 3•4

3 − 6•7

1− 5•63 − …

2 = 1 +1

2 +1

2 +1

2 +1

2 +1

2 +12 + ⋯

EXAMPLES OF CF’S

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Page 8: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

CONTINUED FRACTIONS TYPES

Continued Fractions

Simple

Finite Infinite

Periodic Non-

Periodic

General

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Page 9: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

General Continued Fraction

a1 + 𝑏1

𝑎2+

𝑏2

𝑎3 +

𝑏3

𝑎4 +

𝑏4

𝑎5 + 𝑏5𝑎6…

Simple Continued Fraction (bi = 1)

a1 + 1

𝑎2+

1

𝑎3 +

1

𝑎4+

1

𝑎5 + 1 𝑎6…

= [a1; a2, a3, a4, …] = [a1: a2, a3, a4, …]

Convergents

𝑝𝑖

𝑞𝑖

= [a1; a2 a3, a4, a5,…an]

CONTINUED FRACTIONS FORMS

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Page 10: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, ...]

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, …]

φ = [1; 1, 1, 1, 1, 1, 1, … ]

FAMOUS CONTINUED FRACTIONS

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Page 11: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Consider the fraction 47

13. Can we find the continued fraction (cf.) for this

finite ratio?

We should consider breaking up the fraction into mixed form.

So 47

13 = 3 +

8

13

We need the numerator to be one for our cf. form so we can apply the

following algebraic identity 𝑎

𝑏=1𝑏

𝑎

to move forward.

FINITE RATIO EXAMPLE

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Page 12: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

47

13 = 3 +

113

8

= 3 + 1

1 + 5

8

= 3 + 1

1+ 185

= = 3 + 1

1+ 1

1 + 35

47

13 = 3 +

1

1+ 1

1 + 153

= 3 + 1

1+ 1

1 + 1

1+23

= 3 + 1

1+ 1

1 + 1

1+23

47

13 = 3 +

1

1+ 1

1 + 1

1+132

= 3 + 1

1+ 1

1 + 1

1+1

1+12

FINITE RATIO EXAMPLE

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Page 13: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

This procedure is an example of an algorithm

called the Euclidean Algorithm which was

developed by Euclid in his famous book The

Elements. We can also look at the problem from

geometrical and coding perspectives.

FINITE RATIO EXAMPLE

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Page 14: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

GEOMETRIC CONSTRUCTION

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Page 15: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Geometric Construction

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Page 16: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Let’s consider the calculations already performed for the fraction 47

13 .

47 = 3(13) + 8

13 = 1(8) + 5

8 = 1(5) + 3

5 = 1(3) + 2

3 = 1(2) + 1

2 = 2(1) + 0

EUCLIDEAN ALGORITHM

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Page 17: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

We continue this operation on the two numbers and it will always stop when

the length of the shortest rectangle is one and the remainder after the last

triangle has been divided is zero.

We can find the cf. form from the leading numbers from the list:

47

13= 3 +

1

𝟏+ 1

𝟏 + 1

𝟏+1

𝟏+1𝟐

EUCLIDEAN ALGORITHM

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Page 18: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

We all know that φ is derived from the roots of the quadratic equation

φ 2 - φ - 1 = 0

φ = 1 + 1

φ

φ = 1 + 1

1 + 1

φ

φ = 1 + 1

1+ 1

1 + 1φ

φ = 1 + 1

1+ 1

1 + 11+⋯

φ = [1; 1, 1, 1, 1, 1, 1, 1, …]

PHI DERIVATION

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Page 19: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

We can look at the partial cfs for the fraction 47

13 to find better approximations

to the exact value of this fraction. These values are called convergents.

3, 3 + 1

1 , 3 +

1

1+ 1

1

, 3 + 1

1+ 1

1 + 11

, 3 + 1

1+ 1

1 + 1

1+11

3, 4, 7

2, 11

3, 18

5, 47

13 = 3, 4, 3.5, 3.666.., 3.6, 3.65…

CONVERGENTS

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Page 20: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

We can generalize this procedure. Let ci =𝑝𝑖

𝑞𝑖 represent

the convergents for a given fraction with i ≥ 0.

So c1 =𝑝1

𝑞1

= a1 = 𝑎1

1

c2 = 𝑝2

𝑞2

= a1 + 1

𝑎2

= 𝑎1𝑎2+1

𝑎2

c3 = 𝑝3

𝑞3 = a1 +

1

𝑎2+1

𝑎3

= 𝑎1𝑎2𝑎3+𝑎1+𝑎3

𝑎2𝑎3+1

CONVERGENTS

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Page 21: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

c4 = 𝑝4

𝑞4 = a1 +

1

𝑎2+1

𝑎3+1𝑎4

= 𝑎1𝑎2𝑎3𝑎4+𝑎1𝑎2+𝑎1𝑎4+𝑎3𝑎4+1

𝑎2𝑎3𝑎4+𝑎2+𝑎4

So we can “see” the following recursions for the

convergent terms if we look carefully:

pi = aipi-1 + pi-2

qi = aiqi-1 + qi-2

CONVERGENTS

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Page 22: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Using these recursion formulas, the convergents for the golden ratio φ are

1

1, 2

1,3

2, 5

3, 8

5, 13

18, …..

We can see that the numbers in the numerators and denominators are terms from the Fibonacci sequence!

We should notice that

pi = aipi-1 + pi-2 = (1)pi-1 + pi-2 = pi-1 + pi-2

qi = aiqi-1 + qi-2 = (1)qi-1 + qi-2 = qi-1 + qi-2

and ci = ti+1

𝑡𝑖

where ti are the terms of the Fibonacci sequence.

CONVERGENTS

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Page 23: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

2 = 1 +1

2 +1

2 +1

2 +1

2 +1

2 +12 + ⋯

= [1; 2, 2, 2, 2, 2, 2, … ]

SQUARE ROOT 2

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Page 24: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Let’s see how we can derive this cf representation.

2 = 1 + ( 2 - 1) = 1 +11

2−1

= 1 + 1

2+1

But 2 + 1 = 2 + ( 2 − 1) = 2 + 11

2−1

= 2 +1

2+1

Thus 2 = 1 + 1

2+1

2+1

If we use this rationalization method again repeatedly, we find the

cf. of 2 consists of a single 1 followed entirely by 2’s.

Thus 2 = [1; 2, 2, 2, 2, 2, … ]

SQUARE ROOT 2

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Page 25: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Is there a pattern in the representation?

Can we generalize numbers of the form 𝑛2 + 1?

Let’s give it a go!

SQUARE ROOT 17

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Page 26: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Let’s consider a simple case and suppose

A = [a; b, b, b, b, b,…]

We can rewrite A in the form A = a + 1

[𝑏; 𝑏, 𝑏, 𝑏, 𝑏, 𝑏,… ]

We need to determine the value of B = [b; b, b, b, b, b, …]

Just like for A, we can rewrite B = b + 1

[𝑏; 𝑏, 𝑏, 𝑏, 𝑏, 𝑏,… ]

PERIODIC CF’S

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Page 27: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

So we have B = b + 1

B which can be rewritten as B2 – bB -1 = 0

Using the Quadratic Formula, we have B = 𝑏+ 𝑏2+4

2

Thus A = a + 2

𝑏+ 𝑏2+4 = a -

𝑏− 𝑏2+4

2 = 2𝑎−𝑏

2+𝑏2+4

2

So we have 2𝑎−𝑏

2+𝑏2+4

2 = [a; b, b, b, b, …]

If b = 2a, then 𝑎2+ 1 = [𝑎; 2𝑎, 2𝑎, 2𝑎, 2𝑎, … ]

PERIODIC CF’S

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Page 28: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Let’s try to derive the continued fraction for

3 on the board together!

SQUARE ROOT 3

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Page 29: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

3 = 1 +1

1 +1

2 +1

1 +1

2 +1

1 +12 + ⋯

= [1; 1, 2, 1, 2, 1, 2, … ] .

SQUARE ROOT 3

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Page 30: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

PERIODIC CF’S

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Page 31: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

What about cube roots?

These continued fractions are not periodic.

We will find their cf. representation

from their decimal first.

From the decimal part,

we will repeatedly invert the fractional part.

CUBE ROOTS

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Page 32: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

CUBE ROOTS

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Page 33: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

INFINITE CONTINUED

FRACTIONS

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Page 34: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Π 1/4

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Page 35: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Π

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Page 36: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

CUBE ROOTS

We can find the cube roots using the bisection method.

Easy for high school students to understand.

Can be easily coded in a programming language.

We have some simple Java programs to demonstrate.

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Page 37: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Rational Numbers

Square Root

Cube Root

π

JAVA PROGRAMS

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Page 38: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

How do we evaluate Pi using a formula?

We use an identity that came up recently in our Math club

meeting.

π = 16 arctan 1

5 − 4 arctan

1

239

FORMULA FOR Π

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Page 39: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

Ramanujan proved two connections between π, e and φ:

RAMANUJAN

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Page 40: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

A CONTINUED FRACTION

APPROXIMATION OF THE GAMMA

FUNCTION

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Page 41: Continued Fractions: Invisible Patterns Ramana Andra ......These continued fractions are not periodic. We will find their cf. representation from their decimal first. From the decimal

You Tube video: Continued Fractions - Professor John Barrow

https://www.youtube.com/watch?v=zCFF1l7NzVQ&t=670s

The Topsy-Turvy World of Continued Fractions:

https://www.math.brown.edu/~jhs/frintonlinechapters.pdf

Cube Root of a Number:

http://www.geeksforgeeks.org/find-cubic-root-of-a-number/

Gamma Function:

http://www.sciencedirect.com/science/article/pii/S0022247X12009274

A Continued Fraction Approximation of the Gamma Function:

http://www.sciencedirect.com/science/article/pii/S0022247X12009274

https://ac.els-cdn.com/S0022247X12009274/1-s2.0-S0022247X12009274-main.pdf?_tid=4981f0e4-b07a-11e7-bb6a-

00000aab0f26&acdnat=1507942672_ff44d4811825271fcf37c06314338fe1

Java Code:

https://dansesacrale.wordpress.com/2010/07/04/continued-fractions-sqrt-steps/

Continued Fractions:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html

REFERENCES

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