The 50th edition of the

Kenya Students’ Science Congress.

Theme: Science and Technology for the realization of vision 2030

Category: Mathematics Exhibit

Title: Pi

Presenters: 1.Elias Waweru

2.

School: Ruthimitu mixed secondary school.

Province: Nairobi

Venue:

Date: 19thMay 2012

Declaration -I-The research as in this presentation was designed by us under the guidance of our teachers;

1. Mr. Onyango Ngoye

And

2. Mr. Washington Mutahi

Signed: 1. Elias Waweru……………………………………..

2. ………………………………………….……………….

In this project we have analyzed a mathematical concept used in many calculations in Science and Technology.

Declaration-2-

We supervised the project and the writing of this paper and recommend it for presentation in the 50th edition of the Kenya students’ science congress 2012.

Onyango Ngoye…………………………………………………..

Washington Mutahi……………………………………………

School Stamp

AbstractPi( ) is the ratio of the circumference of a circle to its diameter regardless of the size of the circle.

Pi =c /d

Experiments show that pi is an irrational number which cannot be written as the ratio of two integers. Its decimal representation does not repeat and does not terminate when calculated in base ten.

Experiments also show that pi is a transcendental number which cannot be constructed.

Despite the abstracts in computing a true and accurate pi it is applied largely in mathematics, physics, building construction, mechanics, geomorphology and popular culture.

This paper presents our research to prove the irrationality, transcendentalism and practical application of pi as Kenya faces the year 2030.

AcknowledgementsWe would like to thank the following for their support in the research and writing of this project;

1. Mr. Onyango Ngoye (Science Club Patron)

2. Mr. Washington Mutahi (Mathematics Teacher)

3. Mrs. Doreen mirikau (H.O.D Science)

4. The Principal Ruthimitu Mixed Secondary School.

They took their time to organize us and gave us to funds to fully facilitate this project;

1. Elias Waweru…………………………………………………

And

2. …………………………………………………………………………

Table of content Page

Prologue

(a) Forum, Edition, Theme,

Category, Title and Name of presenters. (i)

(b) Declaration 1 (ii)

(c) Declaration 2 (iii)

(c) Abstract (iv) (d) Acknowledgements (v) (e) Table of Contents (vi)

Chapter 1 (a) Introduction 1

(b) Background information 2

(c) Statement of problem 4

(d) Objectives 5

(e) Relevance of project to theme 6

(f) Limitations 7

(g) Assumptions and Precautions 8

……………………………………………………………………………………………………………… .

Chapter 2 Page

Literature Review (i) Books 9

(ii) Scientific Journals 9

(iii) Internet 9

Chapter 3

Methodology (i) Procedure, & Diagrams 10

(ii) Observations 10

(iii)Data 11

Chapter 4

Data analysis and interpretation 12

Chapter 5

Conclusions and Recommendations 13

List of appendices 15

Chapter 1

IntroductionBackground Information

The Latin name Pi, written in Greek pronounced in English pie, was first adopted for use in mathematics in 1706 by Mathematician William Jones who studied the mathematical concepts used in the architecture of the Great Pyramids of Giza (2589-2566) in Egypt. The pyramids were right angled triangles embedded in circles, so despite the knowledge of Pi, he got a constant 1760/280 which is equal to 2Pi. This was fascinating.

The passage in the Bible dated 970-930 BC, 1 Kings 7:23 and Chronicles4:2 discussing the ceremonial circular pool built by King Solomon with a diameter of ten cubits and a circumference of thirty cubits shows that the designers took the value of Pi to be approximately three units. This was a very good attempt of accurate measurements.

The international pi-day, March 14th (after 3.14) is used in the novel pi by Carl Sagan who imagines God hiding pi deep in the Universe then sending

men to search, dig out and get it whole without cutting off any part. The digging continues endlessly.

The story of pi has therefore been a long story of endless calculations and search with mind boggling figures.

This is why the Nobel Prize winning poet Wislaw Szmborska wrote a poem about pi;

The caravan of digits,

That is pi,

Does not stop,

At the edge of the page,

But runs off the air,

Over the wall,

A leaf,

A bird’s nest,

The clouds,

Straight into the sky,

Through all bloatedness,

And bottomlessness.

Ooh how short,

All but mouse like,

Is the comet’s tail.

Literature Review.The oldest record of mathematically estimated value of pi was done by Archimedes (287-212 B.C) who inscribed regular polygons with 96 sides in circles of different sizes. He measured the perimeter of each polygon and its ‘diameter’ and proved that;

310/71<Pi>310/70

He found this fascinating and kept a record of his finding. He never came up with a representation of pi but concluded that the closest relative of a circle has it diameter about three times its diameter irrespective of the size of the polygon.

Following Archimedes demonstration and the invention of the Venerable compass, Mathematicians agreed in 265AD to use a value 355/113 as the value of pi for calculating area, circumference and radius of circles. This figure was used for 900 years until the second millennium AD when decimals were integrated into Mathematics for use to express fractions.

By the year 1424, the circle was defined and the value of pi calculated to eleven decimal places.

Definition of a cycle

A circle is the locus of points equidistant from a given point called the centre.

The common distance from the centre is called radius.

The longest cord which can be drawn in a circle is called Diameter.

In 1949, a computation that took 70hours gave a2037 digit number and no end was seen.

On 6thAugust 2010, Japanese and American computer experts calculated pie to the Value of 5trillion decimal digits without reaching the end.

Why such painful calculations?

Why do we need pi?

Can we not do without pi?

Does pi really exist?

Could Mathematicians have gone wrong somewhere in the course of time?

Statement of problem1. Mathematicians have created a language that has

mystified many concepts required for everyday use. Pi is one of the Concepts.

2. Many people are shy to apply the concepts because they do not want to be found with mistakes, but because life has to go on anyway, they resort to trial and error which wastes more materials and time and often give wrong results.

Materials and time are the main assets for achieving vision 2030.

3. As Kenya faces the year 2030, eyes are on the growth of the informal sector (Jua Kali) to spur development by first creating enough employment.

In the Jua Kali are Metal Workers, Wood Workers, Textile Workers, Beauticians and a host of other Very vital service providers. This important sector must use Mathematics.

This project unveils a mask covering the irrational and transcendental number called pi. We then proceed to demystify it and recommend its wide and indiscriminate use.

ObjectivesThis project hopes;

1. To Show why every numerate person should know and use pi. 2. To demonstrate the irrationality and transdentality of pi.3. To identify and suggest areas of application of pi.

Relevance of project to theme

Kenya is expected to steadily grow technologically such that by the year 2030, it would be a middle level economy with majority of employment generated from local entrepreneurs. The Jua kali Sector is expected to be one the driving factors.

Building technologists, Mechanics, Textile workers, Beauticians just to name a few, are to play their part in the development of this country by adopting a scientific approach to their trades and therefore saving on time and materials which they use.

The use of pi in calculating and costing materials and fittings cannot therefore be underestimated.

Mathematics must therefore be main-streamed for applications by all the middle level technicians.

Let us save time and money to achieve vision 2030.

Let us start talking pi.

Limitations1. My calculations are based on a scientific calculator configured

to give results up to 10decimal digits.2. All my calculations are in base ten. Other secrets of pi may be

discovered by calculations in other bases.3. When using a ruler, it is correct only to one decimal place.4. When using a Vanier Calipers it is correct to two decimal

places.

Guarantee/assumptions and precautions1. We need pi for practical uses in design, both on plain and three-

dimensional geometry.

2. Theoretical pi is elusive like chasing the wind and the agreed simple value of 3.14 should be made practical for everyday use, it gives accuracy for all practical purposes.

3. No more than 39 digits of pi is functionally useful in calculations by engineers and geomorphologists.

Chapter 3

MethodologyRequirements.

1. Ruler.2. Vanier Calipers.3. Ten cylinders of different sizes.4. Strips of paper.5. Calculator.

Procedure.

1. Make ten strips of paper.2. Wrap the strips of paper around cylinders of different sizes.3. Use a ruler to measure the lengths of each paper.4. Measure the diameter of each cylinder.5. Divide the circumference of each cycle with its diameter.6. Note your results.7. Compare the results and make a conclusion.

Chapter 4

Data Analysis and Interpretation 1. Pi is an irrational number which cannot be written as a

ratio of two natural numbers. 2. The irrationality of pi is because it is a ratio between

two items without natural mathematical relationship; a curve and a straight line.

3. Pi is a transcendental number. It cannot be constructed because it involves both a curve and a straight line. It has proved impossible to construct a square circle.

4. Pi can be applied in calculating the: (a) Area of circles and arcs. (b) Circumference of circles.

(C) Radius of circles.

Chapter 5

Conclusions and Recommendations1. The informal sector (Jua Kali) is the biggest employer in

Kenya.2. Jua Kali is expected to drive the economy of Kenya towards

Industrialization as we move towards the year 2030, when Kenya will be expected to be a middle level income economy.

3. Currently over 98% of those who work in the Jua kali manufacture their wares with minimal application of mathematics. This is because of the scare by the jargons of mathematics which are applied without reasoning out at school level. One such a scare is the pi, which is introduced in the fifth year of primary education without being derived and its future use not spelled out. Pupils are punished when they do not understand it; they hate it and refuse to employ it when time for its use comes.

4. All middle level technologists should be encouraged to use mathematical concept in their endeavors to design and manufacture products in the Jua kali sector. One such a concept is pi which has everyday application noting by the number of curves and lines in the items made at the Juakali.

References1. Heath T.L. (1956). Thirteen books of the elements.

Volume 1. Published by Dover.2. Robert S. (2005). The elements of Euclid.

Published by Elibron Classics. 3. www.wikipedia.org//Tignometry4. www.answer.com

5. www. Edis . ifas .ufl.edu /hs 186

Appendix