Circle’s True Pi Value Equals the Square Root of Ten , livre ebook

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The author of this book has discovered an innovative method of determining the True value of Pi ( = 10 = 3.1623 or 3.16227766016838). This new Pi value is derived from the geometric relationships among the circle’s components with the use of the Circle Theorem and Pythagorean Theorem. Figure 1 contains an inscribed circle in the square consisting of gridlines equally spaced into one-fourth of the side of the square or the diameter of the inscribed circle. The resulting precise Pi value is validated with the use of the Polygon Area formula, Binomial Theorem, and Quadratic Equation. This contemporary approach to finding the true Pi value reputes the traditional method of finding the Pi value. For the past four centuries, many mathematicians have attempted to find the precise Pi value. It began with measuring the circumference and the diameter of a circle and dividing the former by the latter. The erroneous Pi calculation began during the era of Archimedes of Syracuse circa 287–212 before the Christian era (BCE). Archimedes one of the greatest mathematicians of the ancient world introduced the approximate value of Pi as 3.14 (between 3-1/7 and 3-10/17 bound). Since then, humans have been trying to add more digits to the two-decimal placed Pi in an attempt to find a precise Pi which is still an approximation value.

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Publié par	Xlibris US
Date de parution	09 juillet 2023
Nombre de lectures	0
EAN13	9798369402399
Langue	English

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CIRCLE’S TRUE PI VALUE EQUALS THE SQUARE ROOT OF TEN
π = = 3.16227766016838 or π = 3.1623

Albert Vitales Cruz, PhD
Professional Engineer (retired)
United States Citizen
United States Army Veteran

Copyright © 2023 by Albert Vitales Cruz, PhD.
Library of Congress Control Number:
2023912101
ISBN:
Hardcover
979-8-3694-0241-2

Softcover
979-8-3694-0240-5

eBook
979-8-3694-0239-9

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

Any people depicted in stock imagery provided by Getty Images are models, and such images are being used for illustrative purposes only.
Certain stock imagery © Getty Images.

Rev. date: 08/22/2023

Xlibris
844-714-8691
www.Xlibris.com
853624
DEDICATION
Dedicated to: My parents Juan Perez Cruz and Leonora Vitales Cruz;
My deceased wife Linaflor Manantan Cruz; and my siblings Ernesto Vitales Cruz, Josefina Cruz Calayag, Gerardo Vitales Cruz, Erminda Cruz Mora, Corazon Cruz Millena, and Salvador Vitales Cruz
PREFACE
The traditional Pi value of 3.1415 is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Geometry is one of the oldest branches of mathematics. It is concerned with properties of space that are related to distance, shape, size, and relative position of figures. Until the 19 th century, geometry was almost exclusively devoted to the fundamental concepts of Euclidean geometry, which includes the notions of point, line, plane, distance angle surface, and curve. Later in the 19 th century, the scope of geometry has been greatly expanded, and the field has been split into many subfields that were known as combinatorial geometry. Other scholars used various mathematics equations to shape the circumscribed polygon into an inscribed circle by using an ever-increasing number of polygon sides to add more decimal places into the traditional Pi value of 3.1415. Indeed the Pi has been known for almost 4000 years, but even if the number of minutes that elapsed since then the calculated Pi to that number of many decimal places added to the 3.1415 is still only approximating its actual value.
ABSTRACT
The author of this book has discovered an innovative method of determining the True value of Pi ( π = = 3.1623 or 3.16227766016838). This new Pi value is derived from the geometric relationships among the circle’s components with the use of the Circle Theorem and Pythagorean Theorem. Figure 1 contains an inscribed circle in the square consisting of gridlines equally spaced into one-fourth of the side of the square or the diameter of the inscribed circle. The resulting precise Pi value is validated with the use of the Polygon Area formula, Binomial Theorem, and Quadratic Equation. This contemporary approach to finding the true Pi value reputes the traditional method of finding the Pi value which represents the ratio between the circumference and the diameter. For the past four centuries, many mathematicians have attempted to find the precise Pi value. It began with measuring the circumference and the diameter of a circle and dividing the former by the latter. The Pi calculation began during the era of Archimedes of Syracuse circa 287–212 before the Christian era (BCE). Archimedes one of the greatest mathematicians of the ancient world introduced the approximate value of Pi as 3.14 (between 3-1/7 and 3-10/17 bound). Since then, humans have been trying to add more digits to the two-decimal placed Pi to find a more precise approximation of Pi with varying degrees of success. The amateur mathematician William Shanks, for example, calculated Pi by hand to 707 figures in 1873 and died believing so, but decades later it was discovered he had made a mistake at the 528 th decimal place.
Jan de Gier, a professor of mathematics and statistics at the University of Melbourne, says being able to approximate Pi with some precision is important because the mathematical constant has many different practical applications. “Knowing Pi to some approximation is incredibly important because it appears everywhere, from the general relativity of Einstein to corrections in the GPS to all sorts of engineering problems involving electronics,” de Gier says. In maths, Pi pops up everywhere. “You can’t escape it,” says David Harvey, an associate professor at the University of New South Wales. In 1897, the State of Indiana in the US almost did away with fussy strings of decimals altogether. The State’s bill is almost enshrined in the law that π = 3.2 because the trigonometric method to shape a square into a circle is a mathematical impossibility.
The ubiquity of π makes it one of the most widely recognized mathematical constants used in elementary mathematics and scientific research. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines. It is an impressive and time-consuming feat but this may not be necessary. The race to introduce more decimals into the erroneous Pi equals 3.1415 may be useless because this new approach to finding the true Pi value is so precise.
CONTENTS
Dedication
Preface
Abstract
Chapter 1 Introduction
Chapter 2 Research Study
Chapter 3 Literature Review
Chapter 4 Results
Chapter 5 Summary
Chapter 6 Recommendation
Reference
Appendix A Glossary (Terms definitions)
Appendix B Notary Public Document
CHAPTER 1
INTRODUCTION
The author of this book Albert V. Cruz precisely proves that the value of Pi (π) equals the square root of ten ((π = 3.16227766016838) or π = 3.1623 rounded to four decimal places, which is the true ratio between a circumference and a diameter of the circle. To prove this true Pi value, the author utilizes the Circle Theorem and the Pythagorean Theorem, as well as the Binonial and Quadratic Equations. The use of these theorems and equations validates the author’s prior logical statements that the circle’s Arc (πD/4) equals the sum of the isosceles triangle’s equal sides or the Hypotenuse of a Base (full Chord) and the Altitude (half Chord) formed by a 90-degree central angle.
This new Pi value reputes the old traditional approximation of Pi (π = 3.1415). The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of Pi = 3 (i.e., 3r 2 = π r 2 ). One Babylonian tablet (ca. 1900–1680 BCE) indicates a value of 3.125 for Pi, which is a closer approximation. In the Egyptian Rhind Papyrus (ca.1650 BCE), there is evidence that the Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for Pi. The ancient cultures mentioned above found their approximations by actual measurement of the circumference and the diameter of a circle in which the ratio between the former and the latter represents a Pi.
The Ancient Greek mathematician Archimedes (circa 170 BCE) discovered an effective method for approximating the value of Pi (π). With the use of a polygon, he inscribed a regular hexagon in a circle and then circumscribed another regular hexagon in the same circle. He obtained a rough approximation of Pi (π) by dividing the Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons. The polygon is inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave the approximate area of the circle. Archimedes knew that he had not found the true value of Pi. He acknowledged that his Pi was only an approximation within those limits. Archimedes also used a 96-side polygon, which helped him find the closest approximation of Pi (π) as the original straight sides began to shape into a circle. The Archimedes number of 3.14 has remained one of the most prevalent approximations of Pi (π) ever since.
A similar approach was used by Zu Chongzhi who could not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. Zu calculated the value of the ratio of the circumference of a circle to its diameter to be 3.1459 (355/113). To compute this accuracy for Pi, he must have started with an inscribed regular 24,576-sided polygon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places. Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, the use of the symbol was popularized by Euler, who adopted it in 1737. An 18 th -century French mathematician named Georges Buffon introduced a method to calculate Pi based on probability.
Since ancient Babylonian times, humans have been trying to introduce more digits to the approximated constant with varying degrees of success. To find a precise value of Pi, in the mid-20 th century, Swiss researchers spent 108 days calculating traditional Pi to a record of 62.8tn digits. Using a computer, their approximation beat the previous world record of 50tn decimal places and was calculated 3.5 times as quickly. The amateur ma