The area of a circle is one of the first formulas that you learn as a young math student. It is simply taught as, . There is no explanation as to why the area of a circle is this arbitrary formula. As it turns out the area of a circle is not an easy task to figure out by your self. Early mathematicians knew that area was, in general to four sided polygons, length times width. But a circle was different, it could not be simply divided into length and width for it had no sides. As it turns out, finding the measurement to be squared was not difficult as it was the radius of the circle. There was another aspect of the circle though that has led one of the greatest mathematical voyages ever launched, the search of Pi.
One of the first ever documented estimates for the area of a circle was found in Egypt on a paper known as the Rhind Papyrus around the time of 1650 BCE. The paper itself was a copy of an older “book” written between 2000 and 1800 BCE and some of the information contained in that writing might have been handed down by Imhotep, the man who supervised the building of the pyramids.
The paper, copied by the scribe named Ahmes, has 84 problems on it and their solutions. On the paper, in problem number 50 he wrote; “Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as a circle.” Given that we already know that the area of a circle is we find that the early Egyptian estimate for the area of a circle was which simplified to or 3.16049… Though, the papyrus does not go into detail as to how Ahmes derived this estimate. This estimate for Pi given by the ancient Egyptians is less than 1% off of the true value of Pi. Given, there was no standard of measurement in that day and they also had no tools to aid them in such calculations such as compasses or measuring tapes, this is an amazingly accurate value for Pi and the area of a circle.
Another early attempt at the area of a circle is found in the Bible. In the old testament within the book of Kings Vii.23 and also in Chronicles iv.2 a statement is made that says; “And he made a molten sea, ten cubits from one brim to the other; it was round all about and his height was five cubits: and a line of thirty cubits did compass it round about.” From this verse, we come to the conclusion that Pi is 30/10 or simply 3. The book of Kings was edited around the time of 550 BCE. Mu...
... middle of paper ...
...ct, the rest of the mathematical world doesn’t dare question their founding mathematicians, and that they alone, the cyclometer, have discovered the true value of Pi. One circle squarer even went so far as to submit a law in his home state of Indiana that his value of Pi be used as the legal value of Pi. It was passed, but to this day awaits further legislation in regard to its factuality.
In the end, there is still an ongoing search for the true area of a Circle in continued research of the number Pi. Scientists today have reached a record number of decimals of Pi to 206,158,430,000 using a Hitachi Supercomputer. The calculation took 37 hours, 21 minutes and 4 seconds. Using the latest calculation for Pi, if you were to assemble a circle a million miles in diameter, the circle would be less than an inch off. But why the pursuit of a solution that will never end? For many, being that there are no perfect circles even in nature, the perfect circle is an unattainable goal to seek. Through the adventure of discovering new aspects about the circle, other insights may be revealed. The mystery of the circle is an endless pursuit, but for mathematicians, it is the pursuit of perfection.
The Mayans used tons and tons of geometry throughout their creations. Which is obviously figured out just by thinking of the one thing that most of everybody knows and relates to the Mayans, the calendar, and the Aztec’s then took the Mayan calendar and adapted it to come up with their own calendar. They probably used trial and error, I’m sure of. They created many drawings that all involved geometry in one form or another.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
The Wheel of Theodorus is a spiral that illustrates irrational numbers. The wheel is started by first drawing a right triangle with equal legs, and a hypotenuse
Born in c.276 BC, Eratosthenes was a Greek scientific writer, astronomer, and poet. He made the first measurement, which showed Earth’s size. Best known as the first person who calculated the circumference of the Earth, Eratosthenes of Cyrene, living in the time around 276 BC, developed a measuring system using a standard unit of measure called the stadia. This form of standard measurement during the time assisted Eratosthenes with calculating very accurately. He studied the way in which the Sun’s rays fell vertically at noon on the summer solstice while in Alexandria Egypt. Eratosthenes correctly guessed that the distance from the Earth to the Sun was very great and that the Sun’s rays are practically parallel when they reach the Earth.
Areas of the The following shapes were investigated: square, rectangle, kite. parallelogram, equilateral triangle, scalene triangle, isosceles. triangle, right-angled triangle, rhombus, pentagon, hexagon, heptagon. and the octagon and the sand. Results The results of the analysis are shown in Table 1 and Fig.
Italian mathematician Raphael Bombelli is credited with major contributions to both Algebra and geometrical proofs. Emerging from a difficult period in his family’s era, Bombelli became the key figure in understanding imaginary numbers while also taking credit with the invention of complex numbers. He challenged common mathematicians’ thinking and view of mathematics at the time until his works were well known and rightfully praised.
Then in Euclid II, 7, it goes farther to explain that “if a straight line be cut at random, the square on the whole and that on one of the segments both together, are equal to twice the rectangle contained by the whole and said segm...
My research demonstrates how a desire to calculate the area of circular regions for constructions purposes led ancient mathematicians to develop methods from which pi could be derived and why pi is still a major subject today even though the number of digits approximated has long since passed the point of practicality. I separated the research into two parts, one dedicated to the history of pi and another to analyzing a few of the prominent methods of approximating pi, by reading scholarly articles, books, and other resources. The analysis shows that the hunt for pi has persisted because of the challenge it presents to mathematicians and the hope they hold of uncovering other secrets of mathematics in the process. It appears that, while nothing
Around Two thousand five hundred years ago, a Greek mathematician, Pythagoras, invented the Pythagorean Theorem. The Theorem was related to the length of each side of a right-angled triangle. In a right-angled triangle, the square on the hypotenuse, the side opposite to the right angle, equals to the sum of the squares on the other two sides. (148, Poskitt) To know more about this famous theorem, we can look at the other forms of the Pythagorean Theorem, such as it can also be written as c^2-a^2=b^2 which is for reverse operations like finding side b with the data of a and c. Meanwhile, the proofs of the theorem can make us understand more about the invention of the theorem and how Pythagoras figured it out. And with the invention of this theorem, we shall look into where this theorem was used in these days and how important it is.
Pi has a very rich and detailed history since it's creation long ago. It has been a well known ratio for around 4000 years. Many mathematicians have used different ways of calculating Pi but the first to make a huge breakthrough was a man by the name of Archimedes. To calculate the magical number of Pi, Archimedes used the Pythagorean Theorem to find the perimeter of two regular polygons. He used at first a Hexagon, but then thought is not a circle just a polygon with so many sides that you can’t count (He might not have actually said that). So he went on doubling the sides of these polygons (on the left), until he reached a 96-gon as we demonstrated in our model, knowing the more sides the more accurate the number. These polygons were inscribed and circumscribed around a circle as you can see in our model. He knew that he would not get the number he was looking for, but http://i0.wp.comi/techiemathteacher.com/wp-content/uploads/2014/01/pi3.jpg the closest he could get considering the technology back then, So to get to his approximation, he divided the polygons perimeter by the diameter, doing so for both the inscribed and the circumscribed pol...
The Greeks were able to a lot of things with only a compass and a straight edge (although these were not their sole tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found means to construct parallel lines, to bisect angles, to construct various polygons, and to construct squares of equal or twice the area of a given polygon. However, three constructions that they failed to achieve with only those two tools were trisecting the angle, doubling the cube, and squaring the circle.
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started. In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development. Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra. Geometric problems relating to similar figures, area and volume were also studied and values obtained for p.The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea's paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration. The theory of conic sections show a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry. The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe. Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe. The progress in algebra had a major psychologic...
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...