For centuries, mathematicians around the world have struggled in the chase for an increasingly accurate approximation of pi. The first recorded avenue of approximating the value of pi by theoretical means was the polygon approximation method, employed by Archimedes around 250 B.C., which was accurate only to about two decimal places when he performed it (Groleau). This was partly due to the fact that Greek mathematicians at the time had no concept of 0, thus only used fractions, not decimals, and the fact that they had no access to algebra (McKeeman). Now, over 2000 years later, pi has been calculated and verified to more than 12.1 trillion digits thanks, in part, to the Bailey-Borwein-Plouffe (BBP) formula, published in 1996, which allows us to quickly and easily calculate a specific digit of pi (Yee and Kondo; Seife). Though the polygon method and the BBP formula each work in entirely …show more content…
The polygon approximation method uses the perimeters of polygons to approximate pi and “was the first theoretical, rather than measured, calculation of pi,” but while it worked great for Archimedes in his time, it has limited practicality today (Groleau). In order to estimate pi, Archimedes used a circle that had a polygon inscribed in and circumscribed about it, then found the perimeter of each polygon, and used those values as the upper and lower limits of pi (Groleau). He started with a hexagon as the inner polygon of a circle with a diameter of 1 whose perimeter is equal to 6r when r is the radius of the circle (McKeeman). This meant that since the circumference of the circle was 2πr, 6r < 2πr and the lower limit of pi was 3 (McKeeman). He then found the perimeter of the outer hexagon to calculate the upper limit and proceeded to double the number of sides and repeat the
After 3rd century BC, Eratosthenes calculation about Earth's circumference was used correctly in different locations such as Alexandria and syene (Aswan now) by simple geometry and the shadows cast. Eratosthenes's results undertaken in 1ST century by Posidonius, were corroborated in Alexandria and Rhodes by the comparison between remarks is excellent.
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.
Euclidean Geometry has been around for over thousands of years, and is studied the most in high school as well as college courses. In it's simplest form, Euclidean geometry, is concerned with problems such as determining the areas and diameters of two-dimensional figures and the surface areas and volumes of solids. Euclidean Geometry is based off of the parallel postulate, Postulate V in Euclid's elements, which states that, "If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles."
Believing that helping the earth for one day will make a difference, is like believing something because it can be rationalized. Humans, as a species, are programed to rationalize their actions, such as only “helping the earth” on one day because it is better than not helping at all. Even if every single person in the entire world were to “help the earth” at the same time for just one day, the impact would actually be quite minimal. Politicians have tried to agree on a common method to better the earth for current and future generations, and the differences in the methods draws a line. This line draws the perimeter for the two main groups, the liberals and the conservatives. Both groups have a much defined methodology of how they plan to better this earth for all generations. The federal government
Squaring the circle with a compass and straightedge had been a problem that puzzled geometers for years. In his notes under the drawing he recognized that “if you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle” . This excerpt alone shows that Da Vinci had an immense understanding of proportion, as well as geometry. On this page, Da Vinci also wrote the exact proportions that he used, based his own observations and the ones used in Vitruvius’s book. Da Vinci
Even the smallest tasks can impact the world in a significant way. Math, despite its trivial appearance, is large in grandeur that governs our world from the inside and the outside. The many twists and turns that exist in Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that can be utilized to solve things in the Mathematical world that seem too daunting to be calculated by normal means.
[5] Niven, I., A simple proof that π is irrational, Bulletin of the American Mathematical
is convergent and ends up converging to φ, the golden ratio [2]. This curious quantity is just a ratio, so what makes it so special? Why is it so mystifying? While the other major constant in mathematics, pi, is a ratio between a circle's circumference and its diameter, phi (φ) considers a rectangle with height, h, and width, w, and forms the following ratio:
...on of light and the rays are proportions in the Fibonacci sequence. Fibonacci relationships are found in the periodic table of elements used by chemists. Fibonacci numbers are also used in a Fibonacci formula to predict the distant of the moons from their respective planets. A computer program called BASIC generates Fibonacci ratios. “The output of this program reveals just how rapidly and accurately the Fibonacci ratios approximate the golden proportion” (Garland, 50). Another computer program called LOGO draws a perfect golden spiral. Fibonacci numbers are featured in science and technology.
Steinburg, D. H. (2012). Baker's math: Essential calculations for working with dough. Dim Sum Thinking, Inc.
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
Irrational numbers are real numbers that cannot be written as a simple fraction or a whole number. For example, irrational numbers can be included in the category of √2, e, Π, Φ, and many more. The √2 is equal to 1.4142. e is equal to 2.718. Π is equal to 3.1415. Φ is equal to 1.6180. None of these numbers are “pretty” numbers. Their decimal places keep going and do not end. There is no pattern to the numbers of the decimal places. They are all random numbers that make up the one irrational number. The concept of irrational numbers took many years and many people to discover and prove (I.P., 1997).
The history of the computer dates back all the way to the prehistoric times. The first step towards the development of the computer, the abacus, was developed in Babylonia in 500 B.C. and functioned as a simple counting tool. It was not until thousands of years later that the first calculator was produced. In 1623, the first mechanical calculator was invented by Wilhelm Schikard, the “Calculating Clock,” as it was often referred to as, “performed it’s operations by wheels, which worked similar to a car’s odometer” (Evolution, 1). Still, there had not yet been anything invented that could even be characterized as a computer. Finally, in 1625 the slide rule was created becoming “the first analog computer of the modern ages” (Evolution, 1). One of the biggest breakthroughs came from by Blaise Pascal in 1642, who invented a mechanical calculator whose main function was adding and subtracting numbers. Years later, Gottfried Leibnez improved Pascal’s model by allowing it to also perform such operations as multiplying, dividing, taking the square root.
from his tables, which showed powers of 10 with a fixed number used as a base.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.