Essay Sample on Pi

“Pi is an infinite, nonrepeating decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII (American Standard Code for Information Interchange) text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time, and manner of your death, and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those two points. All information that has ever existed or will ever exist, the DNA of every being in the universe, EVERYTHING: all contained in the ratio of a circumference and a diameter.” 

The quote is a from a popular meme that has not been attributed to anyone, but it contains a poignant message about the significance of pi. You would think if more people knew about this meme maybe they would appreciate pi more. Sadly, this is not the case people would rather forget it or exploit it for bragging rights. The story is not all about ignorance, but it is a story to ensure that importance pi will surpass the fame of those who set world records related to pi. The creation of Pi Day, March 14, is a pivotal moment that leads to a greater understanding of pi. Mostly, brainiacs, pi-enthusiast, and mathematicians are in attendance of this event is open to people of all ages and backgrounds. The popularity of the holiday makes the knowledge more accessible to all who desire the knowledge. This knowledge whether it truly encompasses the fabric of our lives does play a major role in our understanding of the universe and everyday life. 

Pi is an irrational and transcendental constant that has a never-ending decimal. Irrational meaning that it cannot be represented as a fraction. In 1761, Johann Heinrich Lambert proved that pi is irrational by using the notion of continuous fraction of  〖 tan〗⁡(x). Lambert showed by contradiction that if pi/4 is rational such, then tan⁡(π/4)=1 is irrational. This is contradiction because 1 is a rational number which implies that pi/4 and pi are both irrational. Transcendental meaning that is not a root of a polynomial with rational coefficients. Ferdinand von Lindemann proved that pi is transcendental in 1882. He used the fact that e is transcendental number to prove that pi is transcendental and therefore the squaring of the circle problem is impossible to solve. The constant characteristic ensures that the number is unchangeable between calculations. The calculation of pi results from taking the ratio of the circumference and the diameter of the circle: Π=C⁄d . This equality holds true for any size circle, due to fact that pi is a constant. 

Pi is usually represented as 3.14 for calculation simplicity, but the infinite decimal does not have a pattern, nor does it repeat. The necessity for simplicity extended beyond the approximation of numbers. There was a need to represent this constant with a symbol like e for 2.71 or i for imaginary numbers. Before the introduction of the symbol for pi, many scholars referred to the ratio as, “quantitas in quam cum multiflicetur diameter, proveniet circumferencia (the quantity which, when the diameter is multiplied by it, yields the circumference).” In 1706, The Welsh mathematician William Jones answered the call to create a symbol for pi. He coined the symbol from the first letter of the Greek word περιϕέρεια meaning perimeter or the circumference of a circle. The symbol was not popularized until Swiss mathematician Leonhard Euler, who was known for introducing an abundance of other mathematical symbols, introduced it into common use. 

Methods/Culture/Times/Contribution

The methods for measurement and calculation spans from antiquity to modern day. The earliest sign of approximation is notated in the Bible, the Egyptian Rhind Papyrus, and the Babylonian’s approximation. “And he [Hiram the Phoenician] made a molten sea, ten cubits from the one rim to the other it was round all about, and a line of thirty cubits did compass it round about. And it was a hand breadth thick.”  Hiram was a Phoenician artisan tasked with designing the Jewish temple and this was needed to complete the temple. The verses denote the diameter as 10 cubits and the circumference as 30 cubits, thus the approximation for pi was 3. The first introduction of pi in Egypt was connected to measurements of the Great Pyramid of Giza. Later, the Egyptians used a formula for the area of a circle=(16/9)^2, included in the Rhind Papyrus, which revealed a closer approximation of 3.1605. The Babylonians utilized the area of a circle as 3 times the square of the radius to approximate pi as 3.125. Although these initial measurements and calculations were not precise calculations of the constant, the approximation was exactly what it needed to be for the people of the time.  —1 Kings, chapter 7, verses 23-26

Around 250 BC, the Greeks were the first to create a rigorous algorithm to calculate pi. Archimedes, a Greek mathematician, used the method of computing the perimeter of circumscribed and inscribed 96-sided polygon to obtain an upper bound and lower bound for pi. He proved that pi was between 3.1408 < π < 3.1429 (or 223/71 < π < 22/7). The polygon method spread to the Eastern world in places like China and India. In 265 AD, Chinese mathematicians Lui Hui and Zu Chongzhi used the polygon to approximate pi with 3,072 and 12,288-sided polygons, respectively. Chongzhi’s 12,288-sided polygon method approximates pi to be between π ≈ 355/113 = 3.14159292035 and π ≈ 22/7 = 3.142857142857, which remained the most accurate for more than 800 years. As the polygon’s sides increased, the approximation of pi becomes more precise.

In the early 16th century, the race to find the most digits of pi commenced. The calculation of pi reached new heights with the created of the infinite series in India around 1500AD and calculus in Europe in 1600AD. European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz popularized the infinite series approach to calculate pi. William Shanks, a British mathematician, held the bragging rights for the largest approximation with the infinite series, which resulted in the calculation of 707 digits. He made a mistake in the 528th digit, so the real record stood at 527th until the mid-20th century. The computer era in the 20th century gave super sonic engines to the already soaring field of pi approximation. Early computers utilized iterative algorithms to calculate thousands of digits, which later replaced the infinite series. The Army’s ENIAC (Electronic Numerical Integrator and Computer) first and only computer in the United States created in 1949 was the beginning of a new era in pi computation. The Hungarian mathematician, John von Neumann championed the use of ENIAC to calculate pi to over 2,000 decimal places to research the statistical randomness of constants with non-terminating decimals. His research sparked a transition from the academic advantage of calculating pi to one of a competitive nature. After 1980, the iterative algorithms were more widely used because they were faster and more accurate than the infinite series method.

Pi as the underdog

With more computing power and manipulations of the iterative algorithm method we are now equipped with 62.8 trillion decimal places, in the matter of 300 days as opposed to the years it took for mathematicians to calculate digits by hand. The number 62.8 trillion sounds like an unfathomable accomplishment, something that is truly beyond human comprehension. Suresh Kumar Sharma currently holds the world record on the Pi World Ranking List for the Most Decimals Places of Pi Memorized. He memorized 70,030 digits by associating the numbers with picture. The world record holder a grand total of 17 hours and 14 minutes to recite the numbers. If you are reading this, then I bet you are wondering what the significance is of finding all these digits. The main objective is to flex computing power and the precision of their algorithm of choice. The calculation of pi is a mere byproduct to the fame of becoming a world record breaker. This is the sentiment of many pi-related record holders because of the infinite nature of decimal. This quality of the constant makes it hard to memorize or compute and easy to forget. The way that mathematical concepts are reduced to symbols or simplifications reflects that some mathematical concepts can be approached, but never reached.

Although pi seems unnecessary, there are plenty practical applications of pi that can dispel this myth. The assumption is that only mathematicians, scientists, and engineers use pi, but this belief is false. Pi is also used in communications, music theory, pendulum designs, navigation, and air travel.  Let’s start at the beginning with the circle, without pi, the notion of the perfect circle would not exist. Circles are what make the world go round, literally, and cars would cease to exist amongst other things. The constant provides an alternative way to describe circles in the units of radians. Before the creation of pi, scholars used 360 degrees to denote rotational angle of any circle. The concept of degrees does not have any foundation in math and is arbitrary, in every sense of the word. The introduction of radians connected the obscure constant to concrete measurement of the circumference of the circle. This is usually the first encounter that students have with the elusive constant, in a high school geometry or pre-calculus class. This is where the appreciation of pi must be fostered, so students can see how it is used in daily life. Pi is to be cherished and not reduced to mere memorization for the sake of a class.