1. Introduction
The ratio of a circle’s circumference to its diameter is a mathematical constant that when applied, can determine unknown properties of a given circle with relative accuracy. Represented by the sixteenth letter in the Greek alphabet, Pi is a timeless and world-renowned number. Its fame, however, is disproportionate to the members of society who truly understand the feat that is this irrational number. The concept of Pi has changed the way humans quantify numbers. In a sense, Pi possesses a dual role. Early in history, the pursuit of the ratio caused mathematical thought to evolve. Later, as mathematics progressed, it turned around and began evolving Pi.
This essay aims to reveal the extent at which Pi has been the cause and the effect of mathematical development. In other words, the purpose is to explain Pi’s role in furthering mathematics as well as how advance mathematics furthered the development of Pi. Understanding this relationship reflects mathematics and its evolution throughout history. Pi’s expression reflects the level of mathematics from the early, algebraic 256/81 to the iterative and transcendent expression through calculus.
In short; To what extent has Pi been the cause and the effect of Mathematical development between the period of 1900BCE-1700CE? To find the answer, a full understanding of Pi’s as well as the history of mathematics is imperative. The structure, therefore, will begin with how Pi furthered mathematics early in algebraic and geometric mathematics and continue to the second section where Pi was furthered through advance mathematics. In essence, the paper will be structured chronologically which will result in a comprehensive analysis of Pi’s role as the impetus for mathematical development and as the product of advanced mathematics.
Part 1: Pi’s role in furthering mathematics
2. Babylonians and Egyptians
Pi and the discovery of the ratio in the ancient world set the stage for mathematical development. Evidence from Babylonia and Egypt reveal knowledge of the number however, understanding of Pi’s application was not demonstrated. This can be seen in historical relics: Babylonian clay tablets and Egyptian Rhind Papyrus.
Babylonians are credited with the first expression of Pi with the discovery of a clay tablet in Susa3 (ca. 1900-1650BCE) . The tablet is impressed with three cuneiform numbers:
fig.1
Translated from Babylonian numbers, assumptions have to be made based on the locational context of the tablet. According to Dyer, 3 is the circumference and using the formulas for a circle’s area and perimeter, Pi can be deduced simply as 3. The Babylonians landed perchance on a rounded expression of Pi. They did not know the implications of the number. Therefore, they did not further their mathematics pursuit of Pi.
The Egyptians demonstrated a deeper understanding of Pi. This is clearly seen in problem 50 of the Rhind Mathematical Papyrus.
Example of a round field of diameter 9 khet. What is its area? Take away 1/9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore, it contains 64 setat of land.
Ahmes, writer of the RMP, assumes that the area of a circular field is equal to the area of a square with sides one ninth the diameter . With an established thought process, Ahmes finds the area. Mathematicians can compare problem fifty and its work to contemporary methods, solving for the Egyptian interpretation of Pi. A possible rationale to the subtraction of 1/9 is illustrated by the diagram below
fig. 2
Assuming that the diameter is twice the radius the formula presented in sentence would be, “subtract one ninth from twice the radius squared.” Mathematically, this would be:
2r-1/9 2r
(8/9 (2r))^2
(64/81 (4r^2 ))
256/981 r^2
fig. 3
When compared to the current formula for finding the area of a circle πr2, Pi is 256/81. In decimal form, this is 3.1604. This is relatively accurate however they did not demonstrate true understanding of Pi. Ahmes could have used a square as an approximation, but he did not clarify his methodology. This leads historians to believe that he, alongside other Egyptian Mathematicians influenced by the Rhind Mathematical Papyrus, truly believed that the area of the reduced square was equal to the original circle
Both the Babylonians and the Egyptians performed mathematics only up to an applicable level. For example, the Egyptians utilized plane geometry when constructing grain silos where the structure had to be measured . Amazingly, both ancient groups encountered Pi however, they possessed only a concrete comprehension of Pi. Lacking thoughts of Pi in the abstract sense sparked no curiosity. Lacking in this area caused no subsequent inquiries as claimed previously. While the Babylonians and Egyptians roughly found Pi, they did not understand the implications of the ratio. Therefore, they did not further their mathematics pursuit of Pi, resulting in the evolution of mathematics.
3. Archimedes
The Greeks are known for their philosophy. Their inquisitive minds inquired, leading to discovery. This was the case with the Greek Philosopher Archimedes and his approximation of Pi.
As stated previously, the Babylonians and the Egyptians set the stage for innovation. While they did not push the frontier of mathematics with their expression of Pi, their contribution of discovery and recognition set the stage for innovators like Archimedes to pioneer mathematics.
Thinkers at the time of Archimedes knew of the ratio between a circle’s perimeter. They found this recurring number through the comparison of areas . Archimedes sought to find the ratio exactly as opposed to an approximation based on recurring numbers. To determine the ratio with precision, he created a new recursive method that would calculate Pi with greater precision. Archimedes used mathematics current to him, Euclidian (ca. 300BCE) , and applied it in a new way to create this iterative method.
According to Euclid’s sixth book of Elements, his 3rd proposition states,
If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and, if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.
Simply put, Euclid proposed the proportionate relationship within triangles when their angles are bisected. Archimedes used Euclid’s proposition of triangles and the ratios of sides in circumscribing polygons. Initially, Archimedes placed a hexagon with a known perimeter in a circle. From there, he bisected the equilateral triangles, consequently creating a new polygon with double the number of sides.
fig. 4
The figure above represents the methodology Archimedes employed when circumscribing a polygon.
“(1) the ratio of OA:AC [= sqrt (3):1] >265:153 and
(2) the OC:AC ratio is [= 2:1] = 306:153.”
Utilizing Euclid’s third proposition, Archimedes proves the relationship between the triangle’s ratios. Summarizing an extensive proof, he calculated the lengths of sides opposite to O, doubled them, and multiplied the value from the previous problem by the number of sides. With the perimeter, Archimedes created a fraction of the perimeter to the diameter. He then repeated this process with an inscribed polygon. Utilizing the same proposition, he created triangles all related to one another and determined the ratio of the polygon’s circumference to its diameter. He circumscribed and inscribed polygons in order to approximate with greater accuracy. He assumed that his values would be greater and less than the true value of the circle thus, forming an inequality.
fig. 5
This was the process for the hexagon. Archimedes began with a polygon of which he knew the ratio of the sides to the perimeter. Establishing the methodology, he then began bisecting angles, creating a 12-sided figure, a 24-sided figure, and so on. He repeated his methodology four times until he reached a 96-figured polygon and determined pi between the values of the circumscribed and inscribed polygons, 3 1/7 < π < 3 10/71. Lacking a decimal system, Archimedes was limited to rational numbers and notation. In decimal form, his approximation comes out to 3.1401. Accurate to two decimal places, this is viewed as one of the greatest advances in Pi.
The significance of this estimation lies not in the number of decimal places. The fact that Archimedes created an iterative method, which theoretically could be repeated to eternity, hundreds of years prior to calculus, shows how the conceptualizing of mathematics was developing. This is the epitome of mathematical evolution. Here, Archimedes took existing mathematics, Euclidian geometry, and innovated a process resembling calculus hundreds of years before Newton and Leibniz. Concerning the Greeks and Archimedes, the pursuit of “[the] ratio of the circumference of any circle to its diameter” innovated existing mathematical methods, resulting in the evolution of mathematics.
4. Chinese
Chinese mathematics were advanced relative to the rest of the world during the first century CE. A Eurocentric claim says that Chinese mathematics was due to intercommunication with European nations however, the math used in China was “so different from that of comparable periods in other parts of the world that the assumption of independent development would appear to be justified.” Particularly unique to the Chinese was their utilization of square roots and how they applied this concept to circles, furthering their conceptualization of geometric Mathematics.
The Chinese combined a geometry as well as binomial expansion to determine the roots of large numbers . A problem from The Nine Chapters on the Mathematical Art (Chiu Chang) provides an example of roots:
There is a [square] field of area 71824 [square] pu [or paces]. What is the side of the square?
To solve this equation, they estimated the root to be a three-digit number and performed trial-and-error until they reached a solution. This would create squares and rectangle within the original square requiring binomial expansion.
fig. 6
While this is an oversimplified explanation of a complex process, the figures show for the Chinese’s understanding of roots from a binomial and geometric point of view. Zu Chongzhi, and his son , would use existing mathematics and innovate a process to solve pi; just as Archimedes took existing Euclidian mathematics and estimated pi.
Zu Chongzhi created polygons inside the circle, again a geometric approach, to find the estimation of pi. Similar to Archimedes, Chongzhi began doubling the number of sides of the inscribed polygon. Unique to the Chinese, Chongzhi used the Chinese understanding of square roots to solve the values of the polygon sides rather than the ratio of the side.
fig.7
The use of square roots resembles Pythagorean theorem however, this similarity could be rooted in the use of square roots rather than intercommunication between regions. According to Cooke, Chongzhi understood that “If s_n is the length of the side of a polygon of n sides inscribed in a circle of a unit radius then, s_n=2-√(2&4-s_n^2 ).” . The number of sides represented by n can be go on to infinity. Theoretically, every digit of pi can be calculated which highlights the advantage of writing.
The Chinese, demonstrated by Zu Chongzhi, took present-day mathematical methods for his time period and applied them to a circle to find Pi. Existing methodology for finding square roots were applied to a circle, similar to the Pythagorean Theorem. This enabled Chongzhi to find the actual value of the side as opposed to Archimedes’ ratios and proportions. From this, he was able to calculate pi to decimal places father than the Greek’s approach with fraction. This innovation demonstrates how Pi and the pursuit of the ratio resulted in the further conceptualization of mathematics. In terms of the Chinese, Pi furthered their understanding of the abstract realm of math.
5. Preliminary Conclusion
Thus far in the investigation, Pi and the pursuit of this perplexing ratio has caused mathematics to evolve. Pi has advanced mathematics in the sense that existing methods had to be reworked, innovated to find the ratio with greater precision. The Babylonians and Egyptians set the stage for other people groups to analyze and inquire about it. Archimedes, based on Euclidian propositions, innovated a process with triangles to create polygons, circumscribing and inscribing them and determining their perimeter. The Chinese were already well versed in square roots and they utilized roots to create polygons.
Both people groups postulated methods of finding the ratio using math available to them. While they did not create a new branch of mathematics, they evolved the subject at the conceptual level. They applied existing methods to solve a new problem. With regards to the ancient world, Pi caused the evolution and progression of mathematics as thinkers at the time innovated existing methods to solve a new problem.
Part 2: Mathematics’ role in furthering the expression of Pi
6. Extending the Investigation
Pi, by researching the ancient world and their methods to finding the ratio, can be said to have caused mathematical development. The math used however, was geometric and algebraic, applicable to the real world. Mathematics took a turn in the mid-17th century with the birth of calculus. This new branch of Mathematics utilizes an iterative process. Where a formula is repeated until the distance from the solution is infinitesimally small enough to be assumed as correct. This evolution causes roles to switch in the investigation. Rather than discuss how Pi caused Mathematics to evolve, this new recursive approach can be turned around and evolve pi, calculating the irrational number exhaustively.
7. Newton
Isaac Newton used mathematics pioneered by him and redefined pi. Mathematicians at the time were aware of an iterative process however, there was a distrust in solutions derived by this method. It was an approximation and did not produce an answer as accurate as algebraic means. Newton however, proved the accuracy of the infinite series and declared that “the analysis by infinite series had the same inner consistencies and was subject to the same general laws as the algebra of finite quantities.”
This evolution and pivotal discovery in the realm of mathematics opened another door for Pi. The ratio would no longer be expressed as an inequality. With Newton’s discovery concerning the infinite series, the value of Pi could be calculated, in theory, to infinity. This is in contrast to the Greeks and Chinese and their estimations. They created an estimation which tested and furthered their mathematical abilities. Newton, in contrast, created and mastered a new branch of mathematics which he then used to modernize the expression of pi.
Through geometrical means, Newton analyzed a semi-circle and derived an infinite series which would calculate pi to endless decimal places. First, found the formula of a circle and rearranged the equation to represent a semi-circle:
(x-1/2 )^2+(4-0)^2=(1/2 )^2
x^2-x+1/4+ y^2=1/4
y=√(x-x^2 )
=√x √(1-x)
= x^(1/2) (1-x)^(1/2)
y=x^(1/2) (1-x)^(1/2)
With the equation of the semi-circle, Newton drew a perpendicular line halfway between the center point C and point B. In simplified terms, he then used the procedure outlined in rules of De Analysi. He created a curve and found the area beneath the curve BA.
fig. 8
He used the rules of binomial expansion to begin creating the infinite series. Newton then created a 30-60-90 special triangle to find the area of the sector, a third of the semi-circle’s area. Treating the y and x values as flowing, he was able to form the equation that would determine pi.
fig. 9
The significance, to reiterate, lies in the application of the infinite series. The evolution of the iterative process, independent of pi, was then applied to the ratio to define it in more precise terms. Newton proved the consistency of the infinite series prior to his redefinition of Pi, an evolution of math. Consequently, Pi evolved as well.
6. Euler
Pi is a number however, what kind of number. It is an irrational number however, it cannot be calculated algebraically. As time progressed, mathematicians began exploring the abstract realm of mathematics deeper. This led to a major grouping of numbers as well as the introduction of another category of numbers. Pi would fall into this new category, redefining it once again. This major concept was set in stone after conjecture by the works of Euler, a Swiss mathematician .
Euler was pivotal in redefining mathematics in the 18th century. One of his greatest contributions is the constant e. Euler arrived at this number exploring the concept brought up by Bernoulli. Exploring the concept of compound interest, Bernoulli presented a hypothetical situation asking whether it would be better to have 100% interest per year, or 50% interest twice a year. He then continued in the pattern: 25% four times a year, 1.923% interest 52 times a year, etc. Bernoulli then asked what the value of the dollar would be if the interest would be compounded continuously/infinitely. Euler calculated this constant and discovered it was an irrational number with the properties essential to growth and rate of change . He called it e however what kind of number was it?
Mathematics handles numbers of all types. They fall into the categories of integers, fractions, irrational, and complex numbers, etc. However, Euler proved that they all fall into the same category, algebraic. They can be manipulated and solved using algebra. Demonstrated by Simon Pampena , he walks through Euler’s proof in defining algebraic numbers. He does this by taking all types of numbers and manipulating them to zero using algebraic functions: addition, subtraction, multiplication, division, and exponentials.
Number Manipulation to 0 Algebraic Expression
10 10-10=0 x-10=0
3/4 4(3/4)-3=0 4x-3=0
√2 (√2 )^2-2=0 x^2-2=0
√(-1) (√(-1))^2+1=0 x^2+1=0
√2+√3 5+2√2 √3
5+2√2 √3-5
(2√2 √3 )^2
4×2×3
24-24
0 (x^2-5)^2-24
x^4-〖10x〗^2+1=0
fig. 10
The question still poses, what type of number would e fall into? Charles Hermite sought to prove e as an algebraic number. To do this, he declared that in order for e to be algebraic, a number would have to exist in between zero and one. This leads to a contradiction. Therefore, Hermite proved that e was a number that transcended mathematics as they knew it. Euler defined numbers where algebra, in its purest form, had no use . He called numbers such as these, transcendental. Could the same be said about Pi?
Carl Louis Ferdinand von Lindemann declared that e raised to any algebraic number is transcendental . What if e was raised to the power Pi? If the solution is transcendental, Pi would fall into the family of algebraic numbers.
e^a=transcendental number such that a is algebraic
e^1=e
The constant e to the power of 1, an algebraic number, is e, a transcendental number. This proof works. Euler created an identity that uses Lindemann’s proposal to prove that pi is transcendental.
e^iπ=-1
Euler’s identity raises the constant e to a value but equals -1. The power of which the constant is being raised to is iπ where i=√(-1). Earlier, Pampena proved that √(-1) is algebraic, meaning that Pi is transcendental.
Combining the works of multiple mathematicians, Euler created an equation that proves Pi as transcendental. It goes beyond the familiar realm of arithmetic. This ratio derived from a circle, an everyday object, transcending the mathematics that has been used for millions of years. The abstract mathematical concept of transcendental numbers was created and Pi was eventually assigned under that category. Another example, just as with Newton, where mathematics evolved and pi was redefined as a result of it.
8. Conclusion
Pi played a pivotal role as the impetus for mathematical development. Ancient thinkers had to employ mathematics available to them and innovate them, expanding their abstract understanding. This set the stage for later thinkers like Newton to develop a new branch of mathematics. On the other hand, the current expression of Pi was the result of advanced mathematics. This dual relationship throughout history raises the question, which factor had the greatest effect on the other: did Pi have a greater impact on math or did advanced mathematics have the greatest impact on Pi?
Based on the research done, the degree at which advanced mathematics affected Pi was greater than the extent Pi influenced the development of ancient arithmetic. Despite the Chinese’s and Greek’s innovation of existing arithmetic, their impact on the practice as a whole was negligible. The effects were conceptual; they developed a new way at looking circles exclusively. The exhaustive method created by Archimedes and Zu Chongzhi were not applied in later problems for they were viewed as an inaccurate and tedious process. Calculus, on the other hand, had a lasting impact on Pi. Calculus determined a method that could theoretically find the infinite solution. Newton’s thought experiments that lead to the infinite series was the cause of this great paradoxical feat. In addition, Pi helped in creating a new realm of values, transcendental numbers. These numbers truly transcended existing mathematics and require advanced methods to theoretically calculate them.
Pi and the pursuit of its value says something about humanity. The amazing number is only a piece within an astounding subject, and the subject only one among the countless others that embody the inquiring, human mind. While Pi received more than it contributed to mathematics, the ratio is still a testament to what distinguishes humans from animals no matter the era.