Three Icons of the Mathematical World
A Brief Introduction
“The most remarkable formula in mathematics,” ei+ 1 =0, as claimed by Richard Feynman, a profound theoretical physicist. Euler’s identity captures the presence of three phenomenal mathematical constants. e, the base of natural logarithms; , a ratio between diameter and circumference; and i, the square root of -1. All these numerical values stem from different origins, but they collectively share a critical part in understanding the laws and principles of the mathematical world. But what exactly are these numbers used for, and where do their roots emerge?
Euler’s Number
Many of us have certain constants in our lives. Whether it be our go-to take-out order or the hairstyle that never fails us, we rely on these constants to keep bringing a sense of stability and comfort. In mathematics, these constants are just as fundamental, aiding us through complex calculations. One such example of this is Euler’s number, often seen as e. This never-ending constant is used widely for calculating growth, decay, and most importantly, natural logarithms.
Though named after Swiss mathematician and physicist Leonhard Euler, its discovery can be traced back to another Swiss mathematician, Jacob Bernoulli. In 1683, Bernoulli set out to calculate the future value of an investment at a yearly interest rate of a hundred percent, compounded at different frequencies. Through this process, he established that as compounding frequency increased, “future value of the investment approached approximately 2.718.”
Leonhard Euler was the first to prove that e was the infinite sum of inverse factorials or e=10!+11!+12!+13!+...+1n!, as n approaches infinity. He started to use the notion e to represent this constant. His work extended far beyond this discovery, as he demonstrated how this could be used in logarithms and exponential functions, laying the groundwork for modern mathematical concepts.
In everyday life, Euler’s number has become the key to many real-world applications such as population growth, exponential decay, and finance. Overall, this constant is embedded into occurrences in which things change at a rate proportional to their current state.
Pi
There are many non-ending things in the world. Time, space, doing the dishes, and our cravings for a mouthwatering piece of pie are just a few examples of this. Similarly, pi, or often seen as , is a number that never ends. This irrational number is seen throughout geometry as it correlates a circle’s circumference to its diameter. This fascination has been dated back almost four thousand years.
Its earliest appearance comes from the Babylonains and their attempts to calculate the area of a circle. They had originally did this by multiplying three times the square of the radius, but in a later tablet, a value of 3.125 was used for . In ancient Egypt, a value of 3.1605 was commonly used. Both of these definite values were just the first steps to truly understand the value of pi.
It was not until Archimedes of Syracuse who estimated that the exact value lay between 3 1071 and 317, giving an average value of 3.1418. He did this through inscribing polygons inside and outside of a circle to approximate the value of pi. Throughout the years many other mathematicians extended on Archimedes work, through many tedious calculations. However, an introduction to the infinite series in the seventeenth century allowed for new and improved ways to find the exact value of . Today, computers have been able to find its true value up to one hundred trillion decimal places.
This irrational constant appears in many fields, such as physics, astronomy, and engineering. Pi is an integral part of both ancient and modern mathematics, governing numerous principles in the world around us. It describes phenomena that are not limited to circles, but waves, rotational motion, and oscillations. The never-ending digits symbolize limitless ends and boundless discoveries in a realm of math and science.
i
We all indulge in our own imaginary world sometimes. We read books where we dive into a mythical world and explore the characters’ lives with deep intrigue. We create fictitious conversations in our heads, imagine alternate realities, and so much more. But what about those abstract concepts, like imaginary numbers which allow us to explore mathematics beyond the tangible world?
Imaginary numbers have been used for roughly six hundred years, and its first discoveries can be traced back to the need to solve cubic equations. The first person to have brought forth the idea of imaginary numbers can be stemmed from Girolamo Cardano, an Italian polymath. In his algebraic book, Ars Manga, published in 1545, Cardano acknowledged the existence of imaginary numbers. However, he did not fully establish the concept into the well-known and regarded one we have today.
It was not until twenty-eight years later in 1572 when Rafeal Bombelli addressed the problem of imaginary numbers. Bombelli’s algebra is regarded as “one of the most important and influential works ever published.” Although possibilities of negative numbers had appeared in the works of Cardano, it was Bombelli rather solidified the topic and provided a clear framework for understanding complex numbers. He created a system for manipulating imaginary numbers in algebraic equations, which had been previously seen as “fictitious” or “undefined.” Bombelli was the key to transform these concepts from a curiosity to a rather useful mathematical tool.
Today, imaginary numbers are used in a variety of different topics. In electrical engineering, they are used in the analysis of AC circuits. They measure amplitudes and phases of electrical oscillations, and allow engineers to convert complex differential equations into simpler algebraic equations. Imaginary numbers are used in quantum mechanics, where many theories can only be expressed with complex numbers. In signal processing, the Fourier transform, a “mathematical model which helps to transform the signals between two different domains,” contains complex numbers that are widely used for data compression and image processing.
Works Cited
Barnes-Svarney, Patricia, and Thomas E. Svarney. The Handy Math Answer Book. Visible Ink Press, 2006. Accessed 22 December 2024.
“Euler's Number (e) Explained, and How It Is Used in Finance.” Investopedia, https://www.investopedia.com/terms/e/eulers-constant.asp. Accessed 22 December 2024.
“The Feynman Lectures on Physics Vol. I Ch. 22: Algebra.” Feynman Lectures, https://www.feynmanlectures.caltech.edu/I_22.html#Ch22-S5. Accessed 22 December 2024.
“Fourier Transform - Definition, Formula, Properties, Applications and Examples.” BYJU'S, https://byjus.com/maths/fourier-transform/. Accessed 22 December 2024.
Jones, William, and Leonhard Euler. “A Brief History of Pi (π).” Exploratorium, https://www.exploratorium.edu/pi/history-of-pi. Accessed 22 December 2024.
Piccotti, Tyler. “Meet Archimedes of Syracuse, The Mathematician Who Discovered Pi.” Biography, 13 March 2024, https://www.biography.com/scholars-educators/a43249494/who-discovered-pi-archimedes-of-syracuse. Accessed 22 December 2024.
“Pi | Definition, Symbol, Number, & Facts.” Britannica, 20 December 2024, https://www.britannica.com/science/pi-mathematics. Accessed 22 December 2024.
“What Is Euler's Number? A Beginner-Friendly Overview.” Mathnasium, 16 September 2024, https://www.mathnasium.com/blog/what-is-eulers-number-in-math. Accessed 22 December 2024.
Pictures Captions:
Image of Swiss mathematician Jacob Bernoulli (1655-1705)
Image of Swiss mathematician and physicist Leonhard Euler (1700-1782)
Image of a Babylonian tablet that includes inscriptions to calculate the area of a circle
Image of ancient Greek mathematician and physicist, Archimedes of Syracuse
Image of Italian polymath Girolamo Cardano (1501-1576)
Image of Italian mathematician Rafael Bombelli (1526-1572)