Section 1.1: The Circle Constant: π

π, often referred to as the circle constant, represents the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. This constant finds application across various mathematical fields, including geometry, trigonometry, and calculus, as well as in probability, statistics, and number theory.

The allure of π has fascinated mathematicians, scientists, and philosophers for centuries, tracing its origins back to ancient Babylon between 1900–1600 BC. The Babylonians approximated π as 3.125, a commendable estimate for the era. Subsequently, around 1650 BC, the ancient Egyptians calculated their own approximation of π, arriving at a value of 3.16, documented in the Rhind Mathematical Papyrus.

The intrigue surrounding this number progressed to ancient Greece, where the distinguished mathematician Archimedes tackled the challenge around 250 BC. He pioneered a systematic method to approximate π, ingeniously bounding it between two values using inscribed and circumscribed polygons. Archimedes computed that 3.1408 < π < 3.1429, marking a significant advancement in the understanding of π. His polygonal algorithm prevailed for over a millennium, earning π the nickname "Archimedes's constant."

As time passed, the quest for π continued. In the 5th century, Chinese mathematician Zu Chongzhi refined Archimedes' technique, calculating π to an impressive seven decimal places: 3.1415926 < π < 3.1415927. This achievement remained unmatched for nearly a thousand years.

During the European Renaissance, advancements in the decimal system and the advent of calculus transformed the calculation of π. In the 17th century, Isaac Newton utilized his newly developed calculus to compute π to 15 decimal places.

In the 20th century, mathematicians harnessed powerful computers to pursue π's elusive digits. John Wrench and Levi Smith used a basic calculator in 1949 to compute π to 1,120 decimal places. Later innovations in computing allowed mathematicians to calculate π to millions and even billions of decimal places.

Today, we understand that π is not only an irrational number but also transcendental, adding to its allure. The exploration for more digits of π continues, as its infinite, non-repeating decimal expansion continues to enthrall mathematicians and enthusiasts alike.

Section 1.2: The First Irrational Number: √2

The number √2, approximately equal to 1.41421, is significant as the first number proven to be irrational. This tale unfolds in ancient Greece, around the 6th century BC, involving the mathematician Pythagoras and his followers, the Pythagoreans. They held the belief that the universe could be understood through whole numbers and their ratios.

A Pythagorean mathematician, Hippasus of Metapontum, made a groundbreaking discovery while examining square numbers. He found that the diagonal of a square with sides of length one could not be expressed as a ratio of whole numbers, indicating that √2 was not a rational number. This revelation destabilized the Pythagorean worldview and reportedly led to Hippasus facing dire consequences for his finding.

Hippasus' discovery marked a pivotal moment in mathematical history, revealing that some numbers could not be represented as simple ratios of integers. This opened the floodgates for further exploration of irrational numbers throughout the ages. Today, √2 remains essential in geometry, number theory, and algebra.

Section 1.3: The Calculus Superstar: e

The number e, approximately 2.71828, serves as the base of natural logarithms and plays a crucial role in calculus, with applications across various mathematical domains, including number theory and complex analysis. It is often referred to as Euler's number, in tribute to the mathematician who highlighted its significance.

The origins of e can be traced back to the concept of compound interest. In the early 17th century, mathematician Jacob Bernoulli pondered the implications of investing money at a 100% annual interest rate compounded more frequently. He discovered that as the number of compounding periods increased, the total approached a limit, a number close to 2.718—an early glimpse of e.

As the understanding of e evolved, its mystical properties began to emerge in other branches of mathematics. In the early 18th century, Leonhard Euler further investigated this constant, deriving a formula for the exponential function e^x. Euler's work revealed that e^x possesses the remarkable property of being its own derivative, making it instrumental in solving differential equations, which form the foundation of theoretical physics and engineering.

Moreover, Euler’s identity showcased a profound connection between e and trigonometric functions, linking it to five of the most significant mathematical constants: e, i, π, 1, and 0. This relationship underscores e’s status as one of the most important numbers in mathematics.

Section 2.1: The Golden Ratio: φ

Known as the golden ratio, φ is approximately 1.61803, and unlike π, e, and γ, it has a closed-form formula involving √5. Our journey with φ begins in ancient Greece, where Pythagoras and his followers recognized unique aesthetic proportions in certain ratios, which they believed were key to understanding the universe.

The earliest recorded mention of the golden ratio appears in Euclid's works around 300 BC. He defined it in his seminal collection, "Elements," stating, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser."

In modern terms, we express φ as x/a, where x and a satisfy (x+a)/x = x/a. The solution can be derived to give φ = (1 + √5)/2. The story of the golden ratio took an intriguing turn in the early 13th century when Italian mathematician Leonardo Fibonacci discovered a sequence while studying rabbit populations. This Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, reveals that the ratio of consecutive numbers converges to φ.

The golden ratio's prevalence extends into nature, appearing in designs such as nautilus shells, sunflowers, and even distant spiral galaxies.

Section 2.2: Apéry's Constant: ζ(3)

Apéry's constant, approximately 1.20206, tells a tale of mathematical surprise, creativity, and the merger of analysis with number theory. This narrative begins in the 18th century with Leonhard Euler, who explored the zeta function, denoted ζ(s). Euler uncovered captivating properties, including the unexpected connection between the zeta function and prime number distribution.

In 1734, Euler famously proved that ζ(2) = π²/6, a revelation that shocked many mathematicians. However, a closed-form formula for odd zeta values remained elusive, becoming a kind of holy grail in analysis.

The plot thickened in the 20th century when Roger Apéry made a groundbreaking discovery. In 1978, he proved that ζ(3) is irrational, an unprecedented finding in the realm of odd zeta values. This achievement, known as Apéry's theorem, was monumental as proving a number's irrationality is notoriously difficult.

While we know little about ζ(3), the quest for a closed-form formula remains a coveted objective among mathematicians.

Section 2.3: Euler's Constant: γ

Euler's constant, sometimes called the Euler-Mascheroni constant, is approximately 0.57721 and holds a special place in the hearts of many mathematicians. The story of γ began in the 18th century, with Leonhard Euler exploring the harmonic series: 1 + 1/2 + 1/3 + 1/4 + ⋅⋅⋅.

Although this series diverges to infinity, Euler sought to understand its rate of divergence. He discovered that the difference between the sum of the first n terms and the natural logarithm ln(n) approaches a constant value—γ.

As time progressed, mathematicians continued to uncover the properties of Euler's constant, revealing its presence in number theory, complex analysis, and asymptotic expansions. However, its true nature remains enigmatic; it is still unknown whether γ is irrational or can be expressed as a fraction.

Section 2.4: The Imaginary Unit: i

The journey of the imaginary unit, defined as the square root of -1, is one of innovation and bold exploration of once-taboo mathematical concepts. This story unfolds in the 16th century during the European Renaissance, as mathematicians grappled with polynomial equations leading to perplexing square roots of negative numbers.

Initially dismissed as meaningless, these "imaginary" numbers gained legitimacy through geometric interpretations and practical applications. Mathematicians like Leonhard Euler and Abraham de Moivre advanced the concept of imaginary numbers and complex numbers in the 18th century.

Today, complex numbers, represented as a + bi, where a and b are real numbers, are indispensable in mathematics, including complex analysis, number theory, and differential equations. Furthermore, they serve as crucial tools in engineering and quantum mechanics, allowing for precise modeling of alternating current circuits and deepening our understanding of the fundamental laws governing the universe.

As we conclude this exploration of mathematical constants, I share my favorite: γ. Its pervasive presence in mathematics, paired with the mystery surrounding it, keeps me fascinated. Proving the irrationality of γ would be a monumental achievement.

This wraps up our discussion. What about you—what's your favorite number? If you enjoyed this article and wish to read similar content, consider joining the community on Medium for full access.