Numbers (1)

The ancient and revered branch of mathematics known as number theory is a realm devoted to the study of integers and integer-valued functions. Often referred to as the "queen of mathematics" by the great German mathematician Carl Friedrich Gauss, number theorists delve into the mysteries of prime numbers and the properties of mathematical objects built from integers. These objects can range from rational numbers to algebraic integers, and the study of number theory often involves the analysis of analytical objects that encode the characteristics of integers, primes, and other number-theoretic elements. In addition to exploring the pure world of integers, number theorists also examine the relationship between real and rational numbers, such as through the process of Diophantine approximation. While the term "number theory" has largely replaced the older term "arithmetic" in modern times, the latter term has seen a resurgence in recent decades, particularly due to its widespread use in French mathematics.

One of the earliest known examples of arithmetic can be found in the form of a broken clay tablet called Plimpton 322, dating back to approximately 1800 BC in Mesopotamia. This tablet, also known as the "Plimpton 322 tablet," contains a list of what are known as "Pythagorean triples," or sets of integers that follow the equation a^2 + b^2 = c^2. It is believed that the large number and size of these triples could not have been determined through mere guesswork, leading scholars to speculate that the tablet's creator used some sort of algorithmic method to derive them. The purpose of the tablet remains a mystery, with some suggesting that it served as a tool for solving school problems, while others propose that it had practical applications in fields like Babylonian astronomy. It is worth noting that, while this tablet represents the only surviving fragment of Babylonian number theory, the civilization's understanding of algebra was highly advanced. It is even rumored that the famous mathematician Pythagoras may have learned some of his mathematical skills from the Babylonians during his travels.

The ancient Pythagoreans placed great significance on the concepts of odd and even numbers, and it was through their studies that the irrationality of the number √2 was first discovered. This revelation marked a turning point in the history of mathematics, as it necessitated a distinction between numbers and lengths or proportions, with the former encompassing integers and rational numbers and the latter encompassing real numbers of all kinds. The Pythagoreans also explored the concept of figurate numbers, which included square, triangular, and pentagonal numbers, among others. While the study of these numbers may not have had immediate practical applications, they would later prove to be important in the development of modern mathematics. In contrast to the rich tradition of mathematics in ancient Greek culture, there is relatively little evidence of significant arithmetical or algebraic progress in ancient Egyptian or Vedic societies. However, the Chinese remainder theorem, a fundamental concept in number theory, can be found in the Sunzi Suanjing, a Chinese text dating back to the 3rd, 4th, or 5th century CE. Chinese mathematics also included elements of numerical mysticism, though these ideas do not seem to have led to significant advancements in the field.

Much of what we know about the mathematics of ancient Greece comes from the writings of non-mathematicians who lived during this time period, as well as from mathematical works produced during the Hellenistic period. One notable figure in the history of Greek mathematics is Pythagoras, who is said to have studied with the Magi in Babylon and the priests in Egypt, as well as with Indian philosophers known as Brahmans. Pythagoras is credited with introducing a wide range of mathematical concepts to the Greeks, including geometry, arithmetic, music, and astrology, among others. According to Aristotle, the philosophy of Plato was heavily influenced by the teachings of the Pythagoreans, a claim that was later echoed by Cicero.

Plato was highly interested in the field of mathematics and made a distinction between arithmetic and calculation. Through the writings of his disciple, Theaetetus, we know that Theodorus was able to prove that √3, √5, and √17, among others, are irrational numbers. Theaetetus, in turn, focused on identifying different types of incommensurable numbers and is considered a pioneer in the study of number systems. Euclid, another important figure in the history of number theory, devoted a significant portion of his work "Elements" to the study of prime numbers and divisibility. He provided a method for calculating the greatest common divisor of two numbers, known as the Euclidean algorithm, and gave the first known proof of the infinite number of prime numbers. In 1773, a letter supposedly written by Archimedes to Eratosthenes was discovered, containing a problem known as Archimedes's cattle problem. The solution to this problem involves solving an indeterminate quadratic equation, a task that was first successfully undertaken by the Indian school of mathematics. It is unclear whether Archimedes himself had a method for solving this type of equation.

Very little is known about the life of Diophantus of Alexandria, but it is believed that he lived in the 3rd century AD. His work, "Arithmetica," is a collection of problems that involve finding rational solutions to systems of polynomial equations. These equations, which are now known as Diophantine equations, typically take the form f(x,y)=z^2 or f(x,y,z)=w^2. Diophantus is considered a pioneer in the field of algebraic geometry, as he focused on finding rational parametrizations of algebraic varieties and rational points on curves and varieties. He also studied equations of non-rational curves, such as elliptic curves, and used tangent and secant constructions to find rational points on these curves. While Diophantus primarily focused on rational solutions, he assumed some results about integer numbers, including that every integer can be expressed as the sum of four squares.

Indian mathematics is thought to have developed independently from Greek astronomy, although it is possible that Greek astronomy had some influence on Indian learning, including the introduction of trigonometry. Āryabhaṭa, who lived in the 5th century AD, developed a method for solving pairs of simultaneous congruences known as the kuṭṭaka, or pulverizer, which is similar to the Euclidean algorithm. Brahmagupta, who lived in the 7th century AD, is credited with the systematic study of indefinite quadratic equations, including the so-called Pell equation. Later Sanskrit authors, including Jayadeva, continued to work on the Pell equation and developed the chakravala, or cyclic method, for solving it. Indian mathematics remained largely unknown in Europe until the late 18th century, when the works of Brahmagupta and Bhāskara were translated into English.

In the early 9th century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works, including Diophantus's Arithmetica, which was translated into Arabic by Qusta ibn Luqa. Al-Karajī's treatise al-Fakhri builds on this work to some extent and it is believed that his contemporary, Ibn al-Haytham, knew what would later be called Wilson's theorem. During the Middle Ages, there was little number theory in western Europe, with the exception of a treatise on squares in arithmetic progression by Fibonacci. In the late Renaissance, there was renewed interest in the works of Greek antiquity and the emendation and translation of Diophantus's Arithmetica into Latin played a significant role in the development of number theory in Europe.