Historia de los Números Reales

History of Real Numbers

The history of real numbers, also known as continuous quantities, dates back thousands of years ago when humans first began counting and measuring objects. Over time, various number system developments led to the creation of our current understanding of real numbers. In this article, we will explore the key milestones in the progression from ancient civilizations' early attempts at quantification to modern mathematics' sophisticated handling of real numbers in number theory.

Development of Number Systems

The earliest evidence of complex mathematical concepts can be traced back to Sumerians around 3500 BCE. They developed the base-60 numbering system, which is still used for angular measurement today. Babylonian mathematicians later refined this system by using sexagesimal notation—a positional numeral system with sixty as its base, making it easier to divide large numbers into smaller parts. This was followed by the Greeks' creation of rational numbers, represented as fractions like (\frac{1}{2}), which could represent ratios of sizes and amounts within their worldview.

Egypt made significant strides too, introducing numerals between one and nine as well as the symbol for zero, though they did not formulate the place value principle until much later. Between 700 CE and 900 CE, Indian mathematicians evolved what would become Hindu–Arabic numerals, including specific symbols for each of the ten digits plus zero, allowing simpler arithmetic operations compared to Roman numerals. This paved the way for our modern decimal system.

In Europe, Arabic scholars transmitted these innovations after the Abbasid revolution, integrating them into Islamic civilization where algebra became particularly influential. By the thirteenth century, European scholars had absorbed many aspects of al-Khwarizmi's work through translations from Latin versions of his texts. Al-Kharizmi himself introduced the concept of negative numbers and laid foundations for number theory.

Evolution of Real Numbers

Real numbers emerged gradually over centuries due to advancements in geometry and algebra. Greek philosophers such as Pythagoreans and Euclidean geometers viewed geometric magnitudes as basic entities rather than derived constructs. As geometry grew, so did the need for more precise measurements leading to the introduction of irrational numbers with square roots falling exactly midway between whole numbers. These included pi ((\pi)), phi ((\phi)), and sqrt(2).

Greek philosopher Aristotle played a crucial role here, distinguishing between discrete 'countable' numbers (like those represented by integers) and continuous quantities—the starting point for our contemporary understanding of infinite decimals. Around 800 BC, Archaic Period Greece established a non-geometric method based on patterns observed among odd versus even numbers; this approach spawned Pythagoreanism and Empirical Mathematics. For instance, the Pythagoreans discovered the theorem bearing their name involving right triangles while studying musical intervals related to prime numbers.

Algebra contributed significantly too. Ancient mathematicians used exponential notation before the invention of Arabic numerals. Early Hindu mathematicians knew how to solve quadratic equations by finding square roots. This allowed them to express solutions without realizing they were doing so implicitly. Eventually, real numbers came to encompass both rationals (fractions) and irrationals (incommensurables).

Contributions to Number Theory

Number theory itself has undergone several phases, beginning with early Greeks who studied only odd perfect squares because it directly corresponded to lengths of rods forming geometric figures. Diophantine equations gained prominence during Hellenistic times. Euclid's Elements outlined methods for solving certain types of diophantine problems, laying groundwork for future advances.

India also greatly influenced number theory through works such as Brahmagupta's treatise on algebra and Bhaskara II's book that could be considered India's equivalent to Fibonacci's Liubre Abaci. Greece continued advancing number theory throughout Medieval period thanks largely to Arabs transmitting ideas via holy books translated into Latin and Hebrew.

By Renaissance era, real numbers were commonly understood as any numerical quantity expressed as an unending sequence of integers after the decimal point. This conception enabled mathematical discoveries ranging from Galileo predicting planetary motion accurately enough he didn't observe Jupiter once per year yet always saw two moons together near the topmost part of sky; Isaac Newton providing calculus formulae illustrating physical laws; and Bernhard Riemann developing abstract models representing all possible geometries. Without knowledge of reals - along with their infinitely small elements called 'infinitesimals', mathematics might have stagnated pre-Renaissance!

Today, the field continues evolving rapidly due partly to computers facilitating exploration beyond human algorithmic limitations. Understanding historical context helps appreciate how far humanity has come since tracking game scores in Mesopotamia via cuneiform tablets.