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Real number: the story on HearLore | HearLore
Real number
The square root of two cannot be written as a simple fraction, yet it represents a physical distance that exists on a ruler. This contradiction haunted ancient mathematicians for centuries, forcing them to confront the existence of numbers that could not be counted or measured by whole units. Around 500 BC, Greek mathematicians led by Pythagoras discovered that the diagonal of a square with side length one could not be expressed as a ratio of two integers. This revelation shattered their belief that all quantities were commensurable, meaning they could be measured by a common unit. The concept of irrational numbers emerged from this crisis, challenging the very definition of what a number could be. For the Greeks, numbers were strictly natural numbers, and quantities were merely proportions or ratios of lengths. The existence of a continuous line filled with points that were not rational numbers was a terrifying and revolutionary idea that would take over two thousand years to fully understand.
The Arithmetic of Infinity
In the 16th century, Simon Stevin created the basis for modern decimal notation, insisting that there was no fundamental difference between rational and irrational numbers in how they were written. This shift allowed mathematicians to treat all numbers as part of a single continuum, yet the rigorous definition of this continuum remained elusive. The term real was introduced in the 17th century by René Descartes to distinguish these numbers from imaginary numbers, which were the square roots of negative numbers. Descartes used the adjective real to describe roots of a polynomial, creating a linguistic boundary between the tangible and the abstract. For centuries, mathematicians used real numbers and limits without defining them rigorously, assuming that every sequence of numbers that got closer and closer to a value actually reached a limit. This assumption worked for practical calculations but lacked logical foundation. The development of a suitable formal definition was a major achievement of 19th-century mathematics, serving as the foundation of real analysis. Without this formalization, the calculus developed by Newton and Leibniz rested on shaky logical ground, relying on intuitive notions of continuity that could not be proven.
The Construction of Gaps
The year 1872 marked a turning point when two independent definitions of real numbers were published, one by Richard Dedekind and the other by Georg Cantor. Dedekind defined real numbers as cuts in the rational numbers, effectively filling the gaps between fractions with new points. Cantor defined them as equivalence classes of Cauchy sequences, treating them as the limit of sequences that get arbitrarily close to each other. These definitions solved the problem of completeness, ensuring that every non-empty set of real numbers with an upper bound had a least upper bound. This property, known as Dedekind completeness, distinguished the real numbers from the rational numbers, where sequences could converge to a point that did not exist within the set. The existence of a continuous number line was considered self-evident, but the nature of this continuity was not understood until the 1800s. The rigor developed for geometry did not cross over to the concept of numbers until this period. The formal definitions also exposed deeper issues in the foundations of mathematics, as they involved infinite sets and quantification on infinite sets, which could not be formalized in classical logic. This led to the development of higher-order logics in the first half of the 20th century to handle the complexities of the real number system.
When were real numbers first defined by Greek mathematicians?
Greek mathematicians led by Pythagoras discovered the existence of real numbers around 500 BC. This discovery occurred when they realized the diagonal of a square with side length one could not be expressed as a ratio of two integers.
Who introduced the term real numbers and when was it used?
René Descartes introduced the term real in the 17th century to distinguish these numbers from imaginary numbers. He used the adjective real to describe roots of a polynomial, creating a linguistic boundary between the tangible and the abstract.
What year did Richard Dedekind and Georg Cantor publish their definitions of real numbers?
The year 1872 marked a turning point when two independent definitions of real numbers were published by Richard Dedekind and Georg Cantor. Dedekind defined real numbers as cuts in the rational numbers, while Cantor defined them as equivalence classes of Cauchy sequences.
When did Georg Cantor prove that real numbers are uncountably infinite?
Georg Cantor showed that the set of all real numbers is uncountably infinite in 1874. He published his famous diagonal argument in 1891, which provided a more elegant demonstration of the same result.
When did Paul Cohen prove the independence of the continuum hypothesis?
Paul Cohen proved in 1963 that the continuum hypothesis is an axiom independent of the other axioms of set theory. This result implies that the real numbers are far more numerous than the integers.
What is the cardinality of the set of all real numbers called?
The cardinality of the set of all real numbers is called the cardinality of the continuum. It equals the cardinality of the power set of the natural numbers.
In 1874, Georg Cantor showed that the set of all real numbers is uncountably infinite, meaning there is no one-to-one function from the real numbers to the natural numbers. This discovery revealed that the infinity of real numbers was strictly larger than the infinity of integers, a concept that defied the intuition of the time. The cardinality of the set of all real numbers is called the cardinality of the continuum, and it equals the cardinality of the power set of the natural numbers. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891, which provided a more elegant demonstration of the same result. The statement that there is no cardinality strictly greater than the cardinality of the integers and strictly smaller than the cardinality of the continuum is known as the continuum hypothesis. Paul Cohen proved in 1963 that this hypothesis is an axiom independent of the other axioms of set theory, meaning it can be chosen as true or false without contradiction. This result implies that the real numbers are far more numerous than the integers, yet the rational numbers, which are dense in the real numbers, are only countably infinite. The irrational numbers are also dense in the real numbers, but they are uncountable and have the same cardinality as the reals themselves.
The Limits of Computation
Electronic calculators and computers cannot operate on arbitrary real numbers because finite computers cannot directly store infinitely many digits or other infinite representations. Instead, computers typically work with finite-precision approximations called floating-point numbers, a representation similar to scientific notation. The achievable precision is limited by the data storage space allocated for each number, whether as fixed-point, floating-point, or arbitrary-precision numbers. Real numbers satisfy the usual rules of arithmetic, but floating-point numbers do not, leading to errors in scientific computation. The field of numerical analysis studies the stability and accuracy of numerical algorithms implemented with approximate arithmetic. A real number is called computable if there exists an algorithm that yields its digits, but because there are only countably many algorithms, almost all real numbers fail to be computable. The equality of two computable numbers is an undecidable problem, meaning that no algorithm can determine whether two computable numbers are equal. Some constructivists accept the existence of only those reals that are computable, rejecting the uncountable infinity of non-computable numbers. This limitation highlights the gap between the theoretical perfection of the real number system and the practical constraints of digital computation.
The Physical Continuum
In the physical sciences, most physical constants and variables are modeled using real numbers, from the universal gravitational constant to the position and speed of a particle. Fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, and general relativity are described using mathematical structures based on the real numbers. Actual measurements of physical quantities are of finite accuracy and precision, yet the theories assume a continuous reality. Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative. The real numbers form a metric space where the distance between two points is defined as the absolute value of their difference. This space is complete, meaning that every Cauchy sequence of real numbers converges to a real number. The real numbers are locally compact but not compact, and they form a contractible, separable, and complete metric space of Hausdorff dimension one. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation, yet both yield the same topological space. The real numbers carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval has measure one.
The Logic of the Real
The real numbers are most often formalized using the Zermelo-Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations. The hyperreal numbers, developed by Edwin Hewitt and Abraham Robinson, extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz and Euler. Edward Nelson's internal set theory enriches the Zermelo-Fraenkel set theory syntactically by introducing a unary predicate standard, where infinitesimals are non-standard elements of the set of the real numbers. The continuum hypothesis posits that the cardinality of the set of the real numbers is aleph-one, the smallest infinite cardinal number after aleph-zero, the cardinality of the integers. The real numbers form a vector space over the field of rational numbers, and Zermelo-Fraenkel set theory with the axiom of choice guarantees the existence of a basis for this vector space. However, this existence theorem is purely theoretical, as such a basis has never been explicitly described. The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed, but the standard ordering of the real numbers is not a well-ordering since an open interval does not contain a least element in this ordering.