A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
The number π is a mathematical constant. Defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics, it is equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is sometimes spelled out as "pi", it is called Archimedes' constant. Being an irrational number, π cannot be expressed as a common fraction. Still, fractions such as 22/7 and other rational numbers are used to approximate π; the digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Π is a transcendental number. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Ancient civilizations required accurate computed values to approximate π for practical reasons, including the Egyptians and Babylonians.
Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques; the first exact formula for π, based on infinite series, was not available until a millennium when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. All scientific applications require no more than a few hundred digits of π, many fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records; the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry those concerning circles and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry, it appears therefore in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in all areas of physics. The ubiquity of π makes it one of the most known mathematical constants both inside and outside the scientific community. Several books devoted to π have been published, record-setting calculations of the digits of π result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits; the symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi, derived from the first letter of the Greek word perimetros, meaning circumference.
In English, π is pronounced as "pie". In mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation; the choice of the symbol π is discussed in the section Adoption of the symbol π. Π is defined as the ratio of a circle's circumference C to its diameter d: π = C d The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will have twice the circumference, preserving the ratio C/d; this definition of π implicitly makes use of flat geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x2 + y2 = 1, as the integral: π = ∫ − 1 1 d x 1 − x 2.
An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841. Definitions of π such as these that rely on a notion of circumference, hence implicitly on concepts of the integral calculus, are no longer common in the literature. Remmert explains that this is because in many modern treatments of calculus, differential calculus precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0; the cosine can be defined independently of geometry as a power series, or as the solution of a differen
Marvin Lee Minsky was an American cognitive scientist concerned with research of artificial intelligence, co-founder of the Massachusetts Institute of Technology's AI laboratory, author of several texts concerning AI and philosophy. Marvin Lee Minsky was born in New York City, to an eye surgeon father, to a mother, an activist of Zionist affairs, his family was Jewish. He attended the Bronx High School of Science, he attended Phillips Academy in Andover, Massachusetts. He served in the US Navy from 1944 to 1945, he received a B. A. in mathematics from Harvard University and a Ph. D. in mathematics from Princeton University. He was on the MIT faculty from 1958 to his death, he joined the staff at MIT Lincoln Laboratory in 1958, a year he and John McCarthy initiated what is known now as the MIT Computer Science and Artificial Intelligence Laboratory. He was the Toshiba Professor of Media Arts and Sciences, professor of electrical engineering and computer science. Minsky's inventions include the confocal microscope.
He developed, with Seymour Papert, the first Logo "turtle". Minsky built, in 1951, the first randomly wired neural network learning machine, SNARC. In 1962, Minsky came up with a 7,4 Turing machine. At that point in time, it was known to be the simplest universal Turing machine–a record that stood for 40 years until Stephen Wolfram published a 2,5 universal Turing machine in his 2002 book, A New Kind of Science. Minsky wrote the book Perceptrons, which became the foundational work in the analysis of artificial neural networks; this book is the center of a controversy in the history of AI, as some claim it to have had great importance in discouraging research of neural networks in the 1970s, contributing to the so-called "AI winter". He founded several other famous AI models, his book A framework for representing knowledge created a new paradigm in programming. While his Perceptrons is now more a historical than practical book, the theory of frames is in wide use. Minsky has written on the possibility that extraterrestrial life may think like humans, permitting communication.
In the early 1970s, at the MIT Artificial Intelligence Lab and Papert started developing what came to be known as the Society of Mind theory. The theory attempts to explain how what we call intelligence could be a product of the interaction of non-intelligent parts. Minsky says that the biggest source of ideas about the theory came from his work in trying to create a machine that uses a robotic arm, a video camera, a computer to build with children's blocks. In 1986, Minsky published The Society of Mind, a comprehensive book on the theory which, unlike most of his published work, was written for the general public. In November 2006, Minsky published The Emotion Machine, a book that critiques many popular theories of how human minds work and suggests alternative theories replacing simple ideas with more complex ones. Recent drafts of the book are available from his webpage. Minsky was an adviser on Stanley Kubrick's movie 2001: A Space Odyssey. Minsky himself is explicitly mentioned in Arthur C. Clarke's derivative novel of the same name, where he is portrayed as achieving a crucial break-through in artificial intelligence in the then-future 1980s, paving the way for HAL 9000 in the early 21st century: In the 1980s, Minsky and Good had shown how artificial neural networks could be generated automatically—self replicated—in accordance with any arbitrary learning program.
Artificial brains could be grown by a process strikingly analogous to the development of a human brain. In any given case, the precise details would never be known, if they were, they would be millions of times too complex for human understanding. In 1952, Minsky married pediatrician Gloria Rudisch. Minsky was a talented improvisational pianist who published musings on the relations between music and psychology. Minsky was an atheist, a signatory to the Scientists' Open Letter on Cryonics, he was a critic of the Loebner Prize for conversational robots, argued that a fundamental difference between humans and machines was that while humans are machines, they are machines in which intelligence emerges from the interplay of the many unintelligent but semi-autonomous agents that comprise the brain. He argued that "somewhere down the line, some computers will become more intelligent than most people," but that it was hard to predict how fast progress would be, he cautioned that an artificial superintelligence designed to solve an innocuous mathematical problem might decide to assume control of Earth's resources to build supercomputers to help achieve its goal, but believed that such negative scenarios are "hard to take seriously" because he felt confident that AI would go through a lot of testing before being deployed.
Minsky died of a cerebral hemorrhage at the age of 88. Minsky was a member of Alcor's Scientific Advisory Board, is believed to have been cryonically preserved by Alcor as'Patient 144', whose cooling procedures began on January 27, 2016. 1967 – Computation: Finite and Infinite Machines, Prentice-Hall 1986 – The Society of Mind 2006 – The Emotion Machine: Commonsense Thinking, Artificial Intelligence, the Future of the Human Mind Minsky won the Turing Award in 1969, the Japan Prize in 1990, the IJCAI Award for Research Exce
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, A contains no greatest element; the set B may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. In other words, A contains every rational number less than the cut, B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number, in neither set; every real number, rational or not, is equated to only one cut of rationals. Dedekind cuts can be generalized from the rational numbers to any ordered set by defining a Dedekind cut as a partition of a ordered set into two non-empty parts A and B, such that A is closed downwards and B is closed upwards, A contains no greatest element.
See completeness. It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers; every cut of reals is identical to the cut produced by a specific real number. In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps. A similar construction to that used by Dedekind cuts was used in Euclid's Elements to define proportional segments. A Dedekind cut is a partition of the rationals Q into two subsets B such that A is nonempty. A ≠ Q. If x, y ∈ Q, x < y, y ∈ A x ∈ A. If x ∈ A there exists a y ∈ A such that y > x. By relaxing the first two requirements, we formally obtain the extended real number line, it is more symmetrical to use the notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".
If the ordered set S is complete for every Dedekind cut of S, the set B must have a minimal element b, hence we must have that A is the interval, B the interval [b, +∞). In this case, we say; the important purpose of the Dedekind cut is to work with number sets. The cut itself can represent a number not in the original collection of numbers; the cut can represent a number b though the numbers contained in the two sets A and B do not include the number b that their cut represents. For example if A and B only contain rational numbers, they can still be cut at √2 by putting every negative rational number in A, along with every non-negative number whose square is less than 2. Though there is no rational value for √2, if the rational numbers are partitioned into A and B this way, the partition itself represents an irrational number. Regard one Dedekind cut as less than another Dedekind cut if A is a proper subset of C. Equivalently, if D is a proper subset of B, the cut is again less than. In this way, set inclusion can be used to represent the ordering of numbers, all other relations can be created from set relations.
The set of all Dedekind cuts is itself a linearly ordered set. Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e. every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a linearly ordered set that does have this useful property. A typical Dedekind cut of the rational numbers Q is given by the partition with A =, B =; this cut represents the irrational number √2 in Dedekind's construction. To est
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Infimum and supremum
In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.
There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.
More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.
However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.
Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.