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
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.
That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.
But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.
For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.
An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to
In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent; the mixing process of a solution happens at a scale where the effects of chemical polarity are involved, resulting in interactions that are specific to solvation. The solution assumes the phase of the solvent when the solvent is the larger fraction of the mixture, as is the case; the concentration of a solute in a solution is the mass of that solute expressed as a percentage of the mass of the whole solution. The term aqueous solution is. A solution is a homogeneous mixture of two or more substances; the particles of solute in a solution cannot be seen by the naked eye. A solution does not allow beams of light to scatter. A solution is stable; the solute from a solution cannot be separated by filtration. It is composed of only one phase. Homogeneous means. Heterogeneous means; the properties of the mixture can be uniformly distributed through the volume but only in absence of diffusion phenomena or after their completion.
The substance present in the greatest amount is considered the solvent. Solvents can be liquids or solids. One or more components present in the solution other; the solution has the same physical state as the solvent. If the solvent is a gas, only gases are dissolved under a given set of conditions. An example of a gaseous solution is air. Since interactions between molecules play no role, dilute gases form rather trivial solutions. In part of the literature, they are not classified as solutions, but addressed as mixtures. If the solvent is a liquid almost all gases and solids can be dissolved. Here are some examples: Gas in liquid: Oxygen in water Carbon dioxide in water – a less simple example, because the solution is accompanied by a chemical reaction. Note that the visible bubbles in carbonated water are not the dissolved gas, but only an effervescence of carbon dioxide that has come out of solution. Liquid in liquid: The mixing of two or more substances of the same chemistry but different concentrations to form a constant.
Alcoholic beverages are solutions of ethanol in water. Solid in liquid: Sucrose in water Sodium chloride or any other salt in water, which forms an electrolyte: When dissolving, salt dissociates into ions. Solutions in water are common, are called aqueous solutions. Non-aqueous solutions are. Counter examples are provided by liquid mixtures that are not homogeneous: colloids, emulsions are not considered solutions. Body fluids are examples for complex liquid solutions. Many of these are electrolytes. Furthermore, they contain solute molecules like urea. Oxygen and carbon dioxide are essential components of blood chemistry, where significant changes in their concentrations may be a sign of severe illness or injury. If the solvent is a solid gases and solids can be dissolved. Gas in solids: Hydrogen dissolves rather well in metals in palladium. Liquid in solid: Mercury in gold, forming an amalgam Water in solid salt or sugar, forming moist solids Hexane in paraffin wax Solid in solid: Steel a solution of carbon atoms in a crystalline matrix of iron atoms Alloys like bronze and many others Polymers containing plasticizers The ability of one compound to dissolve in another compound is called solubility.
When a liquid can dissolve in another liquid the two liquids are miscible. Two substances that can never mix to form a solution are said to be immiscible. All solutions have a positive entropy of mixing; the interactions between different molecules or ions may be energetically favored or not. If interactions are unfavorable the free energy decreases with increasing solute concentration. At some point the energy loss outweighs the entropy gain, no more solute particles can be dissolved. However, the point at which a solution can become saturated can change with different environmental factors, such as temperature and contamination. For some solute-solvent combinations a supersaturated solution can be prepared by raising the solubility to dissolve more solute, lowering it; the greater the temperature of the solvent, the more of a given solid solute it can dissolve. However, most gases and some compounds exhibit solubilities that decrease with increased temperature; such behavior is a result of an exothermic enthalpy of solution.
Some surfactants exhibit this behaviour. The solubility of liquids in liquids is less temperature-sensitive than that of solids or gases; the physical properties of compounds such as melting point and boiling point change when other compounds are added. Together they are called colligative properties. There are several ways to quantify the amount of one compound dissolved in the other compounds collectively called concentration. Examples include molarity, volume fraction, mole fraction; the properties of ideal solutions can be calculated by the linear combination of the properties of
Domain of discourse
In the formal sciences, the domain of discourse called the universe of discourse, universal set, or universe, is the set of entities over which certain variables of interest in some formal treatment may range. The domain of discourse is identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables. Many logicians distinguish, sometimes only tacitly, between the domain of a science and the universe of discourse of a formalization of the science. Giuseppe Peano formalized number theory taking its domain to be the positive integers and the universe of discourse to include all numbers, not just integers. For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals over which the quantifiers range. A proposition such as ∀ x is ambiguous. In one interpretation, the domain of discourse could be the set of real numbers. If the domain of discourse is the set of real numbers, the proposition is false, with x = √2 as counterexample.
The term universe of discourse refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on; the concept universe of discourse is attributed to Augustus De Morgan but the name was used for the first time by George Boole on page 42 of his Laws of Thought. Boole's definition is quoted below; the concept discovered independently by Boole in 1847, played a crucial role in his philosophy of logic in his principle of wholistic reference. In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined; the most unfettered discourse is that in which the words we use are understood in the widest possible application, for them the limits of discourse are co-extensive with those of the universe itself. But more we confine ourselves to a less spacious field.
Sometimes, in discoursing of men we imply that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse. Domain of a function Domain theory Interpretation Term algebra Universe
In typography and lettering, a sans-serif, sans serif, gothic, or sans letterform is one that does not have extending features called "serifs" at the end of strokes. Sans-serif fonts tend to have less line width variation than serif fonts. In most print, they are used for headings rather than for body text, they are used to convey simplicity and modernity or minimalism. Sans-serif fonts have become the most prevalent for display of text on computer screens. On lower-resolution digital displays, fine details like serifs may appear too large; the term comes from the French word sans, meaning "without" and "serif" of uncertain origin from the Dutch word schreef meaning "line" or pen-stroke. Before the term "sans-serif" became common in English typography, a number of other terms had been used. One of these outmoded terms for sans serif was gothic, still used in East Asian typography and sometimes seen in font names like News Gothic, Highway Gothic, or Trade Gothic. Sans-serif fonts are sometimes in older documents, used as a device for emphasis, due to their blacker type color.
For the purposes of type classification, sans-serif designs are divided into three or four major groups, the fourth being the result of splitting the grotesque category into grotesque and neo-grotesque. This group features most of the early sans-serif designs. Influenced by Didone serif fonts of the period and signpainting traditions, these were quite solid, bold designs suitable for headlines and advertisements; the early sans-serif typefaces did not feature a lower case or italics, since they were not needed for such uses. They were sometimes released by width, with a range of widths from extended to normal to condensed, with each style different, meaning to modern eyes they can look quite irregular and eccentric. Grotesque fonts have limited variation of stroke width; the terminals of curves are horizontal, many have a spurred "G" and an "R" with a curled leg. Capitals tend to be of uniform width. Cap height and ascender height are the same to create a more regular effect in texts such as titles with many capital letters, descenders are short for tighter linespacing.
Most avoid having a true italic in favour of a more restrained oblique or sloped design, although at least sans-serif true italics were offered. Examples of grotesque fonts include Akzidenz Grotesk, News Gothic, Franklin Gothic and Monotype Grotesque. Akzidenz Grotesk Old Face, Grotesque No. 9 and Monotype Grotesque are examples of digital fonts that retain more of eccentricities of some of the early sans-serif types. The term realist has been applied to these designs due to their practicality and simplicity; as the name implies, these modern designs consist of a direct evolution of grotesque types. They are straightforward in appearance with limited width variation. Unlike earlier grotesque designs, many were issued in large and versatile families from the time of release, making them easier to use for body text. Similar to grotesque typefaces, neogrotesques feature capitals of uniform width and a quite'folded-up' design, in which strokes are curved all the way round to end on a perfect horizontal or vertical.
Helvetica is an example of this. Others such as Univers are less regular. Neo-grotesque type began in the 1950s with the emergence of the International Typographic Style, or Swiss style, its members looked at the clear lines of Akzidenz Grotesk as an inspiration to create rational neutral typefaces. In 1957 the release of Helvetica and Folio, the first typefaces categorized as neo-grotesque, had a strong impact internationally: Helvetica came to be the most used typeface for the following decades. Other neo-grotesques include Unica and Rail Alphabet, in the digital period Acumin, San Francisco and Roboto; as their name suggests, Geometric sans-serif typefaces are based on geometric shapes, like near-perfect circles and squares. Common features are a nearly-exactly circular capital "O" and a "single-story" lowercase letter "a". The'M' is splayed and the capitals of varying width, following the classical model. Of these four categories, geometric fonts tend to be the least useful for body text and used for headings and small passages of text.
The geometric sans originated in Germany in the 1920s. Two early efforts in designing geometric types were made by Herbert Bayer and Jakob Erbar, who worked on Universal Typeface and Erbar. In 1927 Futura, by Paul Renner, was released to great acclaim and popularity. Geometric sans-serif fonts were popular from the 1920s and 1930s due to their clean, modern design, many new geometric designs and revivals have been created since. Notable geometric types of the period include Semplicità, Nobel and Metro. Many geometric sans-serif alphabets of the period, such as those created by the Bauhaus art school and modernist poster artists, were hand-lettered and not cut into metal type at the time. A separate inspiration for many types considered "geometric" in design has been the simplified shapes of letters engraved or stenciled on metal and plastic in industrial use, which follow a simplified structure and are sometimes known as "rectilinear" for their use of straight vertical and horizontal lines. Designs considered geometric in principles but which are less descended from the Futura/Erbar/Kabel tradition include Bank Gothic, DIN 1451, Eurostile and Handel Gothic, along with man