Misplaced Pages

#207792

97-419: 1 ( one , unit , unity ) is a number , numeral , and glyph . It is the first and smallest positive integer of the infinite sequence of natural numbers . This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement , a determiner for singular nouns, and

194-524: A Sanskrit word Shunye or shunya to refer to the concept of void . In mathematics texts this word often refers to the number zero. In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi , an early example of an algebraic grammar for the Sanskrit language (also see Pingala ). There are other uses of zero before Brahmagupta, though the documentation

291-413: A collection or family , especially when its elements are themselves sets. Roster or enumeration notation defines a set by listing its elements between curly brackets , separated by commas: This notation was introduced by Ernst Zermelo in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in

388-429: A determiner used with singular countable nouns , as in one day at a time . The determiner has two senses: numerical one ( I have one apple ) and singulative one ( one day I'll do it ). One is also a gender-neutral pronoun used to refer to an unspecified person or to people in general as in one should take care of oneself . Words that derive their meaning from one include alone , which signifies all one in

485-525: A numeral is not clearly distinguished from the number that it represents. In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers , rational numbers such as one half ( 1 2 ) {\displaystyle \left({\tfrac {1}{2}}\right)} , real numbers such as the square root of 2 ( 2 ) {\displaystyle \left({\sqrt {2}}\right)} and π , and complex numbers which extend

582-429: A sequence , a tuple , or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set. For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ' ... '. For instance, the set of the first thousand positive integers may be specified in roster notation as An infinite set

679-404: A , b positive and the other negative. The incorrect use of this identity, and the related identity in the case when both a and b are negative even bedeviled Euler . This difficulty eventually led him to the convention of using the special symbol i in place of − 1 {\displaystyle {\sqrt {-1}}} to guard against this mistake. The 18th century saw

776-603: A base 4, base 5 "finger" abacus. By 130 AD, Ptolemy , influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals . Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica ( Almagest ),

873-466: A definition is called a semantic description . Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set F can be defined as follows: F = { n ∣ n  is an integer, and  0 ≤ n ≤ 19 } . {\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.} In this notation,

970-446: A gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number . In digital technology , 1 represents the "on" state in binary code , the foundation of computing . Philosophically, 1 symbolizes

1067-439: A given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing. The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD , when he considered the volume of an impossible frustum of a pyramid . They became more prominent when in

SECTION 10

#1732765488208

1164-589: A global number field , is 1 for all simply connected groups (those that are path-connected with no ' holes '). 1 is the most common leading digit in many sets of real-world numerical data. This is a consequence of Benford’s law , which states that the probability for a specific leading digit d {\displaystyle d} is log 10 ⁡ ( d + 1 d ) {\textstyle \log _{10}\left({\frac {d+1}{d}}\right)} . The tendency for real-world numbers to grow exponentially or logarithmically biases

1261-451: A notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis 's De algebra tractatus . In the same year, Gauss provided the first generally accepted proof of the fundamental theorem of algebra , showing that every polynomial over the complex numbers has a full set of solutions in that realm. Gauss studied complex numbers of

1358-451: A part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. The symbol ∞ {\displaystyle {\text{∞}}}

1455-503: A placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. The use of 0 as a number should be distinguished from its use as a placeholder numeral in place-value systems . Many ancient texts used 0. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting . Indian texts used

1552-423: A result, the square ( 1 2 = 1 {\displaystyle 1^{2}=1} ), square root ( 1 = 1 {\displaystyle {\sqrt {1}}=1} ), and any other power of 1 is always equal to 1 itself. 1 is its own factorial ( 1 ! = 1 {\displaystyle 1!=1} ), and 0! is also 1. These are a special case of the empty product . Although 1 meets

1649-425: A rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz . A modern geometrical version of infinity is given by projective geometry , which introduces "ideal points at infinity", one for each spatial direction. Each family of parallel lines in

1746-441: A set S , denoted | S | , is the number of members of S . For example, if B = {blue, white, red} , then | B | = 3 . Repeated members in roster notation are not counted, so | {blue, white, red, blue, white} | = 3 , too. More formally, two sets share the same cardinality if there exists a bijection between them. The cardinality of the empty set is zero. The list of elements of some sets

1843-465: A set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B ; more formally, a function is a special kind of relation , one that relates each element of A to exactly one element of B . A function is called An injective function is called an injection , a surjective function is called a surjection , and a bijective function is called a bijection or one-to-one correspondence . The cardinality of

1940-679: A system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior Hindu–Arabic numeral system around the late 14th century, and the Hindu–Arabic numeral system remains the most common system for representing numbers in the world today. The key to the effectiveness of the system was the symbol for zero , which was developed by ancient Indian mathematicians around 500 AD. The first known documented use of zero dates to AD 628, and appeared in

2037-479: A way to swap true roots and false roots as well. At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. The first use of negative numbers in a European work was by Nicolas Chuquet during the 15th century. He used them as exponents , but referred to them as "absurd numbers". As recently as

SECTION 20

#1732765488208

2134-444: Is a subset of the next one. So, for example, a rational number is also a real number, and every real number is also a complex number. This can be expressed symbolically as A more complete list of number sets appears in the following diagram. The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. Traditionally, the sequence of natural numbers started with 1 (0

2231-403: Is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B , then the region representing A is completely inside the region representing B . If two sets have no elements in common, the regions do not overlap. A Venn diagram , in contrast, is a graphical representation of n sets in which

2328-422: Is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is and the set of all integers is Another way to define a set is to use a rule to determine what the elements are: Such

2425-506: Is a set with exactly one element; such a set may also be called a unit set . Any such set can be written as { x }, where x is the element. The set { x } and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat. If every element of set A is also in B , then A is described as being a subset of B , or contained in B , written A ⊆ B , or B ⊇ A . The latter notation may be read B contains A , B includes A , or B

2522-435: Is a superset of A . The relationship between sets established by ⊆ is called inclusion or containment . Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B . If A is a subset of B , but A is not equal to B , then A is called a proper subset of B . This can be written A ⊊ B . Likewise, B ⊋ A means B is a proper superset of A , i.e. B contains A , and

2619-415: Is an example of a "base-1" number system, since only one mark – the tally itself – is needed. While this is the simplest way to represent the natural numbers, base-1 is rarely used as a practical base for counting due to its difficult readability. In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval ([0,1]), where 1 represents

2716-579: Is common for the Jain math sutra to include calculations of decimal-fraction approximations to pi or the square root of 2 . Similarly, Babylonian math texts used sexagesimal (base 60) fractions with great frequency. The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800 and 500 BC. The first existence proofs of irrational numbers is usually attributed to Pythagoras , more specifically to

2813-399: Is endless, or infinite . For example, the set N {\displaystyle \mathbb {N} } of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality . Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that

2910-482: Is in B ". The statement " y is not an element of B " is written as y ∉ B , which can also be read as " y is not in B ". For example, with respect to the sets A = {1, 2, 3, 4} , B = {blue, white, red} , and F = { n | n is an integer, and 0 ≤ n ≤ 19} , The empty set (or null set ) is the unique set that has no members. It is denoted ∅ , ∅ {\displaystyle \emptyset } , { }, ϕ , or ϕ . A singleton set

3007-538: Is largely due to Ernst Kummer , who also invented ideal numbers , which were expressed as geometrical entities by Felix Klein in 1893. In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points . This eventually led to the concept of the extended complex plane . Prime numbers have been studied throughout recorded history. They are positive integers that are divisible only by 1 and themselves. Euclid devoted one book of

1 - Misplaced Pages Continue

3104-588: Is not as complete as it is in the Brāhmasphuṭasiddhānta . Records show that the Ancient Greeks seemed unsure about the status of 0 as a number: they asked themselves "How can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of 0 and the vacuum. The paradoxes of Zeno of Elea depend in part on

3201-428: Is not equal to A . A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B . Examples: The empty set is a subset of every set, and every set is a subset of itself: An Euler diagram

3298-405: Is often used to represent an infinite quantity. Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity —the general consensus being that only the latter had true value. Galileo Galilei 's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in

3395-434: Is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as "zero plus a positive number is a positive number, and a negative number plus zero is the negative number". The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply

3492-508: Is transcendental and Lindemann proved in 1882 that π is transcendental. Finally, Cantor showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite , so there is an uncountably infinite number of transcendental numbers. The earliest known conception of mathematical infinity appears in the Yajur Veda , an ancient Indian script, which at one point states, "If you remove

3589-440: Is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P ( S ) will leave some elements of P ( S ) unpaired. (There is never a bijection from S onto P ( S ) .) A partition of a set S is a set of nonempty subsets of S , such that every element x in S is in exactly one of these subsets. That is,

3686-568: The Brāhmasphuṭasiddhānta , the main work of the Indian mathematician Brahmagupta . He treated 0 as a number and discussed operations involving it, including division . By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals , and documentation shows the idea later spreading to China and the Islamic world . Brahmagupta's Brāhmasphuṭasiddhānta

3783-697: The Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic , and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras. In 1796, Adrien-Marie Legendre conjectured

3880-568: The Maghreb and Al-Andalus during the Middle Ages The Arabic numeral, and other glyphs used to represent the number one (e.g., Roman numeral ( I ), Chinese numeral ( 一 )) are logograms . These symbols directly represent the concept of 'one' without breaking it down into phonetic components. In modern typefaces , the shape of the character for the digit 1 is typically typeset as a lining figure with an ascender , such that

3977-783: The Pythagorean Hippasus of Metapontum , who produced a (most likely geometrical) proof of the irrationality of the square root of 2 . The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, and so, allegedly and frequently reported, he sentenced Hippasus to death by drowning, to impede spreading of this disconcerting news. The 16th century brought final European acceptance of negative integral and fractional numbers. By

1 - Misplaced Pages Continue

4074-726: The complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields , and the application of the term "number" is a matter of convention, without fundamental significance. Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks . These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals. A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless, tallying systems are considered

4171-842: The n loops divide the plane into 2 zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A , B , and C , there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist). There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. Z {\displaystyle \mathbf {Z} } ) or blackboard bold (e.g. Z {\displaystyle \mathbb {Z} } ) typeface. These include Each of

4268-595: The prime number theorem , describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture , which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis , formulated by Bernhard Riemann in 1859. The prime number theorem

4365-462: The vertical bar "|" means "such that", and the description can be interpreted as " F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. Philosophy uses specific terms to classify types of definitions: If B is a set and x is an element of B , this is written in shorthand as x ∈ B , which can also be read as " x belongs to B ", or " x

4462-461: The 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolò Fontana Tartaglia and Gerolamo Cardano . It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since they did not even consider negative numbers to be on firm ground at

4559-467: The 17th century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the 19th century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook the scientific study of irrationals. It had remained almost dormant since Euclid . In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine , Georg Cantor , and Richard Dedekind

4656-595: The 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless. It is likely that the concept of fractional numbers dates to prehistoric times . The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus . Classical Greek and Indian mathematicians made studies of

4753-611: The 3rd century AD in Greece. Diophantus referred to the equation equivalent to 4 x + 20 = 0 (the solution is negative) in Arithmetica , saying that the equation gave an absurd result. During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta , in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce

4850-603: The Hellenistic zero had morphed into the Greek letter Omicron (otherwise meaning 70). Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus ), but as a word, nulla meaning nothing , not as a symbol. When division produced 0 as a remainder, nihil , also meaning nothing , was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N,

4947-399: The above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, Q + {\displaystyle \mathbf {Q} ^{+}} represents the set of positive rational numbers. A function (or mapping ) from

SECTION 50

#1732765488208

5044-454: The application. 1 is the value of Legendre's constant , introduced in 1808 by Adrien-Marie Legendre to express the asymptotic behavior of the prime-counting function . The Weil's conjecture on Tamagawa numbers states that the Tamagawa number τ ( G ) {\displaystyle \tau (G)} , a geometrical measure of a connected linear algebraic group over

5141-400: The cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane , and indeed of any finite-dimensional Euclidean space . The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and

5238-470: The cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice . (ZFC is the most widely-studied version of axiomatic set theory.) The power set of a set S is the set of all subsets of S . The empty set and S itself are elements of the power set of S , because these are both subsets of S . For example,

5335-416: The development of Greek mathematics , stimulating the investigation of many problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers , which consist of various extensions or modifications of

5432-411: The digit 7 may be written with a horizontal stroke through the vertical line. In digital technology, data is represented by binary code , i.e., a base -2 numeral system with numbers represented by a sequence of 1s and 0s . Digitised data is represented in physical devices, such as computers , as pulses of electricity through switching devices such as transistors or logic gates where "1" represents

5529-438: The digit is the same height and width as a capital letter . However, in typefaces with text figures (also known as Old style numerals or non-lining figures ), the glyph usually is of x-height and designed to follow the rhythm of the lowercase, as, for example, in [REDACTED] . In old-style typefaces (e.g., Hoefler Text ), the typeface for numeral 1 resembles a small caps version of I , featuring parallel serifs at

5626-599: The distribution towards smaller leading digits, with 1 occurring approximately 30% of the time. One originates from the Old English word an , derived from the Germanic root *ainaz , from the Proto-Indo-European root *oi-no- (meaning "one, unique"). Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things. One is most commonly

5723-461: The earliest known records of a numeral system, is the Sumerian decimal- sexagesimal system on clay tablets dating from the first half of the third millennium BCE. The Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols. By c.  2350 BCE , the older Sumerian curviform numerals were replaced with cuneiform symbols, with 1 and 60 both represented by

5820-422: The elements outside the union of A and B are the elements that are outside A and outside B ). The cardinality of A × B is the product of the cardinalities of A and B . This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true. The power set of any set becomes a Boolean ring with symmetric difference as

5917-456: The final i of a "lower-case" Roman numeral. It is also possible to find historic examples of the use of j or J as a substitute for the Arabic numeral 1. In German, the serif at the top may be extended into a long upstroke as long as the vertical line. This variation can lead to confusion with the glyph used for seven in other countries and so to provide a visual distinction between the two

SECTION 60

#1732765488208

6014-655: The first kind of abstract numeral system. The first known system with place value was the Mesopotamian base 60 system ( c.  3400  BC) and the earliest known base 10 system dates to 3100 BC in Egypt . Numbers should be distinguished from numerals , the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. Roman numerals,

6111-417: The form a + bi , where a and b are integers (now called Gaussian integers ) or rational numbers. His student, Gotthold Eisenstein , studied the type a + bω , where ω is a complex root of x − 1 = 0 (now called Eisenstein integers ). Other such classes (called cyclotomic fields ) of complex numbers derive from the roots of unity x − 1 = 0 for higher values of k . This generalization

6208-463: The form of equations; for example, in Planck units the speed of light equals 1. Dimensionless quantities are also known as 'quantities of dimension one'. In quantum mechanics , the normalization condition for wavefunctions requires the integral of a wavefunction's squared modulus to be equal to 1. In chemistry, hydrogen , the first element of the periodic table and the most abundant element in

6305-744: The general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots". European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci , 1202) and later as losses (in Flos ). René Descartes called them false roots as they cropped up in algebraic polynomials yet he found

6402-579: The idea of a cut (Schnitt) in the system of real numbers , separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker , and Méray. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem ( Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formulas involving only arithmetical operations and roots). Hence it

6499-502: The known universe , has an atomic number of 1. Group 1 of the periodic table consists of hydrogen and the alkali metals . In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe in monotheistic traditions. The Pythagoreans considered the numbers to be plural and therefore did not classify 1 itself as a number, but as the origin of all numbers. In their number philosophy, where odd numbers were considered male and even numbers female, 1

6596-422: The maximum possible value. For example, by definition 1 is the probability of an event that is absolutely or almost certain to occur. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on

6693-506: The modern Western world to represent the number 1 is the Arabic numeral , a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom. It can be traced back to the Brahmic script of ancient India, as represented by Ashoka as a simple vertical line in his Edicts of Ashoka in c. 250 BCE. This script's numeral shapes were transmitted to Europe via

6790-468: The naïve definition of a prime number , being evenly divisible only by 1 and itself (also 1), by modern convention it is regarded as neither a prime nor a composite number . Different mathematical constructions of the natural numbers represent 1 in various ways. In Giuseppe Peano 's original formulation of the Peano axioms , a set of postulates to define the natural numbers in a precise and logical way, 1

6887-662: The number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system , which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system , which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits . In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers ), and for codes (as with ISBNs ). In common usage,

6984-422: The number one as God's number, and the basis for all numbers. Number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals ; for example, "5" is a numeral that represents

7081-410: The power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} . The power set of a set S is commonly written as P ( S ) or 2 . If S has n elements, then P ( S ) has 2 elements. For example, {1, 2, 3} has three elements, and its power set has 2 = 8 elements, as shown above. If S is infinite (whether countable or uncountable ), then P ( S )

7178-509: The properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society, the number 13 is often regarded as unlucky , and " a million " may signify "a lot" rather than an exact quantity. Though it is now regarded as pseudoscience , belief in a mystical significance of numbers, known as numerology , permeated ancient and medieval thought. Numerology heavily influenced

7275-403: The real numbers with a square root of −1 (and its combinations with real numbers by adding or subtracting its multiples). Calculations with numbers are done with arithmetical operations, the most familiar being addition , subtraction , multiplication , division , and exponentiation . Their study or usage is called arithmetic , a term which may also refer to number theory , the study of

7372-537: The same elements are equal (they are the same set). This property is called extensionality . In particular, this implies that there is only one empty set. Sets are ubiquitous in modern mathematics. Indeed, set theory , more specifically Zermelo–Fraenkel set theory , has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Mathematical texts commonly denote sets by capital letters in italic , such as A , B , C . A set may also be called

7469-611: The same symbol [REDACTED] . The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal systems. Surviving Babylonian documents date mostly from Old Babylonian ( c.  1500 BCE ) and the Seleucid ( c.  300 BCE ) eras. The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system. The most commonly used glyph in

7566-471: The sense of being by oneself, none meaning not one , once denoting one time , and atone meaning to become at one with the someone. Combining alone with only (implying one-like ) leads to lonely , conveying a sense of solitude. Other common numeral prefixes for the number 1 include uni- (e.g., unicycle , universe, unicorn), sol- (e.g., solo dance), derived from Latin, or mono- (e.g., monorail , monogamy, monopoly) derived from Greek. Among

7663-620: The set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of N {\displaystyle \mathbb {N} } are called countable sets ; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle \mathbb {N} } ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of N {\displaystyle \mathbb {N} } are called uncountable sets . However, it can be shown that

7760-500: The set of all natural numbers is N , also written N {\displaystyle \mathbb {N} } , and sometimes N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} when it is necessary to indicate whether the set should start with 0 or 1, respectively. Set (mathematics) Sets are uniquely characterized by their elements; this means that two sets that have precisely

7857-489: The subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S . Suppose that a universal set U (a set containing all elements being discussed) has been fixed, and that A is a subset of U . Given any two sets A and B , Examples: The operations above satisfy many identities. For example, one of De Morgan's laws states that ( A ∪ B )′ = A ′ ∩ B ′ (that is,

7954-567: The theory of rational numbers, as part of the general study of number theory . The best known of these is Euclid's Elements , dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra , which also covers number theory as part of a general study of mathematics. The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it

8051-408: The theory was made by Georg Cantor ; in 1895 he published a book about his new set theory , introducing, among other things, transfinite numbers and formulating the continuum hypothesis . In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents

8148-430: The time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation seemed capriciously inconsistent with the algebraic identity which is valid for positive real numbers a and b , and was also used in complex number calculations with one of

8245-478: The top and bottom, while the capital I retains a full-height form. This is a relic from the Roman numerals system where I represents 1. Many older typewriters do not have a dedicated key for the numeral 1, requiring the use of the lowercase letter L or uppercase I as substitutes. The lower case " j " can be considered a swash variant of a lower-case Roman numeral " i ", often employed for

8342-571: The ultimate reality or source of existence in various traditions. The number 1 is the first natural number after 0. Each natural number , including 1, is constructed by succession , that is, by adding 1 to the previous natural number. The number 1 is the multiplicative identity of the integers , real numbers , and complex numbers , that is, any number n {\displaystyle n} multiplied by 1 remains unchanged ( 1 × n = n × 1 = n {\displaystyle 1\times n=n\times 1=n} ). As

8439-578: The uncertain interpretation of 0. (The ancient Greeks even questioned whether  1 was a number.) The late Olmec people of south-central Mexico began to use a symbol for zero, a shell glyph , in the New World, possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar . Maya arithmetic used base 4 and base 5 written as base 20. George I. Sánchez in 1961 reported

8536-638: The value for "on". As such, the numerical value of true is equal to 1 in many programming languages . In lambda calculus and computability theory , natural numbers are represented by Church encoding as functions, where the Church numeral for 1 is represented by the function f {\displaystyle f} applied to an argument x {\displaystyle x} once (1 f x = f x {\displaystyle fx=fx} ) . In physics , selected physical constants are set to 1 in natural unit systems in order to simplify

8633-420: The work of Abraham de Moivre and Leonhard Euler . De Moivre's formula (1730) states: while Euler's formula of complex analysis (1748) gave us: The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received

8730-472: The writings of Joseph Louis Lagrange . Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus first connected the subject with determinants , resulting, with the subsequent contributions of Heine, Möbius , and Günther, in the theory of Kettenbruchdeterminanten . The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e

8827-456: Was brought about. In 1869, Charles Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on

8924-560: Was considered neutral capable of transforming even numbers to odd and vice versa by addition. The Neopythagorean philosopher Nicomachus of Gerasa 's number treatise, as recovered by Boethius in the Latin translation Introduction to Arithmetic , affirmed that one is not a number, but the source of number. In the philosophy of Plotinus (and that of other neoplatonists ), 'The One' is the ultimate reality and source of all existence. Philo of Alexandria (20 BC – AD 50) regarded

9021-539: Was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted. Numbers can be classified into sets , called number sets or number systems , such as the natural numbers and the real numbers . The main number systems are as follows: N 0 {\displaystyle \mathbb {N} _{0}} or N 1 {\displaystyle \mathbb {N} _{1}} are sometimes used. Each of these number systems

9118-434: Was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory . Simple continued fractions , closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler , and at the opening of the 19th century were brought into prominence through

9215-424: Was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 ( cardinality of the empty set , i.e. 0 elements, where 0 is thus the smallest cardinal number ) in the set of natural numbers. Today, different mathematicians use the term to describe both sets, including 0 or not. The mathematical symbol for

9312-512: Was treated as the starting point of the sequence of natural numbers. Peano later revised his axioms to begin the sequence with 0. In the Von Neumann cardinal assignment of natural numbers, where each number is defined as a set that contains all numbers before it, 1 is represented as the singleton { 0 } {\displaystyle \{0\}} , a set containing only the element 0. The unary numeral system , as used in tallying ,

9409-519: Was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol. The abstract concept of negative numbers was recognized as early as 100–50 BC in China. The Nine Chapters on the Mathematical Art contains methods for finding the areas of figures; red rods were used to denote positive coefficients , black for negative. The first reference in a Western work was in

#207792