"≥ ""d""(""m"") for all ""m"" ≤ ""n"". The counting function ""Q""(""x"") of largely composite numbers satisfies for positive ""c"",""d"" with formula_17. Because the prime factorization of a highly composite number uses all of the first ""k"" primes, every highly composite number must be a practical number. Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving fractions. Highly composite number A highly composite number, also known as an anti-prime, is a positive integer with more divisors than any smaller positive integer has."
"Integer An integer (from the Latin ""integer"" meaning ""whole"") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The set of integers consists of zero (), the positive natural numbers (, , , …), also called ""whole numbers"" or ""counting numbers"", and their additive inverses (the negative integers, i.e., −1, −2, −3, …). The set of integers is often denoted by a boldface Z ("""") or blackboard bold formula_1 (Unicode U+2124 ℤ) standing for the German word ""Zahlen"" (, ""numbers""). The integers form"
"If the domain is restricted to then each and every member of has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality. Integer An integer (from the Latin ""integer"" meaning ""whole"") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The set of integers consists of zero (), the positive natural numbers (, , , …), also called ""whole numbers"" or ""counting numbers"", and their additive inverses (the negative integers, i.e.,"
"the initial of ""nulla"" or of ""nihil"" (the Latin word for ""nothing""), in a table of epacts, all written in Roman numerals. Though the Romans used a decimal system for whole numbers, reflecting how they counted in Latin, they used a duodecimal system for fractions, because the divisibility of twelve makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on ten On coins, many of which had values that were duodecimal fractions of the unit , they used a tally-like notational system based on twelfths and halves. A dot (•) indicated an"
"Zero sharp In the mathematical discipline of set theory, 0 (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as , where it was denoted by Σ,"
"Zero-based numbering Zero-based numbering or ""index origin = 0"" is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday ""non-mathematical"" or ""non-programming"" circumstances. Under zero-based numbering, the initial element is sometimes termed the ""zeroth"" element, rather than the ""first"" element; ""zeroth"" is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth"
"and right of a slash mark. (For example, ½ may be read ""one-half"", ""one half"", or ""one over two"".) Fractions with large denominators that are ""not"" powers of ten are often rendered in this fashion (e.g., as ""one over one hundred seventeen"") while those with denominators divisible by ten are typically read in the normal ordinal fashion (e.g., as ""six-millionths"", ""six millionths"", or ""six one-millionths""). A simple fraction (also known as a common fraction or vulgar fraction) is a rational number written as ""a""/""b"" or formula_3, where ""a"" and ""b"" are both integers. As with other fractions, the denominator (""b"")"
"all numbers except zero start with a leading 1; the leading 1 is implicit and doesn't actually need to be stored which gives an extra bit of precision for ""free."" The number zero is represented specially: The number representations described above are called ""normalized,"" meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0. Such numbers are called denormal. They don't include as many significant digits"
"about real numbers that are rational numbers, and is expressed symbolically by writing 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 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"
"emphasize, the imaginary part does not include a factor and , not , is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate , by the ideal generated by the polynomial (see below). A real number can be regarded as a complex number whose imaginary part is 0. A purely imaginary number is a complex number whose real part is zero. As with polynomials, it is common to write for and for . Moreover, when the imaginary part is negative, i.e., it is common to write instead of ;"
"to resolve the ambiguity. Where the zero is before a digit other than the units digit, the explicit zero is not ambiguous and is therefore optional, but preferred. Thus: To construct a fraction, the denominator is written first, followed by 分之 (""parts of"") and then the numerator. This is the opposite of how fractions are read in English, which is numerator first. Each half of the fraction is written the same as a whole number. Mixed numbers are written with the whole-number part first, followed by 又 (""and""), then the fractional part. Percentages are constructed similarly, using 百 (100) as"
"Strictly non-palindromic number A strictly non-palindromic number is an integer ""n"" that is not palindromic in any positional numeral system with a base ""b"" in the range 2 ≤ ""b"" ≤ ""n"" − 2. For example, the number six is written as 110 in base 2, 20 in base 3 and 12 in base 4, none of which is a palindrome—so 6 is strictly non-palindromic. For another example, the number 19 written in base ""b"" (2 ≤ ""b"" ≤ 17) is: None of these are a palindrome so 19 is a strictly non-palindromic number. The sequence of strictly non-palindromic numbers"
"a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word ""fraction"" is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as /2 (see square root of 2) and π/4 (see proof that π is irrational). In"
"entire set of symbols used to represent numbers. In a particular numeral system, a single digit represents a different amount than any other digit, although the symbols in the same numeral system might vary between cultures. In modern usage, the Arabic numerals are the most common set of symbols, and the most frequently used form of these digits is the Western style. Each single digit, if used as a standalone number, matches the following amounts:<br> 0, zero. Used in the absence of objects to be counted. For example, a different way of saying ""there are no sticks here"", is to"
"Fraction (mathematics) A fraction (from Latin """", ""broken"") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ""common"", ""vulgar"", or ""simple"" fraction (examples: formula_1 and 17/3) consists of an integer numerator displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ""common"", including compound fractions, complex fractions, and mixed numerals. We begin"
"floating point arithmetic instead of algebraic arithmetic). Division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as where ""a"" is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives ""a"" (assuming ""a"" ≠ 0), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded"
"Pi The number () is a mathematical constant. Originally 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 approximately equal to 3.14159. It has been represented by the Greek letter """" since the mid-18th century, though it is also sometimes spelled out as ""pi"". It is also called Archimedes' constant. Being an irrational number, cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern). Still, fractions such"
"Irreducible fraction An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction ⁄ is irreducible if and only if ""a"" and ""b"" are coprime, that is, if ""a"" and ""b"" have a greatest common divisor of 1. In higher mathematics, ""irreducible fraction"" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible"
"used with prices. For example: With the exception of ""wari"", these are rarely seen in modern usage. Decimal fractions are typically written with either kanji numerals (vertically) or Arabic numerals (horizontally), preceded by a decimal point, and are read as successive digits, as in Western convention. Note that, in written form, they can be combined with either the traditional system of expressing numerals (42.195 kilometers: 四十二・一九五 キロメートル), in which powers of ten are written, or with the place value system, which uses zero (50.04 percent: 五〇・〇四 パーセント.) In both cases, however, the reading follows the traditional system (yon-jū ni-ten ichi-kyū"
"any real number. The integer ""n"" is called the characteristic and the real number ""m"" is called the ""significand"" or ""mantissa"". However, the term ""mantissa"" may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes ""m"" (as in ordinary decimal notation). In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten. Decimal floating point is a computer arithmetic system closely related to scientific notation. Any given real number can be written"
"0 except that one uses L[x] instead of L. See the section on relative constructibility in constructible universe. Zero sharp In the mathematical discipline of set theory, 0 (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as"
"was one of the earliest scientific books to be ""printed"" in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe. 0 is the integer immediately preceding 1. Zero is an even number because it is divisible by 2 with no remainder. 0 is neither positive nor negative. By most definitions 0 is a natural number, and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of"
"Ostrowski numeration In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers. Fix a positive irrational number ""α"" with continued fraction expansion [""a"",""a""...]. Let (""q"") be the sequence of denominators of the convergents ""p""/""q"" to α: so ""q"" = ""a""""q"" + ""q"". Let ""α"" denote ""T""(""α"") where ""T"" is the Gauss map ""T""(""x"") = {1/""x""}, and write ""β"" = (−1) ""α""α ... ""α"": we have ""β"" = ""a""""β"" + ""β"". Every positive real ""x"" can be written"
"numbers can be written in polar coordinates: Furthermore, The geometric meaning is that we multiply the magnitudes and add the angles. The product of two quaternions can be found in the article on quaternions. However, in this case, formula_14 and formula_15 are in general different. The product operator for the product of a sequence is denoted by the capital Greek letter pi ∏ (in analogy to the use of the capital Sigma ∑ as summation symbol). The product of a sequence consisting of only one number is just that number itself. The product of no factors at all is known"
"Signed zero Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero), regarded as equal by the numerical comparison operations but with possible different behaviors in particular operations. This occurs in the ""sign and magnitude"" and ""ones' complement"" signed number representations for integers, and in most floating-point number representations. The number 0 is usually encoded as +0, but can be"
"to the left of the decimal separator (see also truncation). For a nonnegative decimal, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the ""fractional part"", which equals the difference between the numeral and its integer part. When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example , instead of ). In normal writing, this is generally avoided because of the risk of confusion between the decimal mark and other punctuation. Decimal numerals"
"unique set containing no objects, no names and the empty connection), the named set of natural numbers, in which numbers have names in natural languages and numerical systems (for example, the number ten has names: ""10"", ""ten"", ""1010"", ""diez"", ""dix"", ""zehn"", ""X"", and many others) and the set of real numbers, in which numbers are assigned point of a straight line as their names. All mathematical constructions are named sets or systems of named sets. For example, mathematical structures as diverse as graphs, manifolds, vector spaces, natural and real numbers are identified by formally defined terms that satisfy various (axiomatic)"
"a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number. What the Babylonians had instead was a space (and later a disambiguating placeholder symbol ) to mark the nonexistence of a digit in a certain place value. Babylonian numerals Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astronomical"
"Null graph In the mathematical field of graph theory, the term ""null graph"" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an ""empty graph""). The order-zero graph, formula_1, is the unique graph having no vertices (hence its order is zero). It follows that formula_1 also has no edges. Some authors exclude formula_1 from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including formula_1 as a valid graph is useful depends on context. On the positive side, formula_1 follows naturally from the usual"
"regardless of how many fields this line has, or whether it has more or fewer fields than surrounding lines. $ is actually a unary operator with the highest operator precedence. (If the line has no fields, then ""NF"" is 0, ""$0"" is the whole line, which in this case is empty apart from possible white-space, and so has the numeric value 0.) At the end of the input the ""END"" pattern matches, so ""s"" is printed. However, since there may have been no lines of input at all, in which case no value has ever been assigned to ""s"", it"
"only vertices and no edges is known as an ""edgeless graph"". The graph with no vertices and no edges is sometimes called the ""null graph"" or ""empty graph"", but the terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called ""vertex-labeled"". However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The"
"special case of ""m"" = 0 gives formula_68 The special case of ""m"" = 1 is simply called a harmonic number and is frequently written without the ""m"", as The smallest natural number ""k"" such that ""k"" does not divide the denominator of generalized harmonic number ""H""(""k"", ""n"") nor the denominator of alternating generalized harmonic number ""H′""(""k"", ""n"") is, for ""n""=1, 2, ... : In the limit as for , the generalized harmonic number converges to the Riemann zeta function The related sum formula_71 occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of"
"Fractional part The fractional part or decimal part of a non‐negative real number formula_1 is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than ""x"", called floor of ""x"" or formula_2, its fractional part can be written as: For a positive number written in a conventional positional numeral system (such as binary or decimal), its fractional part hence corresponds to the digits appearing after the radix point. However, in case of negative numbers, there are various conflicting ways to extend the fractional part function to them: It is either defined"