transfinite number, cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874. The cardinal number of the finite set of integers {1, 2, 3, … n} is n, and the cardinal number of any other set of objects that can be put in a one-to-one correspondence with this set is also n; e.g., the cardinal number 5 may be assigned to each of the sets {1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 4, 5, 1, 2}, and {a, b, c, d, e}, since each of these sets may be put in a one-to-one correspondence with any of the others. Similarly, the transfinite cardinal number ℵ₀ (aleph-null) is assigned to the countably infinite set of all positive integers {1, 2, 3, … n, … }. This set can be put in a one-to-one correspondence with many other infinite sets, e.g., the set of all negative integers {-1, -2, -3, … -n, … }, the set of all even positive integers {2, 4, 6, … 2n, … }, and the set of all squares of positive integers {1, 4, 9, … n², … }; thus, in contrast to finite sets, two infinite sets, one of which is a subset of the other, can have the same transfinite cardinal number, in this case, ℵ₀. It can be proved that all countably infinite sets, among which are the set of all rational numbers and the set of all algebraic numbers, have the cardinal number ℵ₀. Since the union of two countably infinite sets is a countably infinite set, ℵ₀ + ℵ₀ = ℵ₀; moreover, ℵ₀ × ℵ₀ = ℵ₀, so that in general, n × ℵ₀ = ℵ₀ and ℵ₀ⁿ = ℵ₀, where n is any finite number. It can also be shown, however, that the set of all real numbers, designated by c (for "continuum"), is greater than ℵ₀; the set of all points on a line and the set of all points on any segment of a line are also designated by the transfinite cardinal number c. An even larger transfinite number is 2^c, which designates the set of all subsets of the real numbers, i.e., the set of all {0,1}-valued functions whose domain is the real numbers. Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering. The transfinite ordinal number of the positive integers is designated by ω.

transcendental number: see number.

real number: see number.

rational number: see number.

prime number: see number theory.

oxidation number or oxidation state: see valence.

octane number, figure of merit representing the resistance of gasoline to premature detonation when exposed to heat and pressure in the combustion chamber of an internal-combustion engine. Such detonation is wasteful of the energy in the fuel and potentially damaging to the engine; premature detonation is indicated by knocking or pinging noises that occur as the engine operates. If an engine running on a particular gasoline makes such noises, they can be lessened or eliminated by using a gasoline with a higher octane number. The octane number of a sample of fuel is determined by burning the gasoline in an engine under controlled conditions, e.g., of spark timing, compression, engine speed, and load, until a standard level of knock occurs. The engine is next operated on a fuel blended from a form of isooctane that is very resistant to knocking and a form of heptane that knocks very easily. When a blend is found that duplicates the knocking intensity of the sample under test, the percentage of isooctane by volume in the blended sample is taken as the octane number of the fuel. Octane numbers higher than 100 are found by measuring the amount of tetraethyl lead that must be added to pure isooctane to duplicate the knocking of a sample fuel. At present three systems of octane rating are used in the United States. Two of these, the research octane and motor octane numbers, are determined by burning the gasoline in an engine under different, but specified, conditions. Usually the motor octane number is lower than the research octane. The third octane rating, which federal regulations require on commercial gasoline pumps, is an average of research octane and motor octane. Under this system a regular grade gasoline has an octane number of about 87 and a premium grade of about 93. Most American-made cars that were built in the 1971 model year or later can use regular gasoline. To prevent knocking, premium grade gasoline must be used in many cars built before 1971 and in some new cars that have high-performance engines.

number-average molecular weight: see molecular weight.

number theory, branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p<1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p₁ · p₂ · p₃ · · · p_n) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard Euler, C. F. Gauss, and Pierre de Fermat. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

number, entity describing the magnitude or position of a mathematical object or extensions of these concepts.

The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their members can be matched in a one-to-one correspondence. Ordinal numbers refer to position relative to an ordering, as first, second, third, etc. The finite cardinal and ordinal numbers are called the natural numbers and are represented by the symbols 1, 2, 3, 4, etc. Both types can be generalized to infinite collections, but in this case an essential distinction occurs that requires a different notation for the two types (see transfinite number).

The Integers and Rational Numbers

To the natural numbers one adjoins their negatives and zero to form the integers. The ratios a/b of the integers, where a and b are integers and b ≠ 0, constitute the rational numbers; the integers are those rational numbers for which b = 1. The rational numbers may also be represented by repeating decimals; e.g., 1/2 = 0.5000 … , 2/3 = 0.6666 … , 2/7 = 0.285714285714 … (see decimal system).

The Real Numbers

The real numbers are those representable by an infinite decimal expansion, which may be repeating or nonrepeating; they are in a one-to-one correspondence with the points on a straight line and are sometimes referred to as the continuum. Real numbers that have a nonrepeating decimal expansion are called irrational, i.e., they cannot be represented by any ratio of integers. The Greeks knew of the existence of irrational numbers through geometry; e.g., 2 is the length of the diagonal of a unit square. The proof that 2 is unable to be represented by such a ratio was the first proof of the existence of irrational numbers, and it caused tremendous upheaval in the mathematical thinking of that time.

The Complex Numbers

Numbers of the form z = x + yi, where x and y are real and i = -1, such as 8 + 7i (or 8 + 7-1), are called complex numbers; x is called the real part of z and yi the imaginary part. The real numbers are thus complex numbers with y = 0; e.g., the real number 4 can be expressed as the complex number 4 + 0i. The complex numbers are in a one-to-one correspondence with the points of a plane, with one axis defining the real parts of the numbers and one axis defining the imaginary parts. Mathematicians have extended this concept even further, as in quaternions.

The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation zⁿ + a_n - 1z^n - 1 + … + a₁z + a₀ = 0, where the coefficients a₀, a₁, … a_n - 1 are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number 2 is algebraic because it is a root of the equation z² + 2 = 0; similarly, i, a root of z² + 1 = 0, is also algebraic. However, F. Lindemann showed (1882) that π is transcendental, and using this fact he proved the impossibility of "squaring the circle" by straight edge and compass alone (see geometric problems of antiquity). The number e has also been found to be transcendental, although it still remains unknown whether e + π is transcendental.

mass number, often represented by the symbol A, the total number of nucleons (neutrons and protons) in the nucleus of an atom. All atoms of a chemical element have the same atomic number (number of protons in the nucleus) but may have different mass numbers (from having different numbers of neutrons in the nucleus). Atoms of an element with the same mass number make up an isotope of the element. Different isotopes of the same element cannot have the same mass number, but isotopes of different elements often do have the same mass number, e.g., carbon-14 (6 protons and 8 neutrons) and nitrogen-14 (7 protons and 7 neutrons).

index number, in econometrics, a figure reflecting a change in value or quantity as compared with a standard or base. The base usually equals 100 and the index number is usually expressed as a percentage. For example, if a commodity cost twice as much in 1970 as it did in 1960, its index number would be 200 relative to 1960. Index numbers are used especially to compare business activity, the cost of living, and employment; one of the most influential indexes in the United States is the Consumer Price Index (see under cost of living). Index numbers enable economists to reduce unwieldy business data into easily understood terms.

imaginary number: see number.

complex number: see number.

atomic number, often represented by the symbol Z, the number of protons in the nucleus of an atom, as well as the number of electrons in the neutral atom. Atoms with the same atomic number make up a chemical element. Atomic numbers were first assigned to the elements c.1913 by H. G. J. Moseley; he arranged the elements in an order based on certain characteristics of their X-ray spectra and then numbered them accordingly. The elements are now arranged in the periodic table in the order of their atomic numbers. Mendeleev's periodic law was originally based on atomic weights. See mass number.

algebraic number: see number.

Reynolds number [for Osborne Reynolds], dimensionless quantity associated with the smoothness of flow of a fluid. It is an important quantity used in aerodynamics and hydraulics. At low velocities fluid flow is smooth, or laminar, and the fluid can be pictured as a series of parallel layers, or lamina, moving at different velocities. The fluid friction between these layers gives rise to viscosity. As the fluid flows more rapidly, it reaches a velocity, known as the critical velocity, at which the motion changes from laminar to turbulent (see turbulence), with the formation of eddy currents and vortices that disturb the flow. The Reynolds number for the flow of a fluid of density ρ and viscosity η through a pipe of inside diameter d is given by R=ρdv/η, where v is the velocity. The Reynolds number for laminar flow in cylindrical pipes is about 1,000.

Mach number [for E. Mach], ratio between the speed of an object and the speed of sound in the medium in which the object is traveling. An airplane that has the velocity of Mach 3.0 is traveling at three times the speed of sound as measured in the prevailing atmospheric conditions.

Avogadro's number [for Amedeo Avogadro], number of particles contained in one mole of any substance; it is equal to 602,252,000,000,000,000,000,000, or in scientific notation, 6.02252×10²³. For example, 12.011 grams of carbon (one mole of carbon) contains 6.02252×10²³ carbon atoms, and 180.16 grams of glucose, C₆H₁₂O₆, contains 6.02252×10²³ molecules of glucose. Avogadro's number is determined by calculating the spacing of the atoms in a crystalline solid through X-ray methods and combining this data with the measured volume of one mole of the solid to obtain the number of molecules per molar volume.

Number of bonds (see bonding) an atom can form. Hydrogen (H) always has valence 1, so other elements' valences equal the number of hydrogen atoms they combine with. Thus, oxygen (O) has valence 2, as in water (H₂O); nitrogen (N) has valence 3, as in ammonia (NH₃); and chlorine (Cl) has valence 1, as in hydrochloric acid (HCl). The valence depends on the number of unpaired electrons in the outermost (and, in transition elements, the next) shell of the atom's structure. The sharing of the unpaired (valence) electrons in a bond mimics the stable configuration of the noble gases, whose outer shells are full. Elements that can achieve stable configurations by various combinations have more than one valence.

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In mathematics, a quantity that can be expressed as a finite or infinite decimal expansion. The counting numbers, integers, rational numbers, and irrational numbers are all real numbers. Real numbers are used in measuring continuously varying quantities (e.g., size, time), in contrast to measurements that result from counting. The word real distinguishes them from the imaginary numbers.

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Number of a chemical element in the systematic, ordered sequence shown in the periodic table. The elements are arranged in order of increasing number of protons in the nucleus of the atom (the same as the number of electrons in the neutral atom), and that number for each element is its atomic number.

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Any positive integer greater than 1 and exactly divisible only by 1 and itself. The sequence of prime numbers begins 2, 3, 5, 7, 11, 13, 17, 19, 23, 29elipsis but follows no discernible pattern. The issues of the regularities and irregularities in the distribution of primes are among the most important questions in number theory. Primes have been recognized at least since Pythagoras. It has been known that there are infinitely many of them at least since Euclid. The prime-number factors of an integer are the prime numbers whose product is that integer (see fundamental theorem of arithmetic).

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Branch of mathematics concerned with properties of and relations among integers. It is a popular subject among amateur mathematicians and students because of the wealth of seemingly simple problems that can be posed. Answers are much harder to come up with. It has been said that any unsolved mathematical problem of any interest more than a century old belongs to number theory. One of the best examples, recently solved, is Fermat's last theorem.

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Basic element of mathematics used for counting, measuring, solving equations, and comparing quantities. They fall into several categories. The counting numbers are the familiar 1, 2, 3 . . . ; whole numbers are the counting numbers and zero; integers are the whole numbers and the negative counting numbers; and the rational numbers are all possible quotients formed by integers, including fractions. These numbers can be symbolically represented by terminating or repeating decimals. Irrational numbers cannot be represented by fractions of integers or repeating decimals and must be represented by special symbols such as 2, math.e, and π. Together, the rational and irrational numbers constitute the real numbers, which form an algebraic field (see field theory), as do the complex numbers. While the counting numbers and rational numbers come about as the means of counting, calculating, and measuring, the others arose as means of solving equations. Seealso transcendental number.

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Among the real numbers, any of those that cannot be represented as quotients of integers. In decimal form, irrational numbers are represented by nonterminating, nonrepeating decimals. Examples include square roots of prime numbers and such transcendental numbers as π and math.e.

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Any number of the form math.bmath.i where math.b is a real number, math.i is the square root of −1, and math.b is not zero. Seealso complex number.

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In fluid mechanics, a number that indicates whether the flow of a fluid (liquid or gas) is absolutely steady (in streamlined, or laminar flow) or on the average steady with small, unsteady changes (in turbulent flow; see turbulence). The Reynolds number, abbreviated N_Re or Re, has no dimensions (see dimensional analysis) and is defined as the size of the flow—as, for example, the diameter of a tube (math.D) times the average speed of flow (math.v) times the mass density of the fluid (ρ)—divided by its absolute viscosity (μ). Osborne Reynolds demonstrated in 1883 that the change from laminar to turbulent flow in a pipe occurs when the value of the Reynolds number exceeds 2,100.

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