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integer

(a) It is clear that every integer is divisible by I!.

10065The efforts which were consequently made in the early days of spectroscopy to discover some numerical relationship between the different wave lengths of the lines belonging to the same spectrum rather disregard the fact that even in acoustics the relationship of integer numbers holds only in special and very simple cases.

58422) he says, of Integer vitae: 'Tis a verse in Horace; I know it well: I read it in the grammar long ago."

5747is less than 2 o o If the numerator of the fraction consists of an integer and 4 - e.g.

5442If s represents the series of integer numbers the distribution of frequency may be represented by C+Bs2, where C and B are constants.

4233t (22P11),; X 1 and 12 each assuming all integer (including zero) values.

3640&c., where p+pq is the quantity whoseTi power or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, m the power, which may be a positive or negative integer or a fraction, and a, b, c, &c., the several terms in order, In a second letter, dated the 24th of October 1676, to Oldenburg, Newton gave the train of reasoning by which he devised the theorem.

2829This process consists in proving that a property involving p is true when p is any positive integer by proving (I) that it is true when p= 1, and (2) that if it is true when p=n, where n is any positive integer, then it is true when p = n+ I.

2830If we wish to be more general, while still adhering to Deslandres' law as a correct representation of the frequencies when s is small, we may write n - A (s+ 1 1) 2 - - a Po+Pi(s + c) -F +pr(s+ c)r' where s as before represents the integer numbers and the other quantities involved are constants.

2525- There are two kinds of approach to a limit, which may be illustrated by the series forming the expansion of (x+h) n, where n is a negative integer and 1> h/x> o.

2425(1-20) The actual form of a perpetuant of degree 0 has been shown by MacMahon to be +1 K0_1+1 K 3+20-4 K2, 01, 0-2, 0-3, ...3, 2), K 0, Ke -1, ...K 2 being given any zero or positive integer values.

2329For practical purposes the number taken as base is so; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, i.e.

2122As really denoted any integer or whole; whence the English word "ace."

2123These works possess considerable originality, and contain many new improvements in algebraic notation; the unknown (res) is denoted by a small circle, in which he places an integer corresponding to the power.

2021In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.

1724r= io, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures.

1622The application of the method to the calculation of (I +x) n, when n= p/q, q being a positive integer and p a positive or negative integer, involves, as in the case where n is a negative integer, the separate consideration of the form of the coefficients b 1, b 2, ...

1523Here n represents an integer which is 3 if the vibration is a simple doublet, but may have a higher integer value.

1421An im 5 proper fraction is therefore equal either to an 2 I integer or to a mixed number.

1016(v.) Since (r) is an integer, (r) is divisible by r!; i.e.

1018The explanation of this property of the base io is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact.

1018For the application of continued fractions to the problem " To find the fraction, whose denominator does not exceed a given integer D, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to G.

1018Comparison with the table of binomial coefficients in ï¿½ 43 suggests that, if m is any positive integer, (I +x)-m =Sr+Rr (25), where Sr=I -m[1]x+m[2]x2...+(-)rm[r]xr (26), Rr_(_)r+1xr+11m[r] (1Fx) - 1+(m - I[r](I+x) m) (27).

1019He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the parabola y=xm, where m is a positive integer, the area of the curves when m is negative or fractional.

1021If, out of every N cases, where N may be a very large number, a is A in pN cases and not-A in (I - p) N cases, where p is a fraction such that pN is an integer, then p is the probability or frequency of occurrence of A.

920The method of electrical images will enable the stream function, )' to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle 7r/m, where m is an integer (R.

817n's N (I +µ) (s+o.)2 Here and N are constants, while s as before is an integer number.

817- When a fraction cannot be expressed by an integral percentage, it can be so expressed approximately, by taking the nearest integer to the numerator of an equal fraction having ioo for its denominator.

817Trunk series: t N = [s +al +b/s 1 [1 5 +a1 +b'/(I.5)2}2 Main Branch Series: t ytr' - I I N [2 + al + 6/29 2 [r+al Side Branch Series: t nT = N [2 +al+6,/22]2 [s+c+d,s92 Here s stands for an integer number beginning with 2 for the trunk and 3 for the main branch, and r represents the succession of numbers 1 5, 5, 3 5, &c. As Ritz points out, the first two equations appear only to be particular cases of the form n I I N +1)2 in which s and r have the form given above.

828are integers, and a is an integer or zero.

725(ii.) To continue the division we may take as our new unit a submultiple of Q, such as Q/r, where r is an integer, and repeat the process.

614The pth root of a number (§43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately.

314The curve is periodic, and cuts the axis of x at the points x= (2n - I)a, n being an integer; the maximum values of y are =a.

10NUMBERS This mathematical subject, created by Euler, though relating essentially to positive integer numbers, is scarcely regarded as a part of the Theory of Numbers (see Number).

10Thus if x= log x =1 - where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have log (E +in) = log,/ Q 2 +7)2) t i (a + 2 n 7r), where a is the numerically least angle whose cosine and sine are (2 +n 2) and 71/,i (t 2 +n 2), and n denotes any integer.

10(s+µ + a/s)Ã‚° In all cases s represents the succession of integer numbers.

10Thus 2 is equal to -, and a is equal to -16Ã‚°, and conversely; in other words, any fractional number is equivalent to the fractional number obtained by multiplying or dividing the numerator and denominator by any integer.

10RESERVE [= n] n indicates a scalar integer constant or default integer-type scalar variable name.

10H Integer Height of the two-dimensional lattice used in the cellular automata.

10The third argument is an integer, whose value is unified with the specified list element.

10Certain relations still hold, the most important being (22) of ï¿½ 44 (ii.), which holds whatever the values of m and of n may be; r, of course, being a positive integer.

00The results in (b) apply also if n is a negative integer.

00The asym ptotes are x= = 2na, n being an integer.

00in the case o f 3-372 - it is uncertain whether we should z g 8 Ioo take the next lowest or the next highest integer.

00., and that if we go up to the factor i ?n4n+1 the product of these factors differs from the true value of the number by less than gngn -{-1 In certain cases two or more factors can be combined so as to produce an expression of the form i where k is an integer.

00integer arithmetic within the decoders does not work, due to the algorithms used.

00Arbitrary precision integer arithmetic is provided by some implementations of the language.

00Image A byte or integer array of either two or three dimensions, containing the image to be written.

00Particles with zero or integer spin are called bosons after Satyendra Bose, who together with Einstein described their kind of statistical mechanics.

00Integer A field described as Integer is a whole number which can be stored in four 8-bit bytes.

00This structure has three members: An integer indicating the reason for invoking the callback.

00Enter the origin and length as long integer constants in either decimal, octal, or hexadecimal (standard C syntax ).

00coprocessor control word as an integer.

00decimal integer in the range from 1 to 255.

00Decimal notation decimal notation Decimal numbers consist of one or more of the following:- An optional integer part to the left of any decimal point.

00digit group separators are allowed in the integer portion.

00As such a function, it is, of course, constant between integers and has a jump discontinuity at each integer where.

00Here det must be an integer which is a multiple of the biggest determinant divisor of A.

00enumerate a way of systematically enumerating feasible solutions such that the optimal integer solution is found.

00It may indicate a failure to declare the index to be an integer.

00floating-point value to be rounded to an integer.

00For each pcp-element g there exists an integer e such that g e is normed.

00When converting a larger size integer to a smaller size integer, only the less significant bytes are moved.

00The expression may contain variables of the form " { i } " where i is a nonnegative integer.

00The corresponding argument is a pointer to an unsigned integer.

00The modifiers of the field written as an 32-bit integer.

00An unsigned long long type is treated as an unsigned long type (unsigned 64-bit integer ).

00integer overflow problem came to our attention shortly thereafter.

00integer operands, OR performs a bitwise inclusive " or " operation and returns the result.

00integer n is a number from 1 to 10, inclusive, indicating the desired level of detail of information.

00integer pixel value that is valid only for a particular screen.

00integer array of either two or three dimensions, containing the image to be written.

00ULONG Set this keyword to return an unsigned longword integer array.

00The skip argument is normally interpreted as a posi- tive decimal integer.

00interpreted as an unsigned 16-bit integer.

00The larger the integer is, the larger is the probability of finding all isomorphisms.

00If supplied, diff (x, k ), where k is a nonnegative integer, returns the k -th differences.

00linear equationribed by linear dynamic equations subject to linear inequalities involving real and integer variables.

00integer literals in the source code are by default treated as int values, which are suitable for use with all integer types.

00Integer Arithmetic By default, Perl assumes that it must do most of its arithmetic in floating point.

00The prime number theorem shows that if we pick an integer n at random, it will be prime with probability.

00The conversion types are: Conversion Meaning Notes d Signed integer decimal. i Signed integer decimal. o Unsigned octal.

00The operators DIV and MOD apply to integer operands only.

00operand of the instruction will specify how the data is to be interpreted -- e.g. signed integer, unsigned integer, character.

00The integer overflow problem came to our attention shortly thereafter.

00parsed as an integer, then null is returned.

00The value of integer options has no effect. PID can refer to any process whose id is known, not necessarily a child process.

00In this case, the internal form is an integer pixel value that is valid only for a particular screen.

00It can be an integer or an irreducible polynomial over the field S.

00For example, consider the radix character, the symbol used to separate the integer portion of a number from the fractional portion.

00Numerical values can be integer, real, or double precision.

00This problem can be formulated and solved as an integer programming problem.

00Parameters: s - the String containing the integer. radix - the radix to be used.

00Returns: the integer represented by the string argument in the specified radix.

00radix character, the symbol used to separate the integer portion of a number from the fractional portion.

00RESERVE [= n] n indicates a scalar integer constant or default integer-type scalar integer constant or default integer-type scalar variable name.

00M An integer scalar or array, - L £ M £ L, specifying the order m of.

00Substantial experience has been accumulated in solving large-scale linear, integer programming problems, and recently stochastic integer programming.

00subscript triplet specifies a regular sequence of integer values.

00Like the Integer House, Osborne House is expected to attract tens of thousands of visitors.

00A subscript triplet specifies a regular sequence of integer values.

00unify third argument is an integer, whose value is unified with the specified list element.

00Integer risus wisi, semper eu, congue quis, lobortis ut, massa.

00The newlimit value must be a positive integer between 1 and the maximum value of a non-long integer on the platform.

00It may be set to any integer value greater than or equal to zero.

00The integer variables LINES and COLS are defined in curses.h and will be filled in by initscr with the size of the screen.

00discrete numerical variates have only a fixed set of possible values - typically integer values, arising from counting things.

00The value can be either an integer value specifying a pre-defined line style, or a two-element vector specifying a stippling pattern.

00It was unusual in that it offered comparatively weak integer performance but very strong floating point performance.

00A general formula by which these numbers could be derived was invented by the Arabian astronomer Tobit ben Korra (836-901): if p = 3.2 m - I, q= 3.2 m - 1 - 1 and r = 9.2 2m - 1 - I, where m is an integer and p,q,r prime numbers, then 2 m pq and 2 m r are a pair of amicable numbers.

00(1) he saw that a point or separatrix was quite enough to separate integers from decimals, and that no signs to indicate primes, seconds, &c., were required; (2) he used ciphers after the decimal point and preceding the first significant figure; and (3) he had no objection to a decimal standing by itself without any integer.

00Comparison with the table of binomial coefficients in Ã¯¿½ 43 suggests that, if m is any positive integer, (I +x)-m =Sr+Rr (25), where Sr=I -m[1]x+m[2]x2...+(-)rm[r]xr (26), Rr_(_)r+1xr+11m[r] (1Fx) - 1+(m - I[r](I+x) m) (27).

00and q is a positive integer, we assume that i /4 = i+bix+b2x2..., and we then (cf.

00Certain relations still hold, the most important being (22) of Ã¯¿½ 44 (ii.), which holds whatever the values of m and of n may be; r, of course, being a positive integer.

00The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = o, and 27rV 2 = 47r--n7r (24), n being an integer.

00Discrete numerical variates have only a fixed set of possible values - typically integer values, arising from counting things.

00In the above example 2 R is an integral real number, which is distinct from a rational integer, and from a cardinal number.

01Thus P = kQ+R, where k is an integer.

01+nr, where r is a positive integer.

01

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