# integers Sentence Examples

• are all positive integers, is called a simple continued f raction.

• If we take aq-bp= +1 we have a general solution in integers of ax+by=c, viz.

• Taking any number n to be represented by a point on a line at distance nL from a fixed point 0, where L is a unit of length, we start with a series of points representing the integers I, 2, 3,.

• Similarly, the other rational integers must be distinguished from the corresponding cardinals.

• Similarly, the other rational integers must be distinguished from the corresponding cardinals.

• do not involve x, and the indices of the powers of x are all positive integers, is called a rational integral function of x of degree n.

• There was, however, no development in the direction of decimals in the modern sense, and the Arabs, by whom the Hindu notation of integers was brought to Europe, mainly used the sexagesimal division in the ' " "' notation.

• Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a+1, put x=a+I/y and form the equation in y; if the equation in y has a root between b and b+i, put y = b + I /z, and so on.

• - A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for numerator and denominator.

• Decimal or Briggian Antilogarithms. - In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable.

• It is evident that, in this case, P ' p2, are two series of positive integers increasing without limit if the fraction does not terminate.

• Two cases have been given by Legendre as follows: If a2, a 31 ..., a n, b 2, b3, .., b n are all positive integers, then I.

• Two cases have been given by Legendre as follows: If a2, a 31 ..., a n, b 2, b3, .., b n are all positive integers, then I.

• The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).

• (iv.) If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p-kq)L, which is both less than pL and less than qL.

• Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

• We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.

• the product of any r consecutive integers is divisible by r!

• (b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+ I consecutive integers is divisible by (p+i)!.

• are integers.

• a number which cannot be expressed as the ratio of two integers.

• It is evident that we may have tones of frequency hn 1 kn 2 hn i - kn 2 hnl+kn2, where h and k are any integers.

• The solution in integers of the indeterminate equation ax+by=c may be effected by means of continued fractions.

• The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa-qb, where p and q are any integers.

• - Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout.

• byte integers before the OR operation.

• The existence of a measurable cardinal also implies some slightly startling things about sets of integers in the constructible universe.

• commutative rings is motivated by the properties of the integers.

• This is done by coding DTDs and integrity constraints with linear constraints on the integers.

• Multiply and divide decimals mentally by 10 or 100, and integers by 1000, and explain the effect.

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

• divisor of two integers.

• We say that integers a and b have a common divisor c if c is a divisor if both a and b.

• To calculate the greatest common divisor of two integers and of two polynomials over a field.

• hexadecimal integers or strings.

• Generally, this is false for floating types, true for unsigned integers, and true for signed integers on most machines.

• This is the most common format in which we store non-negative integers with present-day computers.

• The numbers n can be negative or positive 16-bit integers.

• To quote the first paragraph of the API documentation: Immutable arbitrary-precision integers.

• All values in a trace are either hexadecimal integers or strings.

• Note the way that decimal and octal integers are read in by grammar rules.

• The two operands are internally converted to 4 byte integers before the OR operation.

• integrity constraints with linear constraints on the integers.

• The 19th century mathematician Kronecker famously claimed that " God made the integers, all the rest is the work of man.

• multivariate polynomials in order to derive integers makes no sense!

• A rational is represented as a pair of integers, called numerator and denominator.

• octal integers are read in by grammar rules.

• Clearly writing a class hierarchy starting with multivariate polynomials in order to derive integers makes no sense!

• subtraction of two integers less than 1000, and column addition of more than two such integers.

• Law of If we know the weights a and b of two elements that are reciprocal found in union with unit weight of a third element, then proporwe can predict the composition of the compounds which the first two elements can form with each other; either the weights a and b will combine exactly, or if not, these weights must be multiplied by integers to obtain the composition of a compound.

• This single instance of the use of the decimal point in the midst of an arithmetical process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely in the course of performing an arithmetical operation.

• On p. 8, 10.502 is multiplied by 3.216, and the result found to be 33.77443 2; and on pp. 23 and 24 occur decimals not attached to integers, viz.

• (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.

• Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

• the orders of the quantic and covariant, and the degree and weight of the leading coefficient; calling these 'n, e,' 0, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation n9 -2w = e.

• There Is A Still More General Form Of Seminvariant; We May Have Instead Of 0, 0 Any Collections Of Nonunitary Integers Not Exceeding 0, 0 In Magnitude Respectively, (2 A2 3 A3 ...0 Ae)A(L S 2 G2 3 G3 ...0' Ge') B (12 A2 3 A3 ..0 Ab)A(1 S I 2 G2 3 G3 ...B Ge) B (1 22A23A3 ...0 Ae) A(1822 G2 3 G3 ...0' Ge ') B () 8 (1 8 2 A2 3 A3 ...19'Ã‚°) A(2 G2 3 G3 ...0' ' ') B, Is A Seminvariant; And Since These Forms Are Clearly Enumerated By 1 Z.

• It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers contemplated in Ã¯¿½Ã¯¿½ 14-17 are supposed to be positive integers.

• (ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.

• We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.

• The treatment of roots and of logarithms (all being positive integers) belongs to this subject; a= n and p= log a n being the inverses of n=a P (cf.

• (iv.) If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p-kq)L, which is both less than pL and less than qL.

• the product of any r consecutive integers is divisible by r!

• (b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+ I consecutive integers is divisible by (p+i)!.

• do not involve x, and the indices of the powers of x are all positive integers, is called a rational integral function of x of degree n.

• are integers.

• a number which cannot be expressed as the ratio of two integers.

• Taking any number n to be represented by a point on a line at distance nL from a fixed point 0, where L is a unit of length, we start with a series of points representing the integers I, 2, 3,.

• It is evident that we may have tones of frequency hn 1 kn 2 hn i - kn 2 hnl+kn2, where h and k are any integers.

• Decimal or Briggian Antilogarithms. - In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable.

• be integers, a l ib i, a 2 /b 2,.

• The point separating the integers from the decimal fractions seems to be the invention of Bartholomaeus Pitiscus, in whose trigonometrical tables (1612) it occurs and it was accepted by John Napier in his logarithmic papers (1614 and 1619).

• are all positive integers, is called a simple continued f raction.

• It is evident that, in this case, P ' p2, are two series of positive integers increasing without limit if the fraction does not terminate.

• Lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a+1, put x=a+I/y and form the equation in y; if the equation in y has a root between b and b+i, put y = b + I /z, and so on.

• The solution in integers of the indeterminate equation ax+by=c may be effected by means of continued fractions.

• If we take aq-bp= +1 we have a general solution in integers of ax+by=c, viz.

• The process depends on (ii) of § 45, in the extended form that, if x is a factor of a and b, it is a factor of pa-qb, where p and q are any integers.

• - Addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout.

• - A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for numerator and denominator.

• There was, however, no development in the direction of decimals in the modern sense, and the Arabs, by whom the Hindu notation of integers was brought to Europe, mainly used the sexagesimal division in the ' " "' notation.

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

• Carry out column addition and subtraction of two integers less than 1000, and column addition of more than two such integers.

• the orders of the quantic and covariant, and the degree and weight of the leading coefficient; calling these 'n, e,' 0, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation n9 -2w = e.

• be integers, a l ib i, a 2 /b 2,.

• With integers, besides adding and subtracting, it was easy to double and to multiply by 10:

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

• With integers, besides adding and subtracting, it was easy to double and to multiply by 10: