NUMBERS, kinds and distinctions of. Ma thematicians, considering number under a great many relations, have established the following distinctions. Broken num bers, are the same with fractions. See ARITHMETIC. Cardinal numbers are those which express the quantity of units, as 1, 2, 3, 4, &c. ; whereas ordinal num bers, are those which express order, as 1st, 2d, 3d, &c. Compound number, one divisible by some other number be sides unity ; as 12, which is divisible by 2, 3, 4 and 6. Numbers, as 12 and 15, which,have some common measure be sides unity, are said to be compound num bers among themselves. Cubic number, is the product of a square number by its root : such is 27, as being the product of the square number 9, by its root 3. All cubic numbers whose root is less than 6, being divided by 6, the remainder is the root itself: thus 276 leaves the re mainder 3, its root ; 216, the cube of 6, being divided by 6, leaves no remainder; 343, the cube of 7, leaves a remainder 1, which, added to 6, is the cube root ; and 512, the cube of 8, divided by 6, leaves a remainder 2, which added to 6, is the cube root. Hence the remainders of the divisions of the cubes above 216, divided by 6, being added to 6, always gives the root of the cube so divided, till that remainder be 5, and consequently 11, the cube root of the number divided. But the cubic numbers above this being divided by 6, there remains nothing, the cube root being 12. Thus the remainders of the higher cubes are to be added to 12, and not to 6; till you come to 18, when the remainder of the division must be added to 18 ; and so on ad ininitum. From considering this property of the number 6, with regard to cubic num bers, it has been found that all other numbers, raised to any power whatever, had each their divisor, which had the same effect with regard to them that 6 has with regard to cubes. The general rule is this : " If the exponent of the power of a number be even, that is, if that number be raised to the 2d, 4th, 6th, &c. power, it must be divided by 2 ; then the remainder added to 2, or to a multi. ple of 2, gives the root of the number corresponding to its power, that is the 2d, 4th, and root. But if the exponent of the power of the number be uneven, the 3d, 5th, 7th power, the double of that exponent is the divisor that has the property required.
Determinate number, is that referred to some given unit, as a ternary or three ; whereas an indeterminate one, is that referred to unity in general, and is call ed quantity. Homogeneal numbers, are those referred to the same unit; as those referred to different units are termed heterogeneal. Whole numbers are other wise called integers. Rational number, is one commensurable with unity; as a number, incommensurable with unity, is termed irrational, or a surd. See Sean. In the same manner a rational whole number is that whereof unity is an aliquot part; a rational broken number, that equal to some aliquot part of unity ; and a rational mixed number, that consisting of a whole number and a broken one.
Even number, that which may be divided into two equal parts without any fraction, as 6, 12, &c. The sum, difference, and product of any number of even numbers, is always an even number. An evenly even number, is that which may be mea sured, or divided, without any remainder, by another even number, as 4 by 2. An unevenly even number, when a number may be equally divided by an uneven number, as 20 by 5. Uneven number, that which exceeds an even number, at least by unity, or which cannot be divid ed into two equal parts, as 3, 5, &c. The sum or difference of two uneven num bers make an even number ; but the fac turn of two uneven ones make an uneven number. if an even number be added to an uneven one, or if the one be subtract ed from the other, in the former case the sum, in the latter the difference, is an un even number ; but the factum of an even and uneven number is even. The sum of any even number of uneven numbers is an even number ; and the sum of any uneven number of uneven numbers is an uneven number. Primitive, or prime numbers, are those only divisible by uni ty, as 5, 7, &c. And prime numbers among themselves, are those which have no common measure besides unity, as 12 and 19. Perfect number, that whose ali quof parts added together make the whole number, as 6, 28 ; the aliquot parts of 6 being 3, 2, and 1, = 6 ; and those of 28 being 14, 7,4, 2,1, = 28. Imperfect numbers, those whose aliquot parts, add ed together, make either more or less than the whole. And these are distin guished into abundant and defective ; an instance in the former case is 12, whose aliquot parts 6, 4, 3, 2, 1, make 16; and in the latter case 16, whose aliquot parts 8, 4, 2, and 1, make but 15. Plain num ber., that arising from the multiplication of two numbers, as 6, which is the pro duct of 3 by 2 ; and these numbers are called the sides of the plane. Square number, is the product of any multiplied by itself: thus 4, which is the factum of 2 by 2, is a square number. Every square number added to its root makes an even number. Polygonal, or polygonous numbers, the sums of arith metical progressions beginning with unity : these, where the common differ ence is 1, are called triangular numbers ; where 2, square numbers ; where 3, pen tagonal numbers; where 4, hexagonal numbers ; where 5, heptagonal numbers, &c. See POLYGONAL. Pyramidal num bers: the sums of polygonous numbers, collected after the same manner as the polygons themselves, and not gathered out of arithmetical progressions, are call ed first pyramidal numbers : the sums of the first pyramidals are called second py ramidals, &c. If they arise out of trian gular numbers, they are called triangular pyramidal numbers; if out of pentagons, first pentagonal. pyramidals. From the manner of summing up polygonal num bers, it is easy to conceive how the prime pyramidal numbers are found, viz.
(a-2) n3 + 3 n'-(a-5) n expresses all 6 the prime pyramidals.