NUMBERS, APPELLATIONS OF. Various names have been given to classes of numbers, each expressive of properties common to all in its class : they are pointed out in the following list : The whole scale, 1, 2, 3, &c., is called that of natural numbers ; it is subdivided into the scale of odd numbers, 1, 3, 5, &a, and even numbers, 2, 4, 6, he. These again are subdivided into oddly odd numbers, 3, 7, 11, he. ; evenly odd numbers, 1, 5, 9, he. ; oddly even numbers, 2, 6, 10, he.; and evenly even numbers, 4, 8, 12, he. These latter appellations are not in universal use, though they are very con venient. Thus with reference to division by two and by four, all numbers have names; but not with reference to any higher numbers. The expression of a number which divided by m leaves a remainder n (namely, sax+n, where x is a whole number) is so simple, that it is more easily written than described. When 0 is included in the list, it is considered as divisible without remainder by every number.
The division of numbers into square numbers, 1, 4, 9, 16, Sm. ; cube numbers, 1, 8, 27, 64, he.; fourth powers, 1, 16, 81, 256, he., and so on, may be carried to any extent.
A prime number is any one of the list 1, 2, 3, 5, 7, 11, 13, he., no one of which is divisible by any number except unity and itself. A composite number is any one which is not prime.
A karate number is any one out of the following series, the first excepted, which is only introddeed as a basis.
Numbers were once considered as abundant, perfect, and defective. An abundant number was one in which the sum of all its divisors (unity included, but not itself) exceeds the number : thus 12 is an abundant number, because 1 + 2+3 + 4+6 is greater than 12. A perfect number was one in which the sum of all the divisors was equal to the number : thus 6 is 1 +2+3, and is a perfect number, as is 28, or 1 +2+4 +7+14. A defective number was one in which the sum of the divisors is less than the number, as 10, in which 1 +2+ 5 is less than 10. Whenever 2"-1 is a prime number, then 2 " is a perfect number ; thus or 127, is a prime number, whence 2° or 64 x 127, or 8123, is a perfect number.
Amicable numbers are those each of which is equal to the sum of all the divisors of the other. Such are 284 and 220 17296 and 18416 9363583 and 9437056 Other names have been invented descriptive of classes of numbers ; but the preceding are those which most often occur in the past history of mathematics. With the exception of SQUARE, CUBE, PRIME, even, and odd, the preceding appellatives rarely appear in modern works.