548 



NUMBERS 



NUMERALS 



3i be one of the numbers (a* in every triad of con- 

 secutive numbers one must be a multiple of 3), 

 then the others are either 3m - 2, 3m - 1 ; 3m - 1, 

 3m + 1 ; or 3 + 1, 3m + 2. In the first and third 

 CMOS the proposition is manifest, as (3m - 2) 

 (3m - 1 ) and (3m + 1 ) (3m + 2) are each divisible 

 by 2, and therefore their product into 3m is divisible 

 by 6 ( = 1.2.3). In the second case the product is 

 3m( 3m - 1 ) ( 3m + 1 ), or 3m( 9m' - 1 ), where 3 is a 

 factor, and it is necessary to show that w(0t* - 1) 

 Ls divisible by 2 : if m be even, the thing is proved ; 

 but if o<Ul, then in- is odd, 'Jm- is odd, and !>///-' - 1 

 is even ; hence in this case also the proposition is 

 true. It can similarly be proved that the product 

 of four consecutive numbers is divisible by 24 

 (= 1.2.3.4), of live consecutive numbers by 120 

 (=1.2.3.4.5), and so on generally. These pro- 

 positions form the basis for proof of many properties 

 of numbers, such as that the difference of the 

 squares of any two odd numbers is divisible by 8. 

 The difference between a number and ita cube is 

 the product of three consecutive numbers, and is 

 consequently (see above) always divisible by 6. 

 Any prime number which, when divided by 4, 

 leaves a remainder unity, is the sum of two square 

 numbers : thus, 41 = 25 + 16 = 5 s + 4 s , 233 = 169 

 + 64 = 13 a + 8 s , &c. 



Besides these there are a great many interesting 

 properties of numbers which defy classification ; 

 such as that the sum of the odd numbers beginning 

 with unity is a square number (the square of the 

 number of terms added) i.e. 1 + 3 + 5=9 = 3-', 

 1 + 3 + 5 + 7 + 9 = 25 = Sf, &c. ; and the sum of 

 the cubes of the natural numliers is the square of 

 the sum of the numbers i.e. I 1 + 2* + 3 s = 1+8 

 + 27 = 36 = ( 1 + 2 + 3),1 + V + 3 s + 4' = 100 

 = (1 + 2 + 3 + 4)*, &c. 



Numbers are divided into prime and composite 

 prime numbers being those which contain no factor 

 greater than unity, composite numbers those which 

 are the product of two (not reckoning unity) or 

 more factors. The number of primes is unlimited, 

 and so consequently are the others. The product of 

 any number of consecutive numbers is even, as also 

 are the squares of all even numbers ; while the 

 product of two odd nnml>ers, or the squares of 

 odd numliers, are odd. Every composite number 

 can be put under the form of a product of powers 

 of numbers ; thus, 144 = 2* x 3 2 , or generally, 

 n = af.b'.c , where a, b, and c are prime numbers, 

 and the number of the divisors of such a composite 

 number is equal to the product (/>+!) (q + 1) 



ir + 1 ), nnily and the number itself being included, 

 n the case of 144 the number of divisors would be 

 (4 + 1 ) (2 + 1 ), or 5 x 3, or 15, which we find by 

 trial to be the case. Perfect numbers are those 

 which are equal to the sum of their divisors ( the 

 number itself being of course excepted ) ; thus, 

 6=1+2+3, 28=1 + 2 + 4 + 7 + 14, and 496 

 are perfect numbers. Amicable Humbert are pairs 

 of numbers, cither one of the pair being equal to 

 the sum of the divisors of the other; thus, 220 

 (=1+2 + 4 + 5+10+ 11 + 20 + 22 + 44 + 55 

 + 110 = 284) and 284 ( = 1 + 2 + 4 + 71 + 142 = 

 220) are amicable numbers. For other series of 

 numbers, see FIUIJRATB NUMBERS. 



The most ancient writer on the theory of numbers 

 was Diophantus, who flourished in the 3d century, 

 ami the subject received no further development 

 till the time of Vieta and Fermat (q.v.), who 

 greatly extended it. Euler next added nix quota, 

 and was followed by Lagrange, Legendre, and 

 Gauss, who in turn successfully applied themselves 

 to the study of numbers, ami brought tlie theory 

 to it* present state. Cauchy, Lihrf, and Uill (in 

 America) have also devoted themselves to it with 



See Barlow's Theory of Numbtrt (1811); Legcndre's 



Euni nr la Tkforie da Nombrn (3d ed. Paris, 1830); 

 and Gauss's Ditquuitionci Arithmetics (1801; new ed. 

 1800; Fr. trans. 1807): H. J. 8. Smith, in Brit. An. 

 Report* ( 1859-65) ; Cayley, in Brit. Au. Report* (1875). 



Numerals. The invention of signs to repre- 

 sent numbers is doubtless much older than any 

 form of writing. But the origin of counting, such 

 as would involve the use of signs, is not so ancient 

 as might be thought ; the J>WIT of apprehending 

 even comparatively small numliers comes but late 

 in the development from savage to civilised life. 

 Even yet the aliorigines of Australia work with 

 only the numbers 1 and 2 ; 3 being 2 and 1 or 1 

 and 2 ; 4 being 2 and 2 ; and, as a rule, no 

 Australian black can count as high as 7. The 

 earliest visible signs are doubtless the fingers held 

 up ; and the denary system of notation is due to 

 the fact that we have ten lingers. The rude method 

 of finger-counting has been developed into a highly- 

 complicated system of reckoning, still in use in 

 eastern Europe by pedlars ; various positions and 

 arrangement- of the ten digits allowing of reckon- 

 ing as high as 10,000. For permanent purposes a 

 system of single strokes is the most obvious 

 method ; and series of strokes as high as four or 

 live are found in various countries in old inscrip- 

 tions. But strokes, when numerous, are incon- 

 venient and confusing; hence additional symbols 

 are found to make their appearance, for 5, 10, 100, 

 and 1000. In Babylonian inscriptions two Cunei- 

 forms (q.v.) serve to express all the numbers from 

 1 to 99. The Egyptian scheme is explained and 

 illustrated at HIEHOOLVI-HICS ( Vol. V. p. 707 ) ; and 

 from these hieroglyphs were derived the Phoenician, 

 Palmyrene, and Syriac numerals. 



After alphabetic writing was in use, the alpha- 

 betic signs obviously lent themselves to employ- 

 ment as numerals either following the order of 

 the letters, each having a successively greater value 

 than its predecessor ; or the initial letter of the 

 word for the several numbers might be used. Thus, 

 according to the latter method, the Greek inscrip- 

 tions used I for 1, II (II^Te) for 5, A ( AAo) for 10, 

 H (the old sign for the rough breathing in 'Bicaro*) 

 for 100, X ( Xi>uot ) for 1000, and M ( Mi> ) for 10,000. 

 Then a II with a A' inserilied in it stood for 60 

 (5 x 10), and with H inscrilied (5 x 100) for 500. 

 In this connection the capitals or uncials were used 

 of course. Otherwise, following simply the order 

 of the letters, the twenty four letters of the Ionic 

 alphabet were used for the numbers 1 to 24 ; the 

 iKMiks of the Iliad, for example, are often thus 

 numbered. But a more ingenious method was 

 soon adopted by the Greeks, as also by the 

 Hebrews. The alphabet (cursive) was divided 

 into three groups, of which the first did duty for 

 the units, the second for the tens, the next for 

 hundreds. The Hebrew square character had 

 twenty-two distinct letters, and double forms for 

 five or them, so that three groups, each of nine 

 characters, were available. The Creek aliihalict, 

 as ultimately arranged, had twentv-four letters ; 

 the three additional signs required to make up 

 three nines were obtained by keeping two of the 

 old Phoenician letters F or r (see DIOAMMA) for 6, 

 and 5 or 4 ( koph ) for 90, and adding the super- 

 fluous sibilant <TJ (saintii) for 1MK). Then o to 6 were 

 1 to ll ; from t to koph were 10 to 90; p to wunpi 

 were 100 to 900. The thousands were made by 

 Buliscribing an i lieneath the units ; thus o was 

 1000; ,aw*a is 18111. Sometimes a sort of alge- 

 braic method was employed for larger numbers ; 

 /3M = (2 x 10,000) 20,000.' 



The cumbrous Roman method of using the 

 capitals is familiar enough to ourselves yet*. The 

 C has been understood to ! the initial of centum, 

 ami M of mille. But some (as Canon Taylor) con- 

 tend that the Latins, when they dropped the Greek 



