OPERATIONS WITH INTEGRAL NUMBERS It is at present the custom to refer to four fundamental opera tions with numbers—addition, subtraction, multiplication and division. This classification is convenient but arbitrary. Histor ically, the number was at one time given as nine ; at other times, eight; and at others, seven, six or five. It is often asserted, how ever, that there is only one operation that is fundamental, and that this is counting. To count by two is to add two ; to count three twos is to multiply two by three ; to count backwards by two is to subtract twos; and to count backwards by two, beginning with 16 and ending with o, is to find how many twos are contained in 16. Amongst those who gave nine fundamental operations was Sacrobosco (c. 1250), who listed numeration, addition, subtrac tion, duplation (doubling), mediation (halving), multiplication, division, progressions and the finding (extraction) of roots. The Crafte of Nombrynge (c. 1300, ms. in the British Museum) gave seven, omitting numeration and roots; Glareanus (1538) gave six, omitting roots; most of the i6th century reduced this number to five, omitting progressions; Gemma Frisius (154o) reduced it to four, as is now the custom; and Elia Misrachi (c. 1500) gave only three.
Writing and Reading of Large Numbers.—Since the eye grasp renders it difficult to read a number like 207234698, it is the custom to separate such sequences of numerals into groups. If the Greeks had known such a number system, these groups would probably have been of four figures each, the Greeks having used ten thousand (myriad) as the large unit in counting. Since we commonly read numbers by thousands, or thousand thousands (millions), we usually, as an aid to this reading, separate the figures into groups of three or of six. The world, however, has no uniform symbol for use in making this separation. Certain writers have used a vertical bar (Gemma Frisius, 154o; Recorde, c. and others); others have placed dots over the figures for thou sands, millions, and so on; and some have used full stops (periods) in grouping the figures by threes, or superposed arcs (Fibonacci, 1202) or letters. In Great Britain it is the custom to separate into groups of six, using a comma; in the United States the separation is also by a comma (written) or by a space (printed), there being three figures in a group. The groups have been called "periods" (Ramus, 1569), "regions" (Santa-Cruz, 1594), and "ternaries" (Recorde, c. 1542). Spanish writers, especially in the i6th century, commonly placed U or a symbol resembling the Greek 0 after the figure for thousands, and qs (quentos, cuentos) after the one for millions, as in 160U; 462qsoo9U621 (e.g., Texeda, 1546).
The naming of large numbers is comparatively recent. The word "million" appeared in the 13th century but seems not to have been used by mathematicians till about 1340. Its meaning was not confined to till much later, certain writers in the i6th century taking it to be The word "billion" appeared in the 15th century (French, Jehan Adam, c. 1480, and Chuquet, 1484). It was at first taken to mean and this usage still prevails in Great Britain. By the 17th century certain Dutch and French writers used it to designate ion, the latter taking "milliard" as an equivalent. In the American colonies the usage was the same as in England, but owing to the French influence in the early i9th century the word "billion" was afterwards taken to mean and this usage still prevails. The higher number names have only an historical interest. The need for very large or very small num bers is confined to scientists, and these depend upon symbols rather than names as in writing 1.8 X or 3 X Operations.—The operation of addition has made but few changes through the centuries. The Romans could have added more readily than we do, although the writing of the numbers took longer. This is seen in the following additions:—