ARITHMETIC. (PART 1.) DEFINITIONS. 1. Arithmetic is the art of reckoning, or the study of numbers. 2. A unit is one, or a single thing, as one, one bolt, one pulley, one dozen. 3. A number is a unit, or a collection of units, as one, three engines, five boilers. 4. The unit of a number is one of the collection of units which constitutes the number. Thus, the unit of twelve is one, of twenty dollars is one dollar, of one hundred bolts is one bolt. 5. A concrete number is a number applied to some particular kind of object or quantity, as three grate bars, five dollars, ten pounds. 6. An abstract number is a number that is not applied to any object or quantity, as three, five, ten. 7. Like numbers are numbers which express units of the same kind, as six days and ten days, two feet and five feet. 8. Unlike numbers are numbers which express units of different kinds, as ten months and eight miles, seven wrenches and five bolts. NOTATION AND NUMERATION. 9. Numbers are expressed in three ways: (1) by words; (2) by figures; (3) by letters. 10. Notation is the art of expressing numbers by figures or letters. 11. Numeration is the art of reading the numbers which have been expressed by figures or letters. § 1 For notice of copyright, see page immediately following the title page. 12. The Arabic notation is the method of expressing numbers by figures. This method employs ten different figures to represent numbers, viz.: Figures 0 1 2 3 4 5 6 77 8 9 naught, one two three four five six seven eight nine Names cipher, or zero. The first character (0) is called naught, cipher, or zero, and when standing alone has no value. The other nine figures are called digits, and each has a value of its own. Any whole number is called an integer. 13. As there are only ten figures used in expressing numbers, each figure must have a different value at different times. 14. The value of a figure depends upon its position in relation to other figures. 15. Figures have simple values, and local, or relative, values. 16. The simple value of a figure is the value it expresses when standing alone. 17. The local, or relative, value of a figure is the increased value it expresses by having other figures placed on its right. For instance, if we see the figure 6 standing alone, thus . . . we consider it as six units, or simply six. Place another 6 to the left of it; thus . 6 66 The original figure is still six units, but the second figure is ten times 6, or 6 tens. If a third 6 be now placed still one place further to the left, it is increased in value ten times more, thus making it 6 hundreds A fourth 6 would be 6 thousands . A fifth 6 would be 6 tens of thousands, or sixty thousand. . . 666 6666 66666 A sixth 6 would be 6 hundreds of thousands . . 666666 A seventh 6 would be 6 millions 6666666 The entire line of seven figures is read six millions six hundred sixty-six thousands six hundred sixty-six. 18. The increased value of each of these figures is its local, or relative, value. Each figure is ten times greater in value than the one immediately on its right. 19. The cipher (0) has no value itself, but it is useful in determining the place of other figures. To represent the number four hundred five, two digits only are necessary, one to represent four hundred and the other to represent five units; but if these two digits are placed together, as 45, the 4 (being in the second place) will mean 4 tens. To mean 4 hundreds, the 4 should have two figures on its right, and a cipher is therefore inserted in the place usually given to tens, to show that the number is composed of hundreds and units only, and that there are no tens. Four hundred five is therefore expressed as 405. If the number were four thousand five, two ciphers would be inserted; thus, 4005. If it were four hundred fifty, it would have the cipher at the right-hand side to show that there were no units, and only hundreds and tens; thus, 450. Four thousand fifty would be expressed 4050. 20. In reading figures, it is usual to point off the number into groups of three figures each, beginning at the righthand, or units, column, a comma (,) being used to point off these groups. Billions. Millions. Thousands. Units. 3 2 1 5 In pointing off these figures, begin at the right-hand figure and count-units, tens of units, hundreds of units; the next |
